Path Integrals and Quantum Mechanics

Martin SandstromDepartment Of Physics

Umea UniversitySupervisor: Jens Zamanian

October 1, 2015

Abstract

In this thesis we are investigating a different formalism of non-relativistic quantum me-chanics called the path integral formalism. It is a generalization of the classical least actionprinciple. The introduction to this subject begins with the construction of the path integralin terms of the idea of probability amplitudes whose absolute square gives the probability offinding a system in a particular state. Then we show that if the Lagrangian is a quadraticform one needs only to calculate the classical action besides from a time-dependent normal-ization constant to find the explicit expression of the path integral. We look in to the subjectof two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohmeffect, which shows that the vector potential defined in classical electrodynamics have a phys-ical meaning in quantum mechanics. It is also shown that the path integral formulation isequivalent to the Schrodinger description of quantum mechanics, by deriving the Schrodingerequation from the path integral. Further applications of the path integral are discussed.

1

Contents

1 Introduction 3

2 Least Action in Classical Mechanics 5

3 Least Action in Quantum Mechanics 73.1 Summing over Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Propagator with a Quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 83.3 Separating the Propagator into Multiple Propagations . . . . . . . . . . . . . . . . 103.4 Writing the full Propagator with a Time-Slicing Method . . . . . . . . . . . . . . . 113.5 Orthonormality Condition between Propagators . . . . . . . . . . . . . . . . . . . . 123.6 The Relation of the Path Integral to the Schrodinger Equation . . . . . . . . . . . 143.7 Obtaining the Hamilton-Jacobi Equation from a Wave Function . . . . . . . . . . . 153.8 The Propagator in terms of Solutions of the Schrodinger Equation . . . . . . . . . 16

4 Applications of the Path Integral 164.1 Slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Single Slit of a Square Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.2 Single Slit of Gaussian Type . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 Double Slit of a Gaussian Type . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 The Energies for the Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . . 234.3 The Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Conclusion 29

A Derivation of Propagators 30A.1 The Wave-function in Terms of the Spatial Solutions of the Schrodinger Equation . 30A.2 The Free Particle Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.3 The Propagator for the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 33A.4 The Propagator for an Entangled Path with the Origin Removed . . . . . . . . . . 36

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1 Introduction

In the late 19th century an inconsistency between statistical mechanics and electrodynamics wasdiscovered that showed classical mechanics to be inadequate. The inconsistency between statisticalmechanics and electrodynamics was clear to Einstein, Rayleigh and Jeans in 1905 when theyrealized independently that in thermodynamics all degrees of freedom of a system have an averageenergy of 1

2kT , where k is the Boltzmann constant and T the environmental temperature. Inelectrodynamics the total number of frequency modes inside a cavity is inversely proportional tothe square of the frequency, which followed the Rayleigh-Jeans law1,

Bf (T ) ∝ kTf2 (1.0.1)

where Bf is the spectral radiance defined as the radiance through a surface per unit frequency. Thehigher the frequencies are, the spectral radiance associated with each frequency of the oscillatorsgets larger because of the dependence of the square of the frequency. It is an attempt to describethe spectral radiance of electromagnetic radiation at all wavelengths from a black body at a giventemperature. Since all possible frequencies must be considered, the smaller the wavelength are thespectral radiance associated with each wavelength of the oscillators gets larger because the law isinversely proportional to the the fourth power of the wavelength λ = c/f . A contradiction existedin that when we sum over all the modes in the partition function2

Z =∑E

e−βE , (1.0.2)

with each average energy E equal to kT , the series does not converge. Max Planck resolved thisby assuming that the energy of each harmonic oscillator came in a discrete form En = nhf , orquanta, where n is a non-negative integer, f is the frequency of the oscillator and h a constantof proportionality now known as Planck’s constant. Then if we sum over all integers n the seriesconverges to a finite number

Z =

∞∑n

e−βnhf =1

eβhf − 1, (1.0.3)

called the Planck distribution. Later on the physicist Albert Einstein proposed an experimentthat showed that when light at a certain frequency strikes a metal surface, electrons are releasedfrom it. The idea expressed in mathematical form is

hf = Ek − Ev, (1.0.4)

where hf is the energy of a light quanta, Ek the kinetic energy of the electrons and Ev the energyinvolving the applied voltage for the electric field.

Another experiment, which shows that classical mechanics is insufficient in explaining phenomena,is the famous double-slit experiment. A source of particles are traveling freely to a wall wheretwo separated slits have been carved out. If we designate one of the slits by A and the second byB, the particles have two options when they arrive to the wall: Either they can pass through slitA or slit B. At a distance after the slits we place detectors that counts how many particles thathits them. If we plot the number of detections with the position where the detector is positionedthe shape of two bumps will be noticeable, where the height of them is the number of particlesdetected. Clearly the number of particles, at every position along the screen, vary each time wewish to perform the experiment. So all we can detect is the average number of particles hittingthe screen after we have performed the experiment many times from equal initial conditions. Ifwe move the experiment to a very small scale where the slits and the separation between themare very small our intuition tell us that nothing special will happen and the two bumps will again

1See David Bohm, Quantum Theory, p. 62See for instance David Chandler, Introduction to Modern Statistical Mechanics.

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be visible after performing the same experiments. However, this is not what we see because;when we again plot the total number of detections to position a wave-like pattern will appear inthe distribution, just as what we expect when two water waves interfere with each other. Thiseffect after scaling of an experiment is something that classical theories can not explain and amodification was in need. In the 1920’s a new theory called Quantum Mechanics was developedand this theory agreed with experiment. Many formulations were developed; for example WernerHeisenberg’s theory treats matrices using linear algebra and Erwin Schrodinger’s theory deals withwave-functions commonly noted as ψ. The wave-function ψ contains the description of the stateof a system and the absolute square of ψ is interpreted as the probability to find the system in aparticular state. The wave-function satisfies the equation

i~∂

∂tψ(x, t) = Hψ(x, t), (1.0.5)

known as the Schrodinger equation, where H is the Hamiltonian operator and ψ denotes thewave-function.

4

In this thesis we are going to focus on another equivalent formulation of Quantum Mechanicsthat is based on the influential book of Quantum Mechanics by P.A.M. Dirac3 (and later developedby Richard P. Feynman4). Dirac assumed that a solution to the Schrodinger equation had theform

ψ = A exp iScl/~ = A exp

i

~

∫Lcl dt

, (1.0.6)

where Scl is the classical action. The classical action is found by using Lagrange’s equations andsubstituting the resulting equation of motion back in the action and solve the time integral. Thisform of the wave-function can also be found in a paper from 1926 by Erwin Schrodinger5 wherefor the wave-function ψ to have the statistical property it can be written as a product of wave-functions corresponding to states occurring in succession. As the actions must have the additiveproperty under the statistical property, we can find the relation between them as

S = K log ψ (1.0.7)

where S is the classical action and K is a constant with the dimensions of action. One easily seesthat it can be re-written in the form of (1.0.6) but with K undetermined. If one substitutes thissolution into the Schrodinger equation we obtain a differential equation for Scl and the normal-ization A. If we let ~→ 0 the resulting differential equation must reduce to the Hamilton-Jacobiequation from classical mechanics.

2 Least Action in Classical Mechanics

In this section we are going to look at how we can find the path x(t) that minimizes the action.If we vary from the the path x(t) a little bit then the action does not not change to a first ordervariation of the path and accordingly to the principle of least action we obtain the equation ofmotion. The action is a functional in the sense that it takes the whole form of the path x(t) asinput and outputs a number. For example, the action

S[x(t)] =

∫ t2

t1

dtL(x(t), x(t)) (2.0.8)

is a functional of the whole path x(t) where L is the Lagrangian of the system. A functional canbe viewed as a function of a function i.e. that depends on all the points of another function. Tounderstand better what a functional is we may start with a function that depends only on certainpoints and then fill in more points so to smoothen out a line that can be drawn between the points.Say we have a function F with a finite number of points x(t1), x(t2), · · · , x(tN ), · · · :

F = F (x(t1), x(t2), · · · , x(tN ), · · · ). (2.0.9)

The points x(t1), x(t2), · · · , x(tN ), · · · represent different positions in time and together they rep-resent the motion of the system. Say that we can connect the points by straight lines so that wehave formed a polygonal curve, this curve is then an approximation to the motion of the systemand the motion will be discontinuous at the joints of the curve. If we add more points to thefunction F so that we can remove the discontinuities at the joints then the approximation of themotion gets more improved and we can draw a finer line between the points. We continue thisprocess and add more discontinuities and hence if we formally pass the limit of number of pointson the curve to infinity,

limN→∞

F (x1(t1), x2(t2), · · · , x(tN ), · · · ) = F [x(t)], (2.0.10)

3P.A.M Dirac, Quantum Mechanics, p. 1214Richard P. Feynman, Principles of Least Action in Quantum Mechanics5Annalen der Physik (4), vol. 79, 1926

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the curve gets smooth and we obtain what is known as a functional that depends on the form ofthe function x(t). Now let us go back to the function F , which depends only on the points of thefunction x(t), and suppose we vary the points by a small amount so that we can define a changein F as

∆F = F (x(t1) + σ(t1), x(t2) + σ(t2), · · · , x(tN ) + σ(tN ), · · · ) (2.0.11)

− F (x(t1), x(t2), · · · , x(tN ), · · · )

where σ(t1), σ(t2), · · · , σ(tN ), · · · are small deviations from the points x(t1), x(t2), · · · , x(tN ), · · · .Then if we expand ∆F in a Taylor series expansion with a partial sum including only the N firstpoints of the function F we get

∆F = F (x(t1) + σ(t1), x(t2) + σ(t2), · · · , x(tN ) + σ(tN ))− F (x(t1), x(t2), · · · , x(tN )) (2.0.12)

=

N∑i=1

∂F

∂x(ti)σ(ti) + higher order terms

where we have omitted the trailing terms after the first order term for we are only interested ina lowest order variation of the functional F . If we let N be taken in a limit such that the addedpoints removes the discontinuities of the curve the change in the function F becomes a change inthe functional F by an amount σ(t), the sum becomes an integral and we obtain

F [x(t) + σ(t)] = F [x(t)] +

∫ t

dνδF [x(ν)]

δx(ν)σ(ν) + · · · . (2.0.13)

The integrand term δF [x(ν)]/δx(ν) is called a functional derivative and it is the functional deriva-tive of F with respect to the whole function x(t).

If we replace F by S in (2.0.13) and vary the path x(t) by δx(t) we get to a lowest order variationin S

S[x(t) + δx(t)] = S[x(t)] +

∫ t

dνδS[x(ν)]

δx(ν)δx(ν). (2.0.14)

If we perform a variation of the action S[x(ν)] we obtain

δS[x(ν)] = S[x(ν) + δx(ν)]− S[x(ν)] (2.0.15)

= δ

∫ tb

ta

dν L(x(ν), x(ν)) =

∫ tb

ta

dν

(∂L

∂x(ν)δx(ν) +

∂L

∂x(ν)δx(ν)

)where the motion of the system occur between two endpoints x(ta) and x(tb) and x(ν) = dx(ν)/dν.The second term in the integrand contains a variation of the ν-derivative and we can re-write thisterm as

δx(ν) = δ lim∆ν→0

x(ν + ∆ν)− x(ν)

∆ν= lim

∆ν→0

δx(ν + ∆ν)− δx(ν)

∆ν=

d

dνδx(ν). (2.0.16)

Then we can re-write the second term in (2.0.15)by using the product-rule for derivatives as

∂L

∂x(ν)

d

dνδx(ν) =

d

dν

(∂L

∂x(ν)δx(ν)

)− δx(ν)

d

dν

(∂L

∂x(ν)

). (2.0.17)

If we substitute this in (2.0.15) we get

δS[x(ν)] =

∫ tb

ta

dνd

dν

(∂L

∂x(ν)δx(ν)

)+

∫ tb

ta

dν δx(ν)

− d

dν

(∂L

∂x(ν)

)+

∂L

∂x(ν)

(2.0.18)

=∂L

∂x(ν)δx(ν)

∣∣∣∣tbta

+

∫ tb

ta

dν δx(ν)

− d

dν

(∂L

∂x(ν)

)+

∂L

∂x(ν)

.

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Now if there are no variation at the endpoint x(ta) and x(tb) such that δx(ta) = δx(tb) = 0, thefirst evaluation term vanishes and by comparing the variation of S with the expression (2.0.13)we see that the functional derivative of S with respect to x(ν) is

δS[x(ν)]

δx(ν)= − d

dν

∂L

∂x(ν)+

∂L

∂x(ν). (2.0.19)

We wanted to find the path that makes the action functional an extremum and this occur whenthe action does not change to lowest order when we vary the path. This is the same as letting thefunctional derivative of the action S with respect to x(t) be zero and we obtain the equation ofmotion

− d

dν

∂L

∂x(ν)+

∂L

∂x(ν)= 0. (2.0.20)

This is known as the Euler-Lagrange equation and if we know the Lagrangian one finds an explicitexpression of the equation of motion. Then one can solve for the path x(t), which in this case isthe classical path a particle always follow, and if we substitute the classical path denoted here byxcl(t) into the action S one obtains the classical action Scl.

3 Least Action in Quantum Mechanics

Now we wish to extend the least action principle to quantum mechanics.

3.1 Summing over Paths

Let us again consider the double-slit experiment described in the introduction. We want to explainmathematically why the distribution pattern in the double-slit experiment does not follow theclassical ideas. Let us then consider this: Say we have both slits open, and a screen with detectorscounting the total particles at each point along the screen. To each path a number φ[xj ] is assigneddependent on the whole form of xj where xj describes the classical path j taken by the particle.Here we are assuming that the slit is small enough. Say that the particle takes the path throughslit number 1. Then the probability that a particle will take the path x1 is

P (path 1) =∣∣φ[x1]

∣∣2. (3.1.1)

The number P (path 1) should be interpreted as the probability to find the particle at the point(x1, t1) given that it takes the path 1. Likewise if the particle happens to follow the path 2 thenthe probability of finding the particle at the point (x2, t2) is

P (path 2) =∣∣φ[x2]

∣∣2. (3.1.2)

The classical treatment to get the probability that the particle selects any of the two paths 1 and2 is simply obtained by adding the probabilities as

P (any path) =∣∣φ[x1]

∣∣2 +∣∣φ[x2]

∣∣2 (3.1.3)

since, for example, the probability of rolling either a three or a four on a die is the sum of theprobabilities of rolling a three and a four respectively. If we collect all the measurements at thescreen and plot the probability with position of each slit, with the other closed, we obtain twobell-shaped bumps residing next to each other with the peaks centered with the slit. Also we canclearly identify each bump with the respective slit that the particle entered so the particle cannot end up around a point in the probability for slit 2 given it had entered slit 1. The reasonfor that we have two bumps rather than two spikes centered in line with the slits is that whenthe particle enter, say, slit 1 there is a chance that its path will be refracted by the slit and bedetected as a spread along the screen. Now as we mentioned earlier the wave-like distributioncan not be explained by the probability (3.1.3) since it does not reproduce a wave pattern. The

7

solution to this is to treat each number φ[xj ] as a complex number and instead of absolute squareeach number φ[xj ] and add the probabilities we add the φ[xj ]’s then absolute square. What weobtain is something that is different from (3.1.3) that will contain additional terms. We havesaid that the φ[xj ]’s are complex numbers and we are going to call these numbers probabilityamplitudes and the absolute square of the probability amplitude will be called the probability. Sothe probability amplitude is the sum of all the alternatives that the particle can choose from andin the case of the double-slit experiment we have two alternatives: path 1 and path 2. Then thetotal probability amplitude is

K = φ[x1] + φ[x2] (3.1.4)

where we use the notation of the total probability amplitude as K and we will keep this throughoutthis thesis. Then if we absolute square this and treat each probability amplitude as a complexnumber we get the probability of the particle to take either of the two paths as

P (either path) =∣∣K∣∣2 = K∗K = (φ[x1] + φ[x2])

∗(φ[x1] + φ[x2]) (3.1.5)

=∣∣φ[x1]

∣∣2 +∣∣φ[x2]

∣∣2 + φ∗[x2]φ[x1] + φ∗[x1]φ[x2].

If we compare this with (3.1.3) we see that these two probabilities is different from each other.It seems that there has appeared another term in the probability that can explain the wave-likepattern. Now, we can see that the first two terms remind us of the unperturbed problem whenthe particles can choose either slit and we will see two narrow bumps. We are going to postulatethat the probability amplitudes have the form

φ[xj(t)] ∝ expiS[xj(t)]/~

, (3.1.6)

where S[xj ] is the classical action for the path xj(t) and ~ = h/2π is the reduced Planck’sconstant. It is easy to extend the double-slit experiment to multiple slits. Then we have to sumup all probability amplitudes for each alternative that a particle can take and we can write apartial sum of the total amplitude

K =

j∑k=1

φ[xk] ∝j∑

k=1

expiS[xk]/~

. (3.1.7)

If we create even more holes in the wall of slits and add more walls with holes in them as well forthe particles to pass through and if we pass the limit of amplitudes to infinity with the spacingbetween the paths to zero, as if there were infinitely many holes in the walls, and the amount ofwalls to infinity we can formally write (3.1.7) as

K =

∫Dx(t) exp

iS[x(t)]/~

(3.1.8)

where Dx(t) is a representation of the products of measures which corresponds to each wall, wesum over the paths for every amplitude as denoted by the integral symbol and the action S[x(t)]is the classical action. K is a function of the endpoints of the motion of the system known asthe propagator and, as we will later see, can be used to find the probability of a state to make atransition into another state.

3.2 Propagator with a Quadratic Lagrangian

Now that we know the formal expression of the propagator we are going to show that if theLagrangian is of a quadratic form in its position and velocity all we need is to find the classicalaction besides from a normalization constant depending only on the time at the endpoints wherethe motion occur. If we begin with the action

S[x(t)] =

∫ t2

t1

dtL(x(t), x(t)

), (3.2.1)

8

that is an integral of the Lagrangian of a system going from a state at (x(t1), t1) to a state at(x(t2), t2), then we are using the principle of least action with the general quadratic Lagrangian

L(x(t), x(t)

)= a(t)x2 + b(t)xx+ c(t)x2 + d(t)x+ e(t)x+ f(t) (3.2.2)

and substitute the solutions of the equations of motion back into (3.2.1) to obtain the classicalaction. Now, if a particle follows a particular path x(t) where the dash denotes a classical path letus see what happens if we are moving along a path that is nearby the classical path. Let us callthis path x(t). Then we can write this path as a sum of the classical path x(t) and an arbitrarypath y(t) as

x(t) = x(t) + y(t) (3.2.3)

where y(t) is zero at the endpoints. We can expand the left hand side of (3.2.2) in a Taylor seriesaround y(t) and y(t) as

L(x, x) = L( ˙x, x) +∂L

∂x

∣∣∣∣x

y +∂L

∂x

∣∣∣∣˙x

y +1

2

(∂2L

∂x2y2 +

∂2L

∂x∂xyy +

∂2L

∂x2y2

) ∣∣∣∣x, ˙x

, (3.2.4)

where we have terminated the Taylor expansion to second order since it is assumed that theLagrangian is at most quadratic. The first term on the right hand side is the Lagrangian itselfand the trailing terms are functions of y and y, all evaluated on the classical path. Since theLagrangian is at most quadratic in its terms the Taylor series terminate at second order in y andy. By using the chain rule on the third term on the right side of (3.2.4) we obtain

∂L

∂x

∣∣∣∣˙x

y =d

dt

(∂L

∂x

∣∣∣∣˙x

y

)− d

dt

∂L

∂x

∣∣∣∣˙x

y. (3.2.5)

If we substitute this in (3.2.4) the second and third term becomes

∂L

∂x

∣∣∣∣x

y +∂L

∂x

∣∣∣∣˙x

y =∂L

∂x

∣∣∣∣x

y +d

dt

(∂L

∂x

∣∣∣∣˙x

y

)− d

dt

∂L

∂x

∣∣∣∣˙x

y (3.2.6)

=

(− d

dt

∂L

∂x

∣∣∣∣˙x

+∂L

∂x

∣∣∣∣x

)y +

d

dt

∂L

∂x

∣∣∣∣˙x

y.

One can easily notice that the first term on the right hand side is the Euler-Lagrange equationwhich vanishes since the terms are evaluated at the classical path. By comparing the right handside of (3.2.2) with the right side of (3.2.4) we can identify the partial derivatives after explicitcalculation that

∂2L

∂x2= 2a(t),

∂2L

∂x∂x= b(t),

∂2L

∂x2= 2c(t). (3.2.7)

Now by substituting the last term in (3.2.6) and the explicit forms of the partial derivatives backin equation (3.2.4) we obtain

L(x, x) = L(x, ˙x) +d

dt

(∂L

∂x

∣∣∣∣˙x

y

)+

1

2

(2a(t)y2 + b(t)yy + 2c(t)y2

)∣∣∣∣x, ˙x

. (3.2.8)

If we integrate both sides over the times t1 and t2 we get the action

S[x(t)] =

∫ t2

t1

dtL(x, ˙x) +

∫ t2

t1

dtd

dt

(∂L

∂x

∣∣∣∣˙x

y

)+

∫ t2

t1

dt1

2

(2a(t)y2 + b(t)yy + 2c(t)y2

). (3.2.9)

One can see that on the right hand side there are three terms: The first term is the classical action,the second term an integral over a total time-derivative and it is a boundary term and the thirdterm is a time integral over the path y(t). Since the deviation y(t) vanishes at the endpoints thesecond term vanish when evaluated there and we get

S[x(t)] = Scl +

∫ t2

t1

dt1

2

(2a(t)y2 + b(t)yy + c(t)y2

). (3.2.10)

9

where Scl = S[x(t)] is the classical action which is evaluated at the path that makes the action anextremum. We can see that the second term on the right hand side is a functional of the deviationy and depends on the times at the endpoints and by substitution of the action in the formal pathintegral we get

K(x2, t2;x1, t1) = (3.2.11)∫ x(t2)=x2

x(t1)=x1

Dx(t) exp

i

~S[x(t)] +

i

~

∫ t2

t1

dt1

2

(2a(t)y2 + b(t)yy + 2c(t)y2

).

(3.2.12)

On the left hand side we have used the notation of a particle going from a point (x1, t1) to apoint (x2, t2) as read from right to left. Under the path integration the first term on the righthand side in the integrand can be extracted outside the path integral since it does not dependon the deviation y(t). What we have left is a path integral only dependent of the time at theendpoints and the classical path x(t) Let us denote the time-dependent path integral A(t1, t2)then the propagator for the quadratic Lagrangian is

K(x2, t2;x1, t1) = (3.2.13)

expiS[x(t)]/~

∫ y(t2)=0

y(t1)=0

Dy(t) exp

i

~

∫ t2

t1

dt1

2

(2a(t)y2 + b(t)yy + 2c(t)y2

)= A(t1, t2) exp

iS[x(t)]/~

.

This is a really interesting result. We sought an extension of the classical mechanics to quantummechanics by first assigning a number to each amplitude that a particle can possibly take. Thenby postulating the form of the amplitudes we constructed the propagator that can be used tofind the relation between states of a system. Since it is impossible to know which path a particlefollows we can only talk about the probability that a particle reaches a certain state given thatit has been in a previous state. When the majority of paths resides near the classical path x(t)then if the Lagrangian is of quadratic form it suffice only to calculate the classical action of thepath x(t), besides from a normalization constant A(t1, t2), to find the explicit expression of thepropagator. We shall later see how the propagator can be applied in problems.

3.3 Separating the Propagator into Multiple Propagations

There exists situations where the motion of a system is separated in many propagations for examplea particle might move from a state at (xa, ta) to a state at (xc, tc) and then later on from a stateat (xc, tc) to a state at (xb, tb). First let us consider the propagation from a point (xa, ta) to apoint (xb, tb) then the propagator is

K(xb, tb;xa, ta) =

∫ x(tb)=xb

x(ta)=xa

Dx(t) exp

i

~

∫ tb

ta

dtL(x, x, t)

(3.3.1)

where we have included a possible explicit time dependence in the Lagrangian for generality. Ifwe now separate the motion of the particle as mentioned in the beginning of the section then theintegrand of the propagator can be written as

exp iSab/~ = exp i [Sac + Scb] /~ (3.3.2)

where the notation Sab says that we are integrating the Lagrangian from the time ta to the timetb. First we look at the motion from (x(ta), ta) to (x(tc), tc). To find the propagator we integrate(3.3.2) over all paths from (x(ta), ta) to (x(tc), tc). The action from the point (x(tc), tc) to point(x(tb), tb) remains stationary and can be extracted to the outside of the path integral hence we

10

obtain the expression

exp iScb/~∫ x(tc)=xc

x(ta)=xa

Dx(t) exp iSac/~ (3.3.3)

= exp iScb/~K(xc, tc;xa, ta).

If we Integrate (3.3.3) over all paths between the points (xc, tc) and (xb, tb) we obtain∫ x(tb)=xb

x(tc)=xc

exp iScb/~K(xc, tc;xa, ta). (3.3.4)

What we have obtained is however not the same as (3.3.1) since the propagation is free betweenthe endpoints and the way we have constructed the propagator in separating the motion in twosteps restricts the motion such that when the system moves from (xa, ta) to (xb, tb) it will onlypass through the point (xc, tc). So in order to have this construction of the propagator to agreewith (3.3.1) we must integrate over all points x(tc) = xc:

K(xb, tb;xa, ta) = (3.3.5)

=

∫ ∞−∞

dxcK(xc, tc;xa, ta)

∫ x(tb)=xb

x(tc)=xc

Dx(t) exp iScb/~

=

∫ ∞−∞

dxcK(xb, tb;xc, tc)K(xc, tc;xa, ta).

(3.3.6)

One can realize that if we have more than two events the propagation between two points caneasily be extended to multiple propagations: If we let the system propagate between (xb, tb) anda point, say, (xd, td) then we can multiply both sides of (3.3.5) by K(xd, td;xb, tb) and integrateover (xb, tb) to obtain

K(xd, td;xa, ta) =

∫ ∞−∞

dxbK(xd, td;xb, tb)K(xb, tb;xa, ta) (3.3.7)

=

∫ ∞−∞

∫ ∞−∞

dxb dxcK(xd, td;xb, tb)K(xb, tb;xc, tc)K(xc, tc;xa, ta).

(3.3.8)

The process is easily iterated for any number of times and we can write the general result for Nconsecutive propagations with N − 1 separations between them as

K(xN , tN ;x0, t0) =

∫ ∞−∞

∫ ∞−∞· · ·∫ ∞−∞

N−1∏j=1

dxj

N∏j=1

K(xj , tj ;xj−1, tj−1) (3.3.9)

which is the propagator for a system to move between the point x(t0) = x0 and the point x(tN ) =xN where x0 = xa and xN = xb. In the next subsection we are going to investigate the expression(3.3.9) further.

3.4 Writing the full Propagator with a Time-Slicing Method

From the previous result (3.3.9) we can explicitly calculate the formal path integral (3.1.8) by amethod known as time-slicing. The idea is to break up a propagation into multiple parts whereeach propagation is separated equally in time where the time is a small quantity and then take alimiting procedure of the time separation. Let us define the time separation between the separatepaths as ε = (tb − ta)/N where ta and tb are the times at the endpoints and N the number of

11

Figure 1: The path, from a point a to a pointb, can be separated into multiple paths sepa-rated equally in a time ε. To the right at eachline the numbers 1, 2, j and N−1, denote thedifferent points where the total path betweena and b has been separated. The dots to theright indicates that there are more spacingsin time in between.

separated paths. If ε is small we can approximate the propagator K(xj−1, tj−1;xj , tj) in (3.3.9)as

K(xj−1, tj−1;xj , tj) ≈1

Aexp iεL/~ , (3.4.1)

where we have included a normalization constant A, which is dependent on ε, such that thepropagator tends to a delta distribution, δ(xj−1 − xj), when ε → 0. The Lagrangian must bewritten in a discretized form such that when we let ε → 0 or N → ∞ the propagator coincideswith the form (3.3.1). In Richard Feynman’s PhD thesis6 the use of the following approximationfor the Lagrangian

L = L

(xj − xj−1

ε,xj + xj−1

2,tj + tj−1

2

)(3.4.2)

is made and we have included a possible explicit appearance of time in the Lagrangian. Thefirst term is the average velocity, the second term is the average position and the third termis the average time between two successive states. If we use the expression (3.4.2) in (3.4.1),substitute the result in (3.3.9) and take the limit of ε → 0 we will obtain the exact expression ofthe propagator as

K(xb, tb;xa, ta) = (3.4.3)

limε→0

A−(N−1)

∫· · ·∫ N−1∏

j=1

dxj

N∏j=1

exp

i

~εL

(xj − xj−1

ε,xj + xj−1

2,tj + tj−1

2

)(3.4.4)

where it is known that (x0, t0) = (xa, ta) and (xN , tN ) = (xb, tb). This formula can always be usedto calculate the propagator explicitly but if the Lagrangian is of a quadratic form (3.2.2) then itmight save time to use (3.2.13).

3.5 Orthonormality Condition between Propagators

So far we have found several ways of writing the formal expression of the propagator and now weare looking for a orthonormality condition between different propagators. As we have mentionedbefore the propagator for a system being in a position x(tb) at a time tb given that it was ata position x(ta) at a time ta is written as K(xb, tb;xa, ta) where we are reading this expressionfrom right to left. The wave-function is in some sense more general than the propagator, sincethe wave-function can have any initial conditions while the propagator is always a delta functionat some particular point in time. If we do not know the initial conditions, or maybe not even

6Principle of least action in quantum mechanics, p. 31

12

interested in them, we can change to the notation

K(x, t;x′, t′)→ ψ(x, t) (3.5.1)

and if we apply this to equation (3.3.5) we get in turn the relation

ψ(x, t) =

∫ ∞−∞

dx′K(x, t;x′, t′)ψ(x′, t′). (3.5.2)

Thus the relation between a previous state at (x′, t′) and a present state at (x, t) is related by thepropagator K(x, t;x′, t′). In section 3.1 we postulated that the probability density of finding astate with certain coordinates given that it was in a previous state should be proportional to theabsolute square of the propagator K. Then the probability of finding a system in a position x ata time t, with no knowledge of initial conditions, must be

P (x, t) ∝∣∣ψ(x, t)

∣∣2. (3.5.3)

The probability of finding the system in any state given a time t must be

1 =

∫ ∞−∞

dxP (x, t) =

∫ ∞−∞

dx |ψ(x, t)|2 . (3.5.4)

By substituting (3.5.2) into (3.5.4) we obtain∫ ∞−∞

dx |ψ(x, t)|2

=

∫ ∞−∞

dx

∫ ∞−∞

dyK∗(x, t; y, T )ψ∗(y, T )

∫ ∞−∞

dy′K(x, t; y′, T )ψ(y′, T ) (3.5.5)

=

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

dx dy dy′K∗(x, t; y, T )K(x, t; y′, T )ψ∗(y, T )ψ(y′, T ). (3.5.6)

(3.5.7)

Here ψ∗(x, t) is the complex conjugate of the wave-function. Since we have said that (3.5.4) isindependent of the time it means that the expression on the right hand side of (3.5.4) stay thesame however we change the time:∫ ∞

−∞dx |ψ(x, t)|2 =

∫ ∞−∞

dy |ψ(y, t′)|2 . (3.5.8)

Now we must get (3.5.8) from (3.5.5) and this can be found by using the following trick: If we usethe definition of the Dirac delta function we can write the right hand side of (3.5.8) as∫

dy |ψ(y, t)|2 =

∫ ∞−∞

∫ ∞−∞

dy dy′ δ(y − y′)ψ∗(y, t′)ψ(y′, t′) (3.5.9)

where we have used the fact that the probability of finding a system in a state is invariant if onechanges the time and by comparing this result with (3.5.5) we must have the relation∫ ∞

−∞dxK∗(x, t; y, t′)K(x, t; y′, t′) = δ(y − y′). (3.5.10)

So if we know the propagator it must obey this relation.

13

3.6 The Relation of the Path Integral to the Schrodinger Equation

The equivalence of the path integral formulation to the Schrodinger formulation will soon be evi-dent when we are considering a small time interval and Taylor expand around it. The kinetic partof the Lagrangian is assumed to be quadratic in the velocities and the potential to be dependentof position and, for generality, the time. The described Lagrangian is

L =1

2mx2 − V (x, t). (3.6.1)

From the result of equation (3.5.2) we get

ψ(x, t+ ε) =

∫ ∞−∞

dyK(x, t+ ε; y, t)ψ(y, t). (3.6.2)

where the distance between y and x must be small enough to be valid for non-relativistic consid-erations and ε is the time of propagation. If we assume that ε is small we can approximate thepropagator by using (3.4.2) as

K(x, t+ ε; y, t) =1

Aexp

i

~εL

(x− yε

,x+ y

2,

2t+ ε

2

)=

1

Aexp

i

~

[m(x− y)2

2ε− εV

(x+ y

2, t+

ε

2

)](3.6.3)

where 1/A is a normalization constant determined later on. Replacing (3.6.3) in (3.6.2) andchanging variables to x = y − η gives

ψ(x, t+ ε) =1

A

∫ ∞−∞

dη exp

i

~

[mη2

2ε− εV

(x+

η

2, t+

ε

2

)]ψ(x+ η, t) (3.6.4)

Since εV is small to first order in ε we can write

εV(x+

η

2, t+

ε

2

)≈ εV (x, t) + ε

η

2

∂V

∂x+ε2

2

∂V

∂t≈ εV (x, t) . (3.6.5)

where η is small. Hence we can rewrite (3.6.4) as

ψ(x, t+ ε) =1

A

∫ ∞−∞

dη exp

i

~

[mη2

2ε− εV (x, t)

]ψ(x+ η, t). (3.6.6)

Since ε and η are small quantities we can expand both sides in a Taylor series as

ψ(x, t) + ε∂ψ

∂t+ · · · = (3.6.7)

1

A

∫ ∞−∞

dη

exp

(imη2

2~ε

)[1− (i/~)εV (x, t) + · · · ]

(ψ(x, t) + η

∂ψ

∂x+η2

2!

∂2ψ

∂x2· · ·)

.

The expansion in the potential and the left hand side of (3.6.7) can be terminated after first orderin ε and we see that we must keep at least second order in η since all odd orders in η cancels outunder Gaussian integration. If we neglect theses terms (3.6.7) becomes instead

ψ(x, t) + ε∂ψ

∂t=

1

A[1− (i/~)εV (x, t)]

∫ ∞−∞

dη exp

imη2

2~ε

ψ(x, t) +

η2

2!

∂2ψ

∂x2

(3.6.8)

The wave-function and its second derivative in position can be factored out from the Gaussianintegral since they do not depend on η. The first integral on the right hand side is∫ ∞

−∞dη exp

imη2

2~ε

=

√2πi~εm

. (3.6.9)

14

The second integral can be written as∫ ∞−∞

dηη2

2exp

imη2

2~ε

=

∂

∂(im/~ε)

∫ ∞−∞

dη exp

imη2

2~ε

(3.6.10)

=∂

∂(im/~ε)

(2πi~εm

)1/2

= −1

2

(−im2π~ε

)−3/2(− 1

2π

)=

1

4π

(2πi~εm

)3/2

.

where we have used differentiation under the integral sign. If we substitute the result for theintegrals in (3.6.8) we get

ψ(x, t) + ε∂ψ

∂t=

1

A[1− (i/~)εV (x, t)]

ψ(x, t)

√2πi~εm

+∂2ψ

∂x2

1

4π

(2πi~εm

)3/2

(3.6.11)

We see that if we compare both sides we can identify

ψ(x, t) = ψ(x, t)1

A

√2πi~εm

(3.6.12)

so the normalization constant A is

A =

√2πi~εm

. (3.6.13)

Then after replacing A in (3.6.8) we obtain

ψ(x, t) + ε∂ψ

∂t= [1− (i/~)εV (x, t)]

ψ(x, t) +

i~ε2m

∂2ψ

∂x2

(3.6.14)

= ψ(x, t) +i~ε2m

∂2ψ

∂x2− i

~εV (x, t)ψ(x, t)− ε2

2m

∂2ψ

∂x2V (x, t).

By transposing terms, neglecting higher order in ε, multiplying both sides by 1/ε and let ε → 0we obtain in this limit the exact equation

i~∂

∂tψ(x, t) = − ~2

2m

∂2

∂x2ψ(x, t) + V (x, t)ψ(x, t). (3.6.15)

This is of course the one-dimensional time-dependent Schrodinger equation.

3.7 Obtaining the Hamilton-Jacobi Equation from a Wave Function

One of the fundamentals in quantum mechanics is that when we let ~ → 0 we must go overinto classical mechanics. The wave-function can be written in terms of an exponential beingproportional to the classical action

ψ = A exp(iScl/~) (3.7.1)

where ψ = ψ(x, t) and A is some time-independent constant. Thus by substituting this solutionin the Schrodinger equation we get

−ψ ∂Scl∂t

= −ψ i~2m

∂2Scl∂x2

+i

~

(∂Scl∂x

)2

+ ψV. (3.7.2)

By dropping the wave-function ψ on both sides and letting ~→ 0 we should go over into classicalmechanics. If we do so we obtain

− ∂Scl∂t

=1

2m

(∂Scl∂x

)2

+ V (3.7.3)

which indeed is a Hamilton-Jacobi equation from classical mechanics.

15

3.8 The Propagator in terms of Solutions of the Schrodinger Equation

The propagator can also be represented as a discrete sum over the solutions of the Schrodingerequation as shown in the appendix. If we substitute

cn = anexp(iEnt′/~) = exp(iEnt

′/~)

∫ ∞−∞

dxX∗n(x)ψ(x, t′) (3.8.1)

in (A.1.10) we obtain the result

ψ(x, t) =∑n

cnXn(x)exp(−iEnt/~) (3.8.2)

=∑n

[exp(iEnt

′/~)

∫ ∞−∞

dx′X∗n(x′)ψ(x′, t′)

]Xn(x)exp(−iEnt/~)

=

∫ ∞−∞

dx′∑n

X∗n(x′)Xn(x)exp(− iEn(t− t′)/~

)ψ(x′, t′).

Then by comparing this with (3.5.2) we finally arrive at

K(x, t;x′, t′) =∑n

Xn(x)X∗n(x′)exp(iEn(t− t′)/~

), t > t′. (3.8.3)

This expression of the path integral in terms of the solutions to the Schrodinger equation canfor example be applied in the extraction of the energy levels of the harmonic oscillator as will beshown in a later section.

4 Applications of the Path Integral

So far we have found some useful properties of the propagator. With the help of the resultsobtained from earlier investigations we are going to apply the properties of the propagator to someproblems. For the interested the explicit calculations of the free, harmonic and path-constraintpropagator can be found in the appendix.

4.1 Slits

Now we are going to apply the free particle propagator7

K(x, t;x′, t′) =

√m

2πi~(t− t′)exp

im

2~(x− x′)2

t− t′

(4.1.1)

to three slit problems: The single square slit, the single Gaussian slit and the double Gaussian slit.One can easily generalize the results to higher spatial dimensions since the propagator of higherspatial dimensions is simply the product of propagators for each single dimension. We will see thatthe physics, that can be obtained from the square slit, gets obscured by relatively complicatedmathematics and we will introduce a Gaussian function as to ease the analysis and direct us fasterto the physical results.

4.1.1 Single Slit of a Square Type

Imagine a freely moving particle traveling from a point A to another point B in a given time. Ithas no restrictions in its movement but given its departure from A it must end up at B. Now, saywe obstruct the particles movement by inserting a wall with a small hole in it. The particle is stillmoving between point A and B but it is not moving freely, it must pass the hole in the wall to

7See the appendix for derivation.

16

complete its travel between A and B. How can we translate this idea to the use of propagators?The basic idea is to use the successive propagation rule (3.3.5) to find the state of a system giventhat it has been restricted in its free propagation. Say we have a particle in an initial state at anorigo (0, 0) then we know with complete certainty that the particle was in that state every timewe wish to restart the experiment. Let us start with constructing a free particle propagation usingthe result of (3.3.5). The motion of a particle from an origo (0, 0) to a point (X + y, T ) and asuccessive propagation to the point (X + x′, T + t′) given that it was in the position (X + y, T ) iswritten as the product of the two propagators, K(X + x′, T + t′;X + y, T )K(X + y; 0, 0), whereX is the distance from the point of departure to the slit and y is any point along the slit opening(see figure 2). Now, to obtain the amplitude to be at the point X + x′ at a time T + t′, we mustintegrate over all possible positions y. Hence we obtain

K(X + x′, T + t′; 0, 0) =

∫ ∞−∞

dyK(X + x′, T + t′;X + y, T )K(X + y, T ; 0, 0) (4.1.2)

where the position X is some point residing along the separation y. We are only interested in theparticle which enter through the slit, and by neglecting any particles that do not enter the slit wecan change the integration limits so to exclude the contributing amplitudes for the particles notentering the slit. So let us restrict the propagation to a gap of width 2b where b is some constant.Since the integration interval is now (−b, b) instead of (−∞,∞) we have only allowed the particlespassing through the slit and neglected those who do not enter.

Figure 2: A formal view of the singlesquare slit setup where time is upwardand position is to the right in the figure.The particle travels freely from a point(0, 0) to the slit at the point (X + y, T ).If the particle is passing though the slitit contributes to the amplitude and trav-els further to a point X + x′, T + t′. Asit is uncertain to where in the slit theparticle might have passed through, wemust integrate over the width over slit toobtain the total amplitude. Any parti-cle that do not pass through the slit doesnot contribute to the total amplitude.

The propagator when summing over all amplitudes between −b and b becomes

K(X + x′, T + t′; 0, 0) =

∫ b

−bdyK(X + x′, T + t′;X + y, T )K(X + y, T ; 0, 0). (4.1.3)

The particle is released from the origo at a time t = 0 and travels to the point X + y at a timeT , where it must be found in a position interval (X − b,X + b) since otherwise we have nocontribution to the amplitude. If it is detected in this interval, it contributes to the amplitude andtravels further to the point X + x′ at a time T + t′. Then we sum over all possible propagationswithin the interval −b and b, since it uncertain to where in the position interval the particle mightbe. Since the propagation is free we can replace the propagators on the right hand side with thefree particle propagator, (A.2.10), and obtain an expression of the propagation as

K(x′ +X, t′ + T ; 0, 0) =

∫ b

−bdyK(x′ +X, t′ + T ;X + y, T )K(X + y, T ; 0, 0)

= Cm

2πi~√t′T

∫ b

−bdy exp

im

2~

[(X + y)2

T+

(x′ − y)2

t′

](4.1.4)

17

where we have included a constant C to be determined by normalization. Next, we complete thesquare in the integral to obtain

K(x′ +X, t′ + T ; 0, 0) =

Cm

2πi~√t′T

exp

im

2~

[X2

T+x′2

t′

]exp

im

2~t′

(1 +

t′

T

)−1

(x′ − V t′)2

×∫ b

−bdy exp

im

2~t′

(1 +

t′

T

)−1(y

(1 +

t′

T

)− (x′ − V t′)

)2

(4.1.5)

and by a change of variable

iw =

√im

2~t′(x′ − V t′)− y(1 + t′/T )√

1 + t′/T(4.1.6)

we can rewrite the propagator as

K(x′ +X, t′ + T ; 0, 0) =

C

√m

π2i~T

(1 +

t′

T

)−1/2

exp

im

2~

[X2

T+x′2

t′+

1

t′(x′ − V t′)

(1 +

t′

T

)−1]∫ η+/i

η−/i

dw e−w2

(4.1.7)

where we have defined

η± =x′ − V t′ ± b(1 + t′/T )√

2~t′/im√

1 + t′/T(4.1.8)

and V = X/T . The probability density is the absolute square of this, so

P (x′ +X, t′ + T ; 0, 0) = |K(x′ +X, t′ + T ; 0, 0)|2 = |C|2 m

π2~T

(1 +

t′

T

)−1∣∣∣∣∣∫ η+/i

η−/i

dw e−w2

∣∣∣∣∣2

.

(4.1.9)To find the normalization constant C we may use expression (4.1.5). Hence

P (x′ +X, t′ + T ; 0, 0) = K(x′ +X, t′ + T ; 0, 0)K∗(x′ +X, t′ + T ; 0, 0) = |C|2 m2

4π2~2t′T

×

∣∣∣∣∣∫ b

−b

∫ b

−bdy dy′exp

im

2~t′(y2 + y′2

)(1 +

t′

T

)exp

− im~t′

(y + y′)(x′ − V t′)∣∣∣∣∣

(4.1.10)

so that the total probability of finding a particle at the screen is

1 =

∫ ∞−∞

dx′ P (x′ +X, t′ + T ; 0, 0)

= |C|2 m2

2π~2t′T

∣∣∣∣∣∫ b

−b

∫ b

−bdy dy′ δ

(m(y + y′)

~t′

)exp

im

2~t′(y2 + y′2)

(1 +

t′

T

)+

1

2(y + y′)V t′

∣∣∣∣∣(4.1.11)

where δ(z) is the Dirac delta distribution. By using the property

δ(kz) =1

|k|δ(z) (4.1.12)

18

we find for (4.1.11)

1 = |C|2 m2

2π~2t′T

~t′

m

∣∣∣∣∣∫ b

−bdy exp

im

~t′

(1 +

t′

T

)y2

∣∣∣∣∣(4.1.13)

or that

|C|2 =2π~Tm

∣∣∣∣∣∫ b

−bdy exp

im

~t′

(1 +

t′

T

)y2

∣∣∣∣∣−1

.

Thus the normalized propagator for the square slit is

K(x′ +X, t′ + T ; 0, 0) =

√2

πi

(1 +

t′

T

)−1/2∣∣∣∣∣∫ b

−bdy exp

im

~t′

(1 +

t′

T

)y2

∣∣∣∣∣−1/2

× exp

im

2~

[X2

T+x′2

t′+

1

t′(x′ − V t′)

(1 +

t′

T

)−1]∫ η+/i

η−/i

dw e−w2

.

(4.1.14)

We wish to end the analysis here since when we are considering the Gaussian slit, the results willwe the same as for the Gaussian slit and the mathematics will lead us more quickly to the physics.

4.1.2 Single Slit of Gaussian Type

In this section, we are going to extend the setup of the square slit by introducing a function thatis dependent on the displacement along the slit. In the case of the square slit, we can not find aclosed form expression of the propagator in terms of elementary functions, so we are redirectedto the method of approximation. In contrast to the mathematical complexities arising from theanalysis of the square slit, the Gaussian shaped slit takes us quickly to the physics, as will beshown. A general form of a Gaussian function is

f(y) = exp

−y2

2b2

(4.1.15)

where y is the integration variable and 2b the width of the slit. The constant b is also known asthe standard deviation and it is the half-width of the curve from the center to where the heightof the curve has dropped by a factor of e−1. If the shape of the slit is such that most of the arealies within the width 2b of the bell-shaped curve we can with small errors extend the integrationinterval from (−b, b) to (−∞,∞) and solve a Gaussian integral for which we know a closed form.So if we multiply the integrand in the square slit integral by a function f(y) we obtain theexpression for the Gaussian slit propagator as

K(x′ +X, t′ + T ; 0, 0) = Cm

2πi~√t′T

∫ ∞−∞

dy f(y) exp

im

2~

[(X + y)2

T+

(x′ − y)2

t′

](4.1.16)

= Cm

2πi~√t′T

∫ ∞−∞

dy exp

− y2

2b2

exp

im

2~

[(X + y)2

T+

(x′ − y)2

t′

]where we have included a constant C since introducing the function f(y) will leave the propagatornot normalized. We follow the same ideas as for the single square slit. First we complete thesquare in the exponent as

− y2

2b2+im

2~

[(X + y)2

T+

(x′ − y)2

t′

]=im

2~

[x2

T+x′2

t′+A

(y +

B

A

)2

− B2

A

](4.1.17)

19

Figure 3: A free propagation through a Gaussianslit. The particle moves to a point (x + y, T )where its motion is restricted by the Gaussianslit and moves further to a point (x+ x′, T + t′).The shape of the slit enable us to stretch theintegration limits to −∞ and ∞ with very smallerrors since the contributions to the amplitudewill be negligible.

where we have defined the constants

A =i~mb2

+1

T+

1

t′, (4.1.18)

B =X

T− x′

t′.

Then if we substitute this into (4.1.16) we get

K(x′ +X, t′ + T ; 0, 0) = Cm

2πi~√t′T

× exp

im

2~

[X2

T+x′2

t′− B2

A

]∫ ∞−∞

dy exp

imA

2~

(y +

B

A

)2

(4.1.19)

= C

√2πi~mA

m

2πi~√t′T

exp

im

2~

[X2

T+x′2

t′− B2

A

]= C

√m

2πi~At′Texp

im

2~

[X2

T+x′2

t′− A∗B2

|A|2

],

where we have multiplied and divided the last term in the exponential by the complex conjugateof A, A∗. Then if we absolute square (4.1.19) we get

P (X + x′, T + t′; 0, 0) = |K(X + x′, T + t′; 0, 0)|2 (4.1.20)

=

∣∣∣∣C√ m

2πi~At′T

∣∣∣∣2 ∣∣∣∣exp

im

2~

[X2

T+x′2

t′

]∣∣∣∣2 ∣∣∣∣exp

−imA∗B2

2~|A|2

∣∣∣∣2= |C|2 m

2π~|A|t′T

∣∣∣∣exp

−imA∗B2

2~|A|2

∣∣∣∣2 .If we expand the argument

−imA∗B2

2~|A|2=−im2~

(−i~/mb2 + 1/T + 1/t′

)(X/T − x′/t′)2

~2/(m2b4) + (1/T + 1/t′)2 . (4.1.21)

we can see that the only term surviving is the first one on the right hand side in (4.1.21) in theexponent hence we get

−1/2b2(X/T − x′/t′)2

~2/(m2b4) + (1/T + 1/t′)2. (4.1.22)

20

If we substitute our result in (4.1.19) the probability becomes

P (X + x′, T + t′; 0, 0) = |C|2 m

2π~|A|t′Texp

−(X/T − x′/t′)2

~2(m2b2) + (1/T + 1/t′)2

(4.1.23)

= |C|2 m

2π~|A|t′Texp

−(x′ − V t′)2

(δx)2

,

where we have defined the quantities

X = V T, (4.1.24)

(δx)2 =~2t′2

m2b2+ b2

(t′

T+ 1

)2

,

and V is the average velocity of the particle. The quantity (δx)2 tell us the effective width ofthe distribution as detected some distance away from the slit. The second term is what we wouldexpect to see classically since if we increase the time t′ the width of the distribution increases. Onsmaller scales, however, this is not what we would see experimentally but the width is actuallygreater than expected. This can be explained by the first term and the effective width is obviouslya quantum mechanical effect because of the appearance of Planck’s constant ~. If we integrate(4.1.23) over all x′ we get the probability of the system to be in any state at the screen as

1 =

∫ ∞−∞

dx′P (x′, t′; 0, 0) = |C|2 m

2π~|A|t′T

∫ ∞−∞

dx′exp

− (x′ − V t′)2

(δx)2

= |C|2 m

2π~|A|t′T√

(δx)2π. (4.1.25)

Solving for |C|2 gives, apart from a phase,

|C|2 =2√π~Tmb

(4.1.26)

thus (4.1.23) becomes

P (X + x′, T + t′; 0, 0) =2√π~Tmb

m

2π~|A|t′Texp

−(x′ − V t′)2

(δx)2

=

1√π(δx)

exp

−(x′ − V t′)2

(δx)2

(4.1.27)

where we have used the definition of A to write

|A|2b2t′2 = (δx)2. (4.1.28)

If we use a hand-wave analysis of the probability (4.1.25) by not introducing the constant C, theprobability of the system to be in any state at the screen is∫ ∞

−∞dx′ P (X + x′, T + t′; 0, 0) =

m

2π~Tb√π. (4.1.29)

We can see that the integral over the probability over all x′ is independent of the time t′ asexpected. It is dependent on the mass m and the average time of travel T of the particle. Let usnow ignore for the moment that this has dimension of 1/L. This result can with no problem becompared to the exactly same result that the probability that the particle hits the Gaussian slitwithin the interval ∆y in the range of the slit. If ∆y is small we find that

P (X + y, T ; 0, 0) ∆y = |K(X + y, T ; 0, 0)|2 ∆y (4.1.30)

=

∣∣∣∣√ m

2πi~T

∣∣∣∣2 ∣∣∣∣exp

im

2~T(X + y)2

∣∣∣∣2 ∆y

=m

2π~T∆y.

21

The probability that the particle enter through the slit must be exactly the same as detecting theparticle after passing through it. Then (4.1.29) and (4.1.30) are equal and the effective width ofthe Gaussian slit is b

√π.

4.1.3 Double Slit of a Gaussian Type

The treatment of the single Gaussian slit can be extended to multiple slits allowed to interactwith each other. We are going to use the principle of superposition for the double slit of Gaussiantype by simply adding them together and obtain the total propagator for the system. One shouldsee that the probability distribution contains an additional term that contains information forthe interference between the both slit configurations. Say we simply add another slit to theconfiguration of the single Gaussian slit above. Call the propagator of one of the slits K1 and theother for K2 then from the principle of superposition we shall add K1 to K2 to obtain the finalpropagator K as

K = K1 +K2. (4.1.31)

The complex nature of the propagator will output the probability density

|K|2 = |K1 +K2|2 = |K1|2 + |K2|2 + 2 Re (K∗1K2) (4.1.32)

where the first two terms, corresponding to each slit with no interaction, gives a similar result thatto (4.1.23) but we are interested in the last term. It contains information about the interaction ofthe two systems and is mainly what we are going to treat here.

Figure 4: The configuration of thetwo Gaussian slits with amplitudesshown for each slit. We have addeda similar slit to allow interferencebetween two slits so we obtain aninteraction term in the probabilitydensity.

The propagator K can be written as

K = K1 +K2 = K1(X1 + x′, T1 + t′; 0, 0) +K2(X2 + x′, T2 + t′; 0, 0) (4.1.33)

where the coordinates are chosen just in analogy with the single Gaussian slit. If we now use theresult of (4.1.19) we get the expression

K =

√m

2πi~A1T1t′exp

im

2~

[X2

1

T1+x′2

t′− A∗1B

21

|A1|2

](4.1.34)

+

√m

2πi~A2T2t′exp

im

2~

[X2

2

T2+x′2

t′− A∗2B

22

|A2|2

]where we have defined

Aj =i~mbj

2 +1

Tj+

1

t′(4.1.35)

Bj =Xj

Tj− x′

t′, j = 1, 2.

22

Next we find the probability density of the total system by absolute squaring (4.1.34). Since wealready know the two densities for each individual system from the single slit case we are focusingon the third term on the right hand side of (4.1.32). The interference part can be written as

Re(K∗1K2) ∝ cos

m

2~

[X2

2

T2− X2

1

T1+A1B

21

|A1|2− A∗2B

22

|A2|2

](4.1.36)

where the indicies denotes each slit. By using (4.1.28) and the definition of Bj we can rewrite thesecond term in the argument of the cosine as

A1B21

|A1|2− A∗2B

2

|A2|2=A1b

21 (x′ − V1t

′)2

(δx1)2− A∗2b

22 (x′ − V2t

′)2

(δx2)2(4.1.37)

where we have used the same analogy as for the single Gaussian slit where δxj is the effectivedistribution width for the slit j and Vj = Xj/Tj is the average velocity of the particles beforethe respective slit. If we pass the limit of either slit to become similar to the other, such thatdistances, velocities etc. of either slit becomes equal to each other, this term reduces to[

A1b21 (x′ − V1t

′)2

(δx1)2− A∗2b

22 (x′ − V2t

′)2

(δx2)2

](4.1.38)

→ b2 (x′ − V t′)2

(δx)2[A−A∗] =

2i~ (x′ − V t′)2

m(δx)2. (4.1.39)

The first term in the cosine vanishes when we pass the limit and the interaction part becomes

cos

i (x′ − V t′)2

(δx)2

= cosh

(x′ − V t′)2

(δx)2

. (4.1.40)

Thus the resulting interfering part are a hyperbolic cosine that is the sum of the exponential(4.1.23) and another equal exponential but with opposite sign. This means that there are nopossibility for completely destructive interference, because the hyperbolic cosine is always largerthan zero unless its argument is imaginary and this would result in some of the observables to beimaginary which makes no sense.

4.2 The Energies for the Quantum Harmonic Oscillator

In this section, we are going to show how to obtain the energies for the harmonic oscillator8

using the path integral. The main idea is to integrate the propagator (3.8.3) and extract thetime-dependent terms including the energies En. Then we perform the same integration butreplacing (3.8.3) with the explicit from of the harmonic oscillator propagator, express the resultas a series expansion and compare it with (3.8.3). From the uniqueness of the series expansion wecan compare the terms in the series on each side and identify all energies. The propagator for theharmonic oscillator is9

K(x, t;x′, t′) = (4.2.1)√mω

2πi~ sin(ω(t− t′))exp

imω

2~ sin(ω(t− t′))[(x′2 + x2) cos(ω(t− t′))− 2x′x

]where t > t′ and it is understood that this expression is not valid when ωt = 2πn where n is aninteger. From (3.8.3) we can express the propagator in terms of the solutions to the Schrodingerequation, restated here for convenience, as

K(x, t;x′, t′) =

∞∑n=0

X∗n(x′)Xn(x)exp(−iEn(t− t′)/~) (4.2.2)

8See the appendix, section 39See the Appendix, Section 2

23

where t > t′. Now we wish to extract the time dependence from the propagator and by lettingx′ = x and t′ = 0 in the propagator and integrating over all x we get∫ ∞

−∞dxK(x, t;x, 0) =

∫ ∞−∞

dx

∞∑n=0

X∗n(x)Xn(x)exp(−iEnt/~) (4.2.3)

=

∞∑n=0

exp(−iEnt/~)

∫ ∞−∞

dxX∗n(x)Xn(x) =

∞∑n=0

exp(−iEnt/~)

where we have used the result from (A.1.24) for the integration and now we perform the sameintegration with the explicit form of the propagator for the harmonic oscillator:∫ ∞

−∞dxK(x, t;x, 0) =

∫ ∞−∞

dx

√m

2πi~ sin(ωt)exp

imω

~ sin(ωt)[(cos(ωt)− 1)x2]

(4.2.4)

=

√m

2πi~ sin(ωt)

∫ ∞−∞

dx exp

− 2imω

~ sin(ωt)sin2

(ωt

2

)x2

.

The integral is a Gaussian and can be easily solved and then we get∫ ∞−∞

dxK(x, t;x, 0) =

√mω

2πi~ sin(ωt)

√π~ sin(ωt)

2imω

[sin2

(ωt

2

)]−1/2

(4.2.5)

=1

2i

[sin2

(ωt

2

)]−1/2

=1

2i

[sin

(ωt

2

)]−1

.

By using the definition of the sine function

sin(kt) =exp(ikt)− exp(−ikt)

2i(4.2.6)

from complex analysis we can re-write (4.2.5) as

1

2i

[sin

(ωt

2

)]−1

=1

exp(iωt/2)− exp(−iωt/2)=

exp(−iωt/2)

1− exp(iωt). (4.2.7)

By comparing this with (4.2.7) we clearly see that the series is divergent for |exp(iωt)| = 1 andwe get a pole. We can work around this by shifting the exponential by a small complex numberη such that the condition |exp(iωt)| < 1 is valid, re-write the denominator factor as a convergingsum and then let η → 0 to obtain

exp(−iωt/2)

1− exp(iωt)= exp(−iωt/2)

∞∑n=0

exp(−inωt) =

∞∑n=0

exp[−iω(n+ 1/2)t]. (4.2.8)

By comparing this series to (4.2.3) we get the energies for the quantum harmonic oscillator as

En =

(n+

1

2

)~ω (4.2.9)

where n is an integer.

4.3 The Aharonov-Bohm Effect

The Aharonov-Bohm effect is a great example of how the magnetic vector potential in classicalelectrodynamics plays an independent role in quantum mechanics, meaning that in classical elec-trodynamics we only use the vector potential as a tool, whereas the rotation of it, the B-field, hasthe physical meaning. In quantum mechanics the vector potential itself have a physical meaning

24

Figure 5: Illustration of the experimentalsetup. The source of particles origins fromthe point r1 and finishes the journey at thepoint r2. The view is such that we are look-ing down, or up, along the infinitesimally thinand long Dirac string. The angles for r1 andr2 are also visible, as well as the angle θ be-tween them.

as we will later find out. The experiment that exposes the physical independence of the vectorpotential can be illustrated as follows: Inside a cylinder there exists a magnetic field B. Thecylinder is isolated such that a particle, bypassing the cylinder on the outside, can not be effectedby the magnetic field inside. Even though the magnetic field is zero on the outside of the cylinder,the magnetic vector potential A is non-zero there. This is in agreement with classical theory.Now we are shrinking the radius of the cylinder such that we obtain an infinitely thin and in-finitely long string at the origo in the polar plane that we will call a Dirac string, ”containing”the magnetic field. The particles leaves a source at a point r1 and ends up at a point r2, and arenot allowed to cross the origin. Mathematically, we introduce an expression for each possible patha particle might take, that when summed up gives the probability amplitude given the angulardifference φ = θ2 − θ1 + 2πn between r1 and r2. The propagator is given by

K(r2, t2; r1, t1) =

∫ ∞−∞

dφKφ (4.3.1)

where10

Kφ(r2, t2; r1, t1) =m

2πi~(t2 − t1)

∞∑n=−∞

δ(θ2 − θ1 + φ+ 2πn) (4.3.2)

× exp

im

2~(r2

2 + r21)

t2 − t1

∫ ∞−∞

dλ exp (−iφλ) Iλ

(−imr1r2

~(t2 − t1)

),

Iλ is and asymptotic Bessel function and it is understood that t2 > t1. The delta function ensuresthat the difference in the polar angles between r1 and r1 is an integer multiple of 2π. TheLagrangian used to derive the propagator was

L =1

2mr2 − ~λθ, (4.3.3)

where the second term on the right hand side comes from the derivation in (A.4.10) λ is justan integration variable coming from the integral representation of the Dirac delta. If we nowintroduce a term to the free particle Lagrangian containing a vector potential

LA = qA · r (4.3.4)

where q is the charge, A the vector potential and r the velocity of the charge, the Lagrangianterm (4.3.3) changes the Lagrangian L as

L′ = L+ LA = L0 + qA · r − ~λθ (4.3.5)

10We are leaving the derivation, contained in the appendix, section 3, to the reader since it is rather lengthy.

25

where we have defined L0 to be the Lagrangian of a free charge. Next, we will suggest a form ofthe vector potential such that it is non-zero outside the origo, but the curl operation on it yieldsno magnetic field when r 6= 0, but non-zero when r = 0. A magnetic vector potential satisfyingthese properties could be

A =χ

rz × r (4.3.6)

where χ is a constant. We see that it has an inverse radial dependence and the magnitude getsvery large for small r and it is clearly undefined at the origin. The curl operation of this vectorpotential yields the magnetic field

B = ∇×A = χ∇× (r−1z × r). (4.3.7)

To resolve the right hand side we can use the vector calculus identity

∇× (fg) = ∇f × g + f∇× g (4.3.8)

where f is a scalar and g a vector. If we replace f by r−1 and g by x× r we find that

∇× (r−1z × r) = ∇(r−1)× (z × r) + r−1(∇× (z × r)) (4.3.9)

= −r−2r × (z × r) + r−1∇× (z × r).

The first term on the right hand side is −r−2z since z × r = θ because we have cylindricalsymmetry and we have r × θ which is just z. The second term on the right hand side is

r−1∇× (z × r) = zr−2 ∂r

∂r= zr−2 (4.3.10)

where we have used the definition of the curl in cylindrical coordinates. By substituting our resultsfor the magnetic field we obtain

B = ∇×A = −r−2z + r−2z = 0 (4.3.11)

which is just the condition for the magnetic field outside the origo and this is of course valid onlyfor r 6= 0. The magnetic flux when including the origin, can be written as a line integral evaluatedone revolution in the polar plane perpendicular to the Dirac string as

Φ ≡∫B · dS =

∫∇×A · dS =

∮A · dl =

∮χ

rz × r · r dθ θ =

∫dθ χ = 2πχ (4.3.12)

where we can see that the constant χ is proportional to the magnetic flux. Now let us go back tothe vector potential term in the Lagrangian LA: In the (r, θ)-plane we have the vectors r1 and r2

where ri is defined asri = ri cos(θi)x+ ri sin(θi)y. (4.3.13)

The Lagrangian term for the magnetic vector potential contains the time derivative of this vectorwhich implicitly is

dr

dt= r =

dr

dtr + r

dθ

dtθ. (4.3.14)

By substituting the definition of r in LA we get

LA = qA · r =qχ

r2z × rr ·

(dr

dtr + r

dθ

dtθ

)=qχ

rθ ·

(dr

dtr + r

dθ

dtθ

)= qχθ. (4.3.15)

26

It turns out that this magnetic vector potential Lagrangian only depends on the angular speed θ.So by substituting this in (4.3.5) the Lagrangian L′ becomes

L = L0 + qχθ − ~λθ (4.3.16)

which can be written asL = L0 − ~(λ− qχ/~)θ. (4.3.17)

From this result we can see that the additional effect of a magnetic potential acting on the chargesproduces a shift λ → λ − qχ/~ in the propagator. So if we look back to the propagator (4.3.2)and the times as t1 = 0 and t2 = τ and shift λ→ λ− qχ/~ we get for the integral instead∫ ∞

−∞dλ exp −iφ (λ− qχ/~) Iλ−qχ/~

(−imr1r2

~τ

)(4.3.18)

and the propagator Kφ will instead have the form

Kφ =m

2πi~τexp

im

2~τ(r2

2 + r21)

(4.3.19)

×∞∑

n=−∞δ(θ2 − θ1 + φ+ 2πn)

∫ ∞−∞

dλ exp −iφ(λ− qχ/~) Iλ−qχ/~(− imr2r1

~τ

).

If we now integrate this over all possible values of φ the Dirac delta function selects out thecontributing terms and we obtain

K(r2, τ ; r1, 0) =m

2πi~τexp

im

2~τ(r2

2 + r21)

(4.3.20)

×∞∑

n=−∞

∫ ∞−∞

dλ exp −i(θ2 − θ1 + 2πn)(λ− qχ/~) Iλ−qχ/~(− imr1r2

~τ

),

which is the propagator in terms of the winding numbers n and the constant χ. Next, we wouldlike to resolve the integral appearing in the propagator, but the problem in doing so involves theintegration over a Bessel function, that is dependent on the integration variable. As most of thecontribution to the propagator lies in the classical limit, that is, the fluctuating paths near theclassical path makes the significant contributions, we can analyze the propagator in the limit of~→ 0. The argument of the Bessel function inside the integral above, is assumed to be very smallin this classical limit. Therefore we can terminate the Bessel function after some terms, that is,neglect terms higher than first order in the argument. Then the asymptotic Bessel function, givenfrom (4.3.20), can be written as

Iλ−qχ/~

(− imr1r2

~τ

)≈ (4.3.21)(

i~τ2πmr1r2

)1/2

exp

−imr1r2

~τ− 1

2

[(λ− qχ/~)2 − 1

4

]i~τmr1r2

.

By substitution of this approximation of the asymptotic Bessel function in the integral in (4.3.20)we get

Υ ≡(

i~τ2πmr1r2

)1/2 ∫ ∞−∞

dλ exp −i(∆θ + 2πn)(λ− qχ/~) (4.3.22)

× exp

−imr1r2

~τ− 1

2

[(λ− qχ/~)2 − 1

4

]i~τmr1r2

.

27

This can be re-written as(i~τ

2πmr1r2

)1/2

exp

iqχ(∆θ + 2πn)

~+

iq2χ2τ

2~mr1r2− imr1r2

~τ+

i~τ8mr1r2

(4.3.23)

× exp

i~τ

2mr1r2

(qχ− mr1r2(∆θ + 2πn)

~τ

)2∫ ∞−∞

dλ exp

− i~τ

2mr1r2(λ− c)2

where c = qχ − (∆θ + 2πn)/~τ . The integral is just the inverse of the first factor in (4.3.23) sothey cancel out each other and if we neglect all terms linear in ~ as ~ → 0, as they will have avery small contribution, we approximately get

Υ ≈ exp

i(qχ

~− 1)

(∆θ + 2πn) +imr1r2

2~τ(∆θ + 2πn)

2+

iq2χ2τ

2~mr1r2− imr1r2

~τ

(4.3.24)

If we substitute the result in (4.3.20) we obtain for the propagator K

K(r2, τ ; r1, 0) =m

2πi~τexp

im

2~(r2 − r1)2

τ

× exp

iq2χ2τ

2~mr1r2

(4.3.25)

×∞∑

n=−∞exp

imr1r2

2~τ(∆θ + 2πn)2 + i

(qχ~− 1)

(∆θ + 2πn)

.

For the particle to pass clockwise in a plane seen from above the cylinder the winding number nmust be 0 and if the particle pass counter-clockwise the winding number is −1. Let us then, forsimplicity, neglect the particles that passes around the cylinder, having winding numbers n otherthan n = 0,−1. Then the propagator reduces to

K0,−1(r2, τ ; r1, 0) = K0 +K−1 =m

2πi~τexp

im

2~τ(r2 − r1)2

× exp

iq2χ2τ

2~mr1r2

(4.3.26)

×[exp

imr1r2

2~τ(∆θ)2 + i

(qχ~− 1)

∆θ

+ exp

imr1r2

2~τ(∆θ − 2π)2 + i

(qχ~− 1)

(∆θ − 2π)

]where the subscript denotes which terms in the expansion we are considering. The first termcorresponds to the winding number n = 0 and the second term corresponds to the windingnumber n = −1. Classically, there would be no effect since the magnetic field is perfectly isolatedand the vector potential is non-physical. But as one can easily see, the wave function will pick upa phase, containing the constant χ which is precisely the flux through a loop around the cylinder.In other words, in quantum theory, there must be some non-classical mechanism that mediatesinformation that would otherwise not be obtained. We can therefore conclude that the vectorpotential must have a physical meaning in a quantum theory. We wish to see what happens if wetravel along the path corresponding to the winding number n = −1 with an initial state ψ(r1, 0)to a state ψ(r2, τ) in a time τ and then back to the state by the path with winding number n = 0in an equal time. The propagator for the entire process is

K0(r1, 2τ ; r2, τ)K−1(r2, τ ; r1, 0) = K∗0 (r1, 0; r2, τ)K−1(r2, τ ; r1, 0) (4.3.27)

where the complex conjugate of K0 undoes whatever K−1 did, and we must return to the initialstate apart from a phase factor. By the definition of how we compose propagators, and since wemust return to the initial state, the product on the right hand side of (4.3.27) can be though ofas a unit propagator, hence

1 = K∗0 (r1, 0; r2, τ)K−1(r2, τ ; r1, 0) = exp

−i2πqχ

~+imr1r2

2~τ[(∆θ − 2π)2 − (∆θ)2

],

apart from a normalization constant. If 2~τ >> mr1r2 or that ∆θ = π, we obtain

qχ = ~k, k = 0, ±1, ±2, · · · . (4.3.28)

Since χ is constant, the charge q must be quantized in integers of k, and this means also that themagnetic flux, which is proportional to χ, must be quantized.

28

5 Conclusion

The path integral formulation has been shown to be a promising and very useful tool on treat-ing quantum mechanical problems. It is no different from the formulations of, say, Heisenberg,Schrodinger or Dirac since the theories predict the same laws of nature. However all formulationshave their place in the quantum theory since it can be advantageous to switch between them. Theprocess of using path integrals can be most difficult and often involves much time to solve thembut offers much information about the system even when not being directly solved. The pathintegral have been proved to be a vital instrument in the development of quantum field theorywhere it exposes relations between important observables such as in scattering experiments. Onealso notice that when viewing the equations, a direct relation between classical mechanics andquantum mechanics can be seen, and by specifying a classical theory in terms of Hamiltonian orLagrangian mechanics, it is armed for any direct quantization.

Acknowledgments

Many thanks goes to the supervisor for helping out with ideas, stressing the scientific writinglanguage and clarify issues with formulations so they are viewed as clear as possible.

29

A Derivation of Propagators

The theory in this paper does not force the reader to know how the explicit form of the propa-gators arose but how to use them to obtain physics from them. However in this formulation ofquantum mechanics one will constantly be faced upon the path integrals and most of the timehave to calculate them explicitly. So for the curious one we here give the derivations of two basicpropagators and the more difficult propagator relating winding numbers of the different paths ofa system. The basic idea is to use the principle of least action to simplify the evaluation of theaction since most of the paths resides very close to the classical path of a system. The propagatorfor quadratic Lagrangians is therefore the exponential of the classical action.

A.1 The Wave-function in Terms of the Spatial Solutions of the SchrodingerEquation

There is another way of expressing the propagator where it is written in terms of solutions tothe Schrodinger equation. We begin by finding a specific form of the wave-function, the non-relativistic solutions where the positions and times are independent variables. Let us thereforewrite the wave-function as a product of a function dependent only on position, where the positiondoes not explicitly or implicitly depend on time, and a function of time only:

ψ(x, t) = X(x)T (t). (A.1.1)

If we substitute this solution in the Schrodinger equation (3.6.15) we get

i~X(x)dT (t)

dt= − ~2

2mT (t)

d2X(x)

dx2+ V (x, t)X(x)T (t) (A.1.2)

where we have changed the partial derivatives to normal derivatives. The next step is to move theterms dependent only on the position x on one side and the terms dependent on the time t to theother side then we obtain instead

i~1

T (t)

dT (t)

dt= − 1

2m

1

X(x)

d2X(x)

dx2+ V (x). (A.1.3)

where we have restricted us only to stationary potentials V (x) dependent only on the position.We see that the left hand side is dependent only on time and the right hand side is dependentonly on position. In fact both sides must be equal to a constant since if we make any change toone side the other side remains unaffected by it. Let us set this constant to E and then we gettwo differential equations

i~1

T (t)

dT (t)

dt= E (A.1.4)

HX(x) = EX(x).

where H is the Hamiltonian operator

H = − ~2

2m

d2

dx2+ V (x). (A.1.5)

The solution to the differential equation in time is

T (t) = exp(−iEt/~). (A.1.6)

(A.1.7)

Then a solution to the Schrodinger equation can be written as

ψ(x, t) = X(x)T (t) = X(x)exp(−iEt/~). (A.1.8)

30

One can also see that the Schrodinger equation is a linear and homogeneous differential equationso if we can find different solutions to the equation then any linear combination of these is itselfa solution to the Schrodinger equation. If we write the solutions to (A.1.4)as

Tn(t) = exp(−iEnt/~), (A.1.9)

HXn(x) = EnXn(x)

where the n’s are real numbers larger than zero and Xn are assumed to form a complete set. Thegeneral solution to the Schrodinger equation is therefore

ψ(x, t) =∑n

cnXn(x) exp(−iEnt/~), (A.1.10)

where the cn’s are coefficients left to be determined by a method known as the Fourier’s trick bythe following: The wave-function (A.1.10) with t replaced by t′ is

ψ(x, t′) =∑n

cnXn(x) exp(−iEnt′/~) =∑n

anXn(x) (A.1.11)

where we have definedan = cn exp(−iEnt′/~). (A.1.12)

Multiplying both sides of (A.1.11) by the complex conjugate of Xm(x) and integrate over all xgives ∫ ∞

−∞dxX∗m(x)ψ(x, t′) =

∫ ∞−∞

dx∑n

anX∗m(x)Xn(x) (A.1.13)

=∑n

an

∫ ∞−∞

dxX∗m(x)Xn(x).

Before we can evaluate the integral we must prove a condition between the functions Xn and Xm

known as the orthonormal condition∫ ∞−∞

dxX∗m(x)Xn(x) = δmn (A.1.14)

where δmn is the Kronecker delta defined as

δmn =

1, m = n

0, m 6= n(A.1.15)

Say we have two functions of position f(x) and g(x) and the Hamiltonian operator H. Thedefinition of H being a Hermitian operator is∫ ∞

−∞dx f(x)Hg(x) =

∫ ∞−∞

dx g(x)H∗f(x). (A.1.16)

If we look at the differential equations

HXn(x) = EnXn(x), (A.1.17)

H∗X∗m(x) = E∗mX∗m(x) (A.1.18)

and multiply the top equation by X∗m(x), the bottom equation by Xn(x) and integrate bothequations over all x we obtain∫ ∞

−∞dxX∗m(x)HXn(x) = En

∫ ∞−∞

dxX∗m(x)Xn(x) (A.1.19)∫ ∞−∞

dxXn(x)H∗X∗m(x) = E∗m

∫ ∞−∞

dxXn(x)X∗m(x).

31

If we subtract the first equation by the second equation we obtain∫ ∞−∞

dxX∗m(x)HXn(x)−∫ ∞−∞

dxXn(x)H∗X∗m(x) = (En − E∗m)

∫ ∞−∞

dxX∗m(x)Xn(x). (A.1.20)

Using the property (A.1.16) we get for m = n

0 = (En − E∗n)

∫ ∞−∞

dxX∗n(x)Xn(x) (A.1.21)

and since the integral is non-zero we conclude that the energy levels are real. If n 6= m we get

0 = (En − Em)

∫ ∞−∞

dxX∗m(x)Xn(x) (A.1.22)

and since the states are non-degenerate, the energy levels are not equal and the integral must bezero. If n = m the integral ∫ ∞

−∞dxX∗n(x)Xn(x) (A.1.23)

is less than infinity and it can be interpreted as the total probability of finding a particle somewhereif it is normalized to ∫ ∞

−∞dxX∗n(x)Xn(x) = 1. (A.1.24)

So (A.1.14) holds. Now, we are ready to evaluate (A.1.13) by the orthonormality condition of thefunctions Xn and Xm. The right hand side of (A.1.13) is∑

n

an

∫ ∞−∞

dxX∗m(x)Xn(x) =∑

anδmn = am (A.1.25)

since the Kronecker delta, δmn, kills all the integrals where m 6= n. So am ends up as

am =

∫ ∞−∞

dxX∗m(x)ψ(x, t′). (A.1.26)

By substitution of am in (A.1.11) we obtain

ψ(x′, t′) =

∫ ∞−∞

dx∑n

X∗n(x)Xn(x′)ψ(x, t′) (A.1.27)

where we have interchanged x and x′. We see that this is an operation which propagates thewave-function ψ(x′, t′) to ψ(x, t′) through the spatial solutions of the Schrodinger equation.

A.2 The Free Particle Propagator

The most basic propagator must be the propagator for which there are no forces acting upon aparticle i.e. a free particle. The Lagrangian for a free particle in one dimension is

L =1

2mx2 (A.2.1)

where x is the time derivative of the position. From equation (3.2.13) we can clearly see that wehave a quadratic Lagrangian and all we need to do is to find the classical action of the particlemultiply by i/~ and exponentiate it apart from some normalization constant. The next thing todo is to find the equations of motion retrieved from the principle of least action that minimizesthe action for a system. Thus one will get Lagrange’s equations if the boundary remains fixedunder an arbitrary variation. The equation of motion from Lagrange’s equations is

d

dt(mx) = 0, mx = constant (A.2.2)

32

and if we substitute this in the action S we obtain the classical action Scl as

Scl =

∫ t2

t1

dt1

2mx2 =

1

2mx2t

∣∣∣∣x(t2)=x2

x(t1)=x1

=1

2m

(x2 − x1)2

(t2 − t1)2(t2 − t1) =

1

2m

(x2 − x1)2

t2 − t1(A.2.3)

where we assume that the mass of the particle does not change under its motion. Hence bymultiplying this by i~ and exponentiate we get the propagator for the free particle as

K(x2, t2;x1, t1) = A(t1, t2) exp iScl(x2, x1)/~ = A(t1, t2) exp

im

2~(x2 − x1)2

t2 − t1

(A.2.4)

where we have multiplied the exponential with and normalization constant A accordingly to(3.2.13). Let us now determine the constant A: First we change variables t − t1 = ε then thefree propagator can be written as

K(x2, ε;x1, 0) = A(ε) exp

im

2~(x2 − x1)2

ε

. (A.2.5)

In the limit of ε → 0 the propagator must reduce to a delta function δ(x2 − x1) where the deltafunction is defined here by

δ(x2 − x1) =1√π

limε→0

[1√εexp

[− (x2 − x1)2

ε

]]. (A.2.6)

Thus by taking the limit as ε goes to zero of the propagator we get

limε→0

K(x2, ε;x1, 0) = limε→0

A(ε)

exp

[−m(x2 − x1)2

2i~ε

]. (A.2.7)

The next step is to rewrite the just mentioned limit such that it looks like (A.2.6). By changingε = mτ/2i~ and introduce factors of

√π and

√τ we get instead

limτ→0

K(x2, τ ;x1, 0) =1√π

limτ→0

A(τ)√πτ

1√τ

exp

[− (x2 − x1)2

τ

](A.2.8)

and one can easily see that we must choose A(τ) as

A(τ) =

√1

πτ. (A.2.9)

If we now change back τ = 2i~ε/m = 2i~(t2 − t1)/m and substitute for the previously unknownnormalization constant A we obtain for the free particle propagator

K(x2, t2;x1, t1) =

√m

2πi~(t2 − t1)exp

im

2~(x2 − x1)2

t2 − t1

. (A.2.10)

A.3 The Propagator for the Harmonic Oscillator

The structure to obtain the propagator for the harmonic oscillator follow exactly the same lines asfor the free particle case. From classical mechanics one discovers that the Lagrangian is quadraticin kinetic energy and potential energy. So all we have to consider is finding what equations ofmotion that minimizes the action and a normalization constant. The Lagrangian for the harmonicoscillator is stated as

L(x(t), x(t)) =1

2mx2 − 1

2mω2x2 =

1

2mx2 − 1

2kx2 (A.3.1)

33

where k = mω2 and ω is the frequency of the system. Lagrange’s equations give us the equationof motion as

x(t) + ω2x(t) = 0. (A.3.2)

This is a second order differential equation and a solution to this can always be written as

x(t) = C sin(ωt+ α) (A.3.3)

x(t) = ωC cos(ωt+ α)

where C and α are arbitrary constants. If we substitute (A.3.2) in (A.3.1) we obtain the classicalLagrangian and by substituting this in the action we obtain the classical action as

Scl[x(t)] =

∫ t2

t1

dt

[1

2mx2 +

1

2mxx

]. (A.3.4)

The integrand can be rewritten using integration by parts as

1

2mx2 +

1

2mxx =

1

2m

[d

dt(xx)− xx

]+

1

2mxx (A.3.5)

=1

2md

dt(xx)

and then we can evaluate the classical action as

Scl[x(t)] =

∫ t2

t1

dt1

2md

dt(xx) =

1

2mx(t)x(t)

∣∣∣∣x(t2)=x2

x(t1)=x1

=1

2m[x(t2)x(t2)− x(t1)x(t1)

](A.3.6)

where m is again assumed to be constant and the boundaries invariant under a variation. Nowwe will rewrite the solution (A.3.3) and its time derivative as

x(t) = C sin(ω(t− t1,2) + ωt1,2 + α

)(A.3.7)

= C sin(ω(t− t1,2)

)cos(ωt1,2 + α

)+ C cos

(ω(t− t1,2)

)sin(ωt1,2 + α

).

x(t) = Cω cos(ω(t− t1,2) + ωt1,2 + α

).

= Cω cos(ω(t− t1,2)

)cos(ωt1,2 + α

)− Cω sin

(ω(t− t1,2)

)sin(ωt1,2 + α

)where we have introduced the times at the endpoints. From this one can easily see that

x(t1) = C sin(ωt1 + α) (A.3.8)

x(t1) = Cω cos(ωt1 + α)

x(t2) = C sin(ωt2 + α)

x(t2) = Cω cos(ωt2 + α)

hence the equations in (A.3.7) becomes after substituting the above equations

x(t) =x(t1,2)

ωsin(ω(t− t1,2)

)+ x(t1,2) cos

(ω(t− t1,2)

)(A.3.9)

x(t) = x(t1,2) cos(ω(t− t1,2)

)− ωx(t1,2) sin

(ω(t− t1,2)

).

In order to get rid of the time derivatives in (A.3.6) we substitute the above equation in the bottomone so we get

x(t) =ωx(t)− ωx(t1,2) cos

(ω(t− t1,2)

)sin(ω(t− t1,2))

cos(ω(t− t1,2)

)− ωx(t1,2) sin

(ω(t− t1,2)

)(A.3.10)

34

and the products in (A.3.6) becomes by substituting the above equations

x(t2)x(t2) =ωx2

2 − ωx1x2 cos(ω(t2 − t1)

)sin(ω(t2 − t1)

) cos(ω(t2 − t1)

)− ωx2x1 sin

(ω(t2 − t1)

)(A.3.11)

x(t1)x(t1) =−ωx2

1 + ωx1x2 cos(ω(t2 − t1)

)sin(ω(t2 − t1)

) cos(ω(t2 − t1)

)+ ωx2x1 sin

(ω(t2 − t1)

).

Thus the classical action is after some manipulation

Scl[x(t)] =1

2m(x(t2)x(t2)− x(t1)x(t1)

)(A.3.12)

=mω

2 sin(ω(t2 − t1)

)×x2

2 cos(ω(t2 − t1)

)− x1x2 cos2

(ω(t2 − t1)

)− x1x2 sin2

(ω(t2 − t1)

)+ x2

1 cos(t2 − t1

)− x1x2 cos2(ω(t2 − t1))− x1x2 sin2

(ω(t2 − t1)

)=

mω

2 sin(ω(t2 − t1)

) (x21 + x2

2

)cos(ω(t2 − t1)

)− 2x1x2

and since the Lagrangian is quadratic we obtain for the propagator

K(x2, t2;x1, t1) = (A.3.13)

A(t2 − t1) exp

imω

2~ sin(ω(t2 − t1)

) [(x21 + x2

2

)cos(ω(t2 − t1)

)− 2x1x2

], ω(t2 − t1) 6= nπ, n = 0, ±1 ± 2, · · ·

Next step is to determine the normalization constant A. We know that in the limit of ω → 0the propagator must reduce to the free particle propagator. But because of the cosine in theexponential it is not obvious if we can rewrite it as to resemble the delta function we definedearlier, if even possible. So we follow a different approach to find the normalization constant:First let us change t− t1 = T . Then the propagator is again

K(x2, T ;x1, 0) = A(T ) exp

imω

2~ sin(ωT )

[(x2

1 + x22) cos(ωT )− 2x1x2

]. (A.3.14)

Next we use the property (3.5.10) to write

δ(x1 − x′1) =

∫ ∞−∞

dxK∗(x, T ;x′1, 0)K(x, T ;x1, 0) (A.3.15)

where the propagator and its complex conjugate are given by

K∗(x, T ;x′1, 0) = A(T )∗ exp [−iScl(x, T ;x′1, 0)/~] , (A.3.16)

K(x, T ;x1, 0) = A(T ) exp [iScl(x, T ;x1, 0)/~] .

By substituting the explicit forms back in (A.3.15) we get

δ(x1 − x′1) = |A(T )|2∫ ∞−∞

dx exp i[Scl(x, T ;x1, 0)− Scl(x, T ;x′1, 0)]/~ (A.3.17)

= |A(T )|2∫ ∞−∞

dx exp

i

~∂S′cl

∂(x1 − x′1)(x1 − x′1)

35

where we have expanded the different actions to a first order in the difference of the positions andwhere

S′cl = Scl(x, T ;x1, 0)−Scl(x, T ;x′1, 0) =mω

2 sin(ωT )

(x2

1 − x′21) cos(ωT )− 2x(x1 − x′1)

. (A.3.18)

The partial derivative in the exponential is

∂S′cl∂(x1 − x′1)

=∂

∂(x1 − x′1)

[mω

2 sin(ωT )

(x2

1 − x′12) cos(ωT )− 2x(x1 − x′1)

](A.3.19)

= − mωx

sin(ωT )+mω cos(ωt)

2 sin(ωt)(x1 + x′1)

and if we substitute this in (A.3.15) we get

δ(x1 − x′1) = |A(T )|2 exp

imω cos(ωt)

2 sin(ωt)(x2

1 − x′21 )

∫ ∞−∞

dx exp

− imωx

~(x1 − x′1)

sin(ωT )

. (A.3.20)

A change of variable y = −mωx/~ sin(ωT ) gives

δ(x1 − x′1) = |A(T )|2 exp

imω cos(ωt)

2 sin(ωt)(x2

1 − x′21 )

~ sin(ωT )

mω

∫ ∞−∞

dy exp[iy(x1 − x′1)] (A.3.21)

= 2π|A(T )|2 exp

imω cos(ωt)

2 sin(ωt)(x2

1 − x′21 )

~ sin(ωT )

mωδ(x1 − x′1)

= 2π|A(T )|2 ~ sin(ωT )

mωδ(x1 − x′1)

(A.3.22)

where we have used another definition of the Delta function. By dropping the delta functions onboth sides we have

A(T ) =

√mω

2π~ sin(ωT )(A.3.23)

with an undetermined phase. This phase can however be determined by knowing that the harmonicoscillator propagator must reduce to the free particle propagator when ωT → 0. Hence the phasewe are looking for is 1/

√i and with this the propagator becomes

K(x2,t2;x1, t1) = (A.3.24)√mω

2πi~ sin(ω(t2 − t1))exp

imω

2~ sin(ω(t2 − t1))

[(x2

1 + x22) cos(ω(t2 − t1))− 2x2x1

],

ω(t2 − t1) 6= nπ.

(A.3.25)

A.4 The Propagator for an Entangled Path with the Origin Removed

When we derived the previous propagators we dealt with Lagrangians that were quadratic inposition and velocity. The classical action obtained from the classical equations of motion couldsimply be exponentiated and the propagator be derived. In this section we are going to beginwith the propagator for a free system and introduce certain restriction of its motion. We are alsoextending the space to three dimensions and the configuration acquires cylindrical symmetry sowe are working in the plane. Since we are working in a plane the choice of coordinates is (r, θ).In the (r, θ) plane we have two vectors r1 and r2 where each of these vectors can be described bythe angles θ1 and θ2 measured from an origo at r = 0 and lengths r1 = |r1| and r2 = |r2| withit. When the particle moves from r1 to r2 we must consider all possible ways a system can movefrom one state to another because of the principle of superposition of states. We are also going

36

Figure 6: A simplified view of the propaga-tion from a state at (r1, t1) to a state at(r2, t2). The propagation is in a counter-clockwise direction but might as well be ina clockwise direction.

to remove the origin so that the system may not ever be in a state at that point and allow theparticles encircle the origin any way they like.This ensures the restriction of the angles as∫ t2

t1

dtdθ

dt= Θ + 2πn, n = 0, ±1, ±2, · · · (A.4.1)

where the limits of integration have been changed from the angles to the times. Instead of thisrestriction suppose we write the right hand side as∫ t2

t1

dtθ = φ (A.4.2)

where φ is a restriction on the angles we wish to show is equal to Θ + 2πn. For a free particle inthree dimensional space plus time the propagator is simply just

K(r2, t2; r1, t1) ≡ K =

∫ r(t2)=r2

r(t1)=r1

Dr(t) exp(iS[r(t)]/~) (A.4.3)

where we have defined a shorthand for the propagator as K. Under the restriction (A.4.2) whatoperation selects out only the contributing probability amplitudes to the propagator K? TheDirac delta function have just that property that when integrated selects out the contributingterms to the integral. So by our new restriction the propagator can be written as

Kφ =

∫Dr(t) δ

(φ−

∫ t2

t1

dtθ

)exp(iS[r(t)]/~) (A.4.4)

where the subscript on the left hand side indicates that the propagator depends on what we chooseφ as. But from a integral definition of the Delta function

δ

(∫ t2

t1

dtθ − φ)

=1

2π

∫ ∞−∞

dλ exp

iλ

(φ−

∫ t2

t1

dtθ

). (A.4.5)

we get the φ dependent propagator as

Kφ =1

2π

∫ ∞−∞

dλKλ exp iλφ (A.4.6)

that need to be integrated over all φ to get the contributing terms that constructs K and Kλ isdefined as

Kλ =

∫ r2

r1

Dr(t) exp

i

~S[r(t)]− iλ

∫ t2

t1

dt θ

. (A.4.7)

37

The action for the free particle can be explicitly written as

S[r(t)] = S0[r(t)] =

∫ t2

t1

dt1

2mr2. (A.4.8)

where r is the velocity of the radial vector r. If we substitute the explicit action in the exponentialof Kλ we obtain

exp

i

~

∫ t2

t1

dt1

2mr2 − iλ

∫ t2

t1

dt θ

= exp

i

~

∫ t2

t1

dt

(1

2mr2 − ~λθ

). (A.4.9)

If we define

S′[r(t)] = S[r(t)]0 −∫ t2

t1

dt ~λθ (A.4.10)

where S0 is the free particle action we see that the free particle action has been reduced by aterm with units of action. The idea of resolving the explicit form of the propagator Kλ is veryeasy in principle but the mathematics leading to result is painstakingly difficult as will be clearas we progress. We begin by following the time-slicing method since the Lagrangian is not simplyquadratic in position and velocity as it seems by the following: For small times ε, the action canbe resolved in discrete parts in number of N = (t2 − t1)/ε. The effective Lagrangian is thereforediscretized as

L′ = L′(θj − θj−1

ε,rj − rj−1

ε,rj + rj−1

2

)(A.4.11)

or explicitly

L′ =m

2

(rj − rj−1

ε

)2

− ~λ(θi − θj−1)

ε(A.4.12)

Kλ can now be written as

Kλ = limN→∞

CN

∫· · ·∫ N−1∏

j=1

drj exp

i

~

N∑j=1

S[rj , rj−1]

. (A.4.13)

where CN is a normalization constant,

S[rj , rj−1] ≡ m

2ε(∆rj)

2 − ~λ∆θj (A.4.14)

is the discrete partial action and the averages as ∆rj ≡ rj − rj−1 and ∆θj ≡ θj − θj−1. Since weare working in polar coordinates

(∆rj)2 = r2

j + r2j−1 − 2rjrj−1 cos(θj − θj−1) (A.4.15)

and (A.4.14) can therefore be written as

S[rj , rj−1] =m

2ε(r2j + r2

j−1)− m

εrjrj−1 cos(θj − θj−1)− ~λ(θj − θj−1) (A.4.16)

which clearly show that the Lagrangian is non-linear. The main goal is to separate the partialaction such that we can integrate over each variable independently and resolve any cross terms.There are some non-trivial tricks that are used but will be explained as we move along. Webegin by resolving the cosine term. It can be found inside a Taylor expansion of a cosine with aundetermined constant term a as

cos(∆θj + aε) = cos(∆θj)− aε∆θj −1

2a2ε2 (A.4.17)

38

where we have terminated the series after second order in ε. If we now compare this to the partialaction we can identify two terms that exists within the expansion so let us write the expansion ina more suggestive way so to resemble the terms in the discrete action as

m

εrjrj−1 cos(∆θj + aε) +

mrjrj−1

2εa2 =

m

εrjrj−1 cos(∆θj)− amrjrj−1∆θj . (A.4.18)

where each term has been multiplied by a factor of (mrjrj−1)/ε. So we can identify a as

a =−~λ

mrjrj−1(A.4.19)

and by substituting the left hand side of (A.4.18) in the partial action we obtain

S[rj , rj−1] =m

2ε(r2j + r2

j−1)− m

εrjrj−1 cos

(∆θj −

~λεmrjrj−1

)− ε~2λ2

2mrjrj−1. (A.4.20)

So the cross term with an angular and radial term is resolved but now we have introduced a cosinethat has both angular and radial dependence. This term when exponentiated is

exp

− imrjrj−1

~εcos

(∆θj −

~λεmrjrj−1

)(A.4.21)

The exponential can be expanded in a modified Bessel function of an asymptotic form for small εas 11

exp(uε

cos(z))

=

∞∑lj=−∞

exp(iljz) Ilj

(uε

)(A.4.22)

where (−imrjrj−1)/~ = u, z = ∆θj − (λε)/iu and I is the asymptotic modified Bessel function.The Bessel function can be expanded approximately as

Ilj

(uε

)≈( ε

2πu

)1/2

exp

u

ε− 1

2

(lj

2 − 1

4

)ε

u+O(ε2)

(A.4.23)

where O is the big-o notation collecting terms higher than second order in ε. By adding the lastterm, quadratic in λ, from equation (A.4.20) and the linear term in λ, from the definition of z, tothe expansion of the asymptotic modified Bessel function we get after some algebra

∞∑lj=−∞

exp(ilj∆θj)( ε

2πu

)1/2

exp

(u

ε− 1

2

[(lj + λ)2 − 1

4

]ε

u+O(ε2)

)(A.4.24)

=

∞∑lj=−∞

exp(ilj∆θj)Ilj+λ

(uε

).

where we have changed the lower index of the modified Bessel function to lj +λ with no loss sinceif this expansion of the Bessel function were in fact not equal in terms higher than first order inε in (A.4.23) the terms of second order in ε and higher would be neglected anyways and the newform can be contracted in the limit of ε→ 0. The complete exponential of the discretized actionis now

exp

i

~

N∑j=1

S[rj , rj−1]

=

N∏j=1

∞∑lj=−∞

exp(ilj∆θj)Rlj+λ(rj , rj−1) (A.4.25)

where we have defined the radial function

Rlj−λ = exp

im

2~ε(r2j + r2

j−1)

Ilj+λ

(−imrjrj−1

~ε

). (A.4.26)

11doi: 10.1063/1.523588, p. 2319

39

In equation (A.4.25) we can interchange the summation and product and by substituting the resultin (A.4.13) with drj = rjdrjdθj we obtain for Kλ

Kλ(r2, t2; r1, t1) ≡ Kλ = limN→∞

CN

∞∑lj=−∞

∫· · ·∫ N−1∏

j=1

rjdrjdθj

N∏j=1

(A.4.27)

× exp(ilj∆θj)Rlj+λ(rj , rj−1).

At this point we have successfully separated the angular and radial terms. What we see here isa convolution of angular and radial integrals respectively and we begin by resolving the angularintegrals: Let us define the resulting number after we have solved all angular integrals as

Ω ≡∫· · ·∫ N−1∏

j=1

dθj

N∏j=1

exp [ilj(θj − θj−1)] . (A.4.28)

Since per definition of the integral representation of the Kronecker delta∫dθ exp[i(lj − l′j)θ] = 2πδlj l′j (A.4.29)

we can solve Ω by integrate the convoluting integrals up to the N − 1 term and the result of Ω is

Ω = (2π)N−1exp[i(θ2 − θ1)lN ] (A.4.30)

where we have changed the angular endpoints as θN = θ2 and θ0 = θ1. By substituting this in Kλ

we obtain

Kλ = limN→∞

CN

∞∑lj=−∞

(2π)N−1exp[i(θ2 − θ1)lj ] (A.4.31)

×∫· · ·∫ N−1∏

j=1

rjdrj

N∏j=1

Rlj−λ(rj , rj−1)

=

∞∑lj=−∞

exp[i(θ2 − θ1)lj ]Qlj+λ(r2, r1)

where we have defined the Q-function as

Qlj−λ(r2, r1) ≡ limN→∞

(2π)N−1CN

∫· · ·∫ N−1∏

j=1

rjdrj

N∏j=1

Rlj+λ(rj , rj−1). (A.4.32)

Then (A.4.6) becomes by substitution of Kλ as

Kφ(r2, t2; r1, t1) = (2π)−1

∫ ∞−∞

dλ exp(iφλ)Kλ (A.4.33)

= (2π)−1

∫ ∞−∞

dλ exp(iφλ)

∞∑lj=−∞

exp(i(θ2 − θ1)lj)Qlj+λ(r2, r1)

= (2π)−1

∫ ∞−∞

dλ

∞∑lj=−∞

exp[i(θ2 − θ1)lj + iφλ]Qlj+λ(r2, r1).

A shift of lambda, λ→ lj − λ gives instead

Kφ(r2, t2; r1, t1) = (2π)−1

∫ ∞−∞

dλ

∞∑lj=−∞

exp[i(θ2 − θ1 + φ)lj − iφλ]Qλ(r2, r1). (A.4.34)

40

This can with the help of the identity

∞∑lj=−∞

exp[i(θ2 − θ1 + φ)lj ] = 2π

∞∑n=−∞

δ(θ2 − θ1 + φ+ 2πn) (A.4.35)

be written as

Kφ(r2, t2; r1, t1) =

∞∑n=−∞

δ(θ2 − θ1 + φ+ 2πn)

∫ ∞−∞

dλ exp (−iφλ)Qλ(r2, r1). (A.4.36)

The angular part has now been sorted out therefore we shall solve for the radial integrals. Now(A.4.32) with R explicitly shown is

Qλ(r2, r1) = limN→∞

(2π)N−1CN

∫· · ·∫ N−1∏

j=1

rjdrj

N∏j=1

exp

im

2~ε(r2j + r2

j−1)

(A.4.37)

× Iλ

(−imrjrj−1

~ε

).

The problem now is how to integrate over the convoluted Bessel functions and there exist a paperwhere just this type of integral is proved to have a closed form 12. From this paper we can writethe general form of the convoluted integrals as∫ ∞

0

· · ·∫ ∞

0

N−1∏j=1

rjdrjexp

iαN−1∑j=1

r2j

N∏j=1

Iυ (−iBrj−1rj) (A.4.38)

=

N−1∏j=1

(i

2Aj

)exp

−ir2

0

N−1∑j=1

B2

4Aj+r2NB

2

AN

Iυ (−iBNr0rN )

where Aj , AN BN and B are constants left to be determined by the iterated equations

A1 = A, Aj+1 = A− B2

4Aj, (A.4.39)

B1 = B, Bj+1 = B

j∏k=1

B

2Ak. (A.4.40)

We have a freedom to choose the form of the final result to imitate the form of Qλ but with thecost of introducing two functions fN and gN such that (A.4.38) will be equal to

aN exp(ifNr

20 + igNr

2N

)Iυ(−iBNr0rN ). (A.4.41)

If we compare this to Qλ then Qλ will contain two extra terms

exp

im

2~ε(r2N + r2

0)

(A.4.42)

that has not been taken into account in (A.4.38) and it is understood that rN = r2 and r0 = r1. Ifwe expand (A.4.37) we see that all the intermediate points, between r1 and r2, appear twice whichin fact is accounted for in (A.4.38). By identifying the constants B, fN , gN and aN in (A.4.38)we obtain the complementing equations as

aN =

j∏k=1

(i

2Aj

), fN =

1

2B −

N−1∑j=1

B2j

4Aj, (A.4.43)

gN =1

2B − B2

AN12Peak, Inomata, doi: 10.1063/1.1664984

41

and B/2-terms have been added for the presence of the endpoints missing out in (A.4.38) andthese equations let us determine the unknowns. It is evident that A = B hence we can expandthe terms of Aj+1 in (A.4.39) and obtain

A2 = A−1

4A1 =

2 + 1

4A, (A.4.44)

A3 = A−1

3A2 =

3 + 1

6A,

A4 = A−3

5A3 =

4 + 1

8A,

A5 = A−2

5A4 =

5 + 1

10A,

...

From this it easy to see that

Aj =j + 1

2jA (A.4.45)

and by using this result in the equation for Aj+1 found in (A.4.39) gives

Bj = A

j−1∏k=1

A

2

2k

A(k + 1)=

(j − 1)!

j!A =

A

j. (A.4.46)

Then by substituting the expressions for Aj and Bj in (A.4.43) we obtain for fN ,gN and aN :

fN =1

2A− 1

4

N−1∑j=1

A2

j2

2j

A(j + 1)(A.4.47)

=1

2A− 1

2A

N−1∑j=1

1

j(j + 1)=

1

2A− 1

2A

(1− 1

N

)=

A

2N,

gN =1

2A− A2

4AN=

1

2A− 1

2A

N

N + 1=

1

2A

(1

N + 1

),

aN =

N−1∏j=1

(i

2Aj

)=

N−1∏j=1

i

2A

2j

j + 1=

(i

A

)N−11

N.

gN can be approximated by substituting N + 1 ≈ N for large N and by substituting fN , gN , aNinto (A.4.41) we get

aN exp(ifNr

20 + igNr

2N

)Iυ(−iBNr0rN ) =

(i~εm

)N−11

Nexp

(im

2~τ(r22 + r2

1

))(A.4.48)

where we have defined τ = Nε = t2 − t1 and identified rN = r2 and r0 = r1. If we substitute thisin (A.4.38) one should get for Qυ

Qυ(r2, τ ; r1, 0) = limN→∞

CN

(i~εm

)N−1

N−1(2π)N−1exp

im

2~τ(r2

2 + r21)

Iυ

(−imr1r2

~τ

)(A.4.49)

= limN→∞

CN

(2πi~εm

)N ( m

2πi~τ

)exp

im

2~τ(r2

2 + r21)

Iυ

(−imr1r2

~τ

).

For Qυ to be finite in the limit N →∞, the normalization constant must be

CN =( m

2πi~ε

)−N(A.4.50)

42

and the path integral will be defined. Finally we obtain by changing υ = λ the explicit expressionfor Qλ as

Qλ(r2, τ ; r1, 0) =m

2πi~τexp

im

2~τ(r2

2 + r21)

Iλ

(−imr1r2

~τ

). (A.4.51)

Then (A.4.36) becomes

Kφ(r2, t2; r1, t1) =m

2πi~(t2 − t1)

∞∑n=−∞

δ(θ2 − θ1 + φ+ 2πn) (A.4.52)

× exp

im

2~(r2

2 + r21)

t2 − t1

∫ ∞−∞

dλ exp(−iφλ) Iλ

(−imr1r2

~(t2 − t1)

)and we obtain K by integrating Kφ over all configurations φ. So far we have use relatively easymethods to separate the radial parts from the angular parts and the result we have obtainedfor Kφ is very appealing but the integral over a Bessel function remains to be resolved. In theAharonov-Bohm effect we use this result and solve the integral by limiting it in a semi-classicalway by neglecting all terms linear in ~. The exact solution to this integral remains still unknownhowever and it may not even be possible to find a closed form.

43

References

[1] D. Peak A. Inomata. Summation over feynman histories in polar coordinates. Journal ofMathematical Physics, pages 1422–1428, 1969. doi: 10.1063/1.1664984.

[2] Vijay A. Singh A. Inomata. Path integrals with a periodic constraint: Entangled strings.Journal of Mathematical Physics, pages 2318–2323, 1978. doi: 10.1063/1.523588.

[3] David Bohm. Quantum Theory. Dover Publications, 1989.

[4] David Chandler. Introduction to Modern Statistical Mechanics. Oxford University Press, 1987.

[5] P. A. M. Dirac. Quantum Mechanics, Third Edition. Oxford, 1947.

[6] Richard P. Feynman. Principles of Least Action in Quantum Mechanics. PhD thesis, Princeton,1942.

[7] Albert R. Hibbs Richard P. Feynman. Quantum Mechanics and Path Integrals, EmendedEdition. Dover, 2010.

[8] L.S. Schulman. Techniques and Applications of Path Integration, Second Edition. Dover, 2005.

[9] M. Reuter W. Dittrich. Classical and Quantum Dynamics, Second Edition. Springer, 1996.

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