Path Integral Formulation of Qm

7/31/2019 Path Integral Formulation of Qm

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Path Integral Formulation of Quantum Mechanics

Notes

Hemanta Bhattarai

CDP,TU

October 31, 2012


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1

This notes is based on the things that Ive learnt about the path integralformulation of quantum mechanics from the book Quantum Mechanics and

Path integrals by R.P Feynman and A R Hibbs.This note is just the basicabout path integral. Id like to say that this note is just a prerequirementto study above mentioned book and other book on path integral approach toquantum mechanics.


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Contents

1 General Introduction: 31.1 Thought experiment: . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Probablity concept: . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Prof s Nightmare: . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Quantum mechanical law of motion: 72.1 Classical Action: . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Quantum Amplitude: . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Sum over path: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Quantum limits to classical motion: . . . . . . . . . . . . . . . . 122.5 Event occuring in succession: . . . . . . . . . . . . . . . . . . . . 12

3 Schrodinger Equation: 14

4 Perturbration theory: 164.1 First order pertubration: . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 First order pertubration interpretation: . . . . . . . . . . 184.2 Second Order pertubration: . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 Interpretion of the second order perturbation: . . . . . . . 204.3 Integral expression for Kv . . . . . . . . . . . . . . . . . . . . . . 204.4 Wave function expansion: . . . . . . . . . . . . . . . . . . . . . . 21

2


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Chapter 1

General Introduction:

1.1 Thought experiment:

Richard P. Feynman in his, Lecture on Physics Vol III, has wisely indicatedthe misterious nature of the quantum mechanics.He has shown this bizzare na-ture of QM via the thought experiment which is explained below. I think weare familar with the Youngs Double Slit experiment in our high school whilestudying the interference of light waves. In this experiment a source is placed infront of the double slit barrier and a screen behind it.If the light is illuminatedthen we can see the interference pattern in the screen.Let us consider a light source be replaced by a electron gun. The interferencepatten is seen as in the case of the light but if we determine the slit though whichthe electron passes using a greater wavelenght photon the interference pattern

vanishes. So, we can deduce that the interference is seen before measurementand it vanishes after measurment. This is most bizzare nature of the quantumparticle.For detail study of this thought experiment its good to go through Lecture onPhysics Vol III by Richard P.Feynman.

1.2 Probablity concept:

The probablity concept in quantum mechanics is not a new concept it is similarto that of the classical probablity concept,but only the difference is the methodfor the calculation of the probablity amplitude. Let the probablity amplitudeof finding a partilce at position x be (x) and the probablity density of findingthe particle at x is given by |(x)|2.The method of obtaining (x) is dealt inupcomming chapter. In this section we are just learning concept to combinedifferent probablity amplitude.

3


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CHAPTER 1. GENERAL INTRODUCTION: 4

Figure 1.1: Figure of double slit experiment of electron

Let,

1. (S O) = ,be the probablity amplitude of finding the particle at O.

2. (S A1 O = 1, be the probablity amplitude of finding the particleat O passing through the slit A1 when the slit A2 is closed.

3. (S A2 O) = 2, be the probablity of finding the particle at Opassing through the slit A2 when the slit A1 is closed.

When slit A1 is closed then: = 1

When slit A2 is closed then: = 2

When both slits are open:

= 1 + 2

Probablity density of getting particle at O when slit A2 is closed is |1|2and the probablity density of getting particle at O when slit A1 is closedis |2|2. When both slit is open the probablity density of getting particleat O is given by:

||2 = |1 + 2|2 = |1|

2 + |2|2


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So, interference pattern is seen when both the slits are open.

Figure 1.2: With more slits added

In the above figure;

(S A1 B4 O) = (S A1)(A1 B4)(B4 O)More generally

(S Ai Bj O) = (S Ai)(Ai Bj)(Bj O) And theprobablity density for a partilce to be found in O is(S O) =

i

j (S Ai Bj O)

(S O) =

i

j (S Ai)(Ai Bj)(Bj O)


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1.3 Profs Nightmare:

When I was reading Quantum Field Theory In NutShell by A.Zee Iwas very interested on reading a section named Professors Nightmare:A wise guy in a class,here Im going to share it.When a Prof.Joe was giving lecture about the bizzare nature of quantummechanics. He told his students that the probablity amplitude of gettinga particle at O is given by:(S O) = (S A1 O) + (S A2 O)Then,a curious student stood up and asked a question What,if we drilla third hole?. The prof replied Inlcude the propabablity ampliude ofthe particle passing from that point in the sum. The student again askedWhat, if we drill fourth or fifth or sixth hole?.Prof. in angry voice repliedNo matter how many holes you drill, just add the individual probabality

amplitude..The class was silent for sometime. Again, the boy raised hishand and asked What,if we drill infinite number of holes, which meansthere wasnt the slit at all?.The class was silent and the prof. was in thedeep thought about the logic posed by the curious boy.

According to the curious student logic, the probablity amplitude of theparticle found at any point is just the sum over all the probablity amplitudeof the particle passing through the individual hole(though hypothetical incase of free space).


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Chapter 2

Quantum mechanical law

of motion:

2.1 Classical Action:

The mechanics of a classical particle is governed by a simple rule calledHamiltons Principle which states that:

The motion of the system from time t1 to time t2 is such that

the line integral (called the action)

S = t2

t1

Ldt (2.1)

has a stationary value for the actual path of the motion.

S = 0 (2.2)

Here L is the lagrangian of the system.

Considering L(q, q, t) and using (2.1),(2.2) and variational principle we getthe Euler-Lagrange equation:

d

dt

L

q

L

q= 0 (2.3)

Any solution to (2.3) fully describes the system whose lagrangian is L(q, q, t)For example: let us consider we are trying to describe the mechanics ofsimple pendulum of lenght a, the lagrangian of the simple pendulumL(, , t) is:

L =1

2ma22 mga(1 cos)

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CHAPTER 2. QUANTUM MECHANICAL LAW OF MOTION: 8

From (2.3)and above lagrangian we get;

+g

asin = 0 (2.4)

This equation describes the dynamics of the simple pendulum and forsmall angle approximation sin .So,

+g

a = 0

This shows the motion is simple harmonic and the time period is ;

T = 2

a

g

And the motion of the pendulum is the solution of(2.4) i.e

= Asin(t + )

where =

ga

and A, are obtained from the initial condition of thependulum. Consider a free particle, we are going to determine its stationary ac-tion.Let us suppose x(ta) = xa and x(tb) = xb. Here the lagrangian

L =1

2mx2 (2.5)

Using (2.5) and(2.3) (qx) we get;

x = 0 (2.6)

On solving this equation and using the condition x(ta) = xa and x(tb) = xbwe get;

x =xa xbta tb

(2.7)

x =xa xbta tb

t +xatb xbta

ta tb(2.8)

Now the stationary action can be found from (2.1)and (2.7) i.e

S=

tb

ta

1

2m

x2

Scl =1

2

m(xb xa)2

(tb ta)(2.9)


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Then we can easily show that

p(xa) =L

xx=xa

= Sclxa

(2.10)

E(xa) =

L xa

L

x

x=xa

=Scl

ta(2.11)

Path integral approach is the approach to indroduce the fundamentalconcept of classical mechanics, the action, to the quantum mechanics viaquantum amplitude.

2.2 Quantum Amplitude:

In classical mechanics only the path that has the action stationary de-scribes the motion of a classical particle but in the quantum mechanicsaction of all the possible path contributes to the motion of the quan-tum partilce. The information of the quantum particle can be obtainedsolely from the quantum amplitude whose sqaure gives the probablity den-sity.Let,K(b,a) denotes the quantum amplitude for a particle to start fromposition a and found at the position b.Then;

K(b, a) =

possiblepaths

[x(t)] (2.12)

And[x(t)] = Ae(

h )S[x(t)] (2.13)

where A is a constant choosen to normalize K(b,a).

From (2.12) and (2.13) we can deduce that all probablity amplitude fordifferent possible path contributes equally but all of these are at differentphase. The phase of each probablity amplitude is equal to action on thatpath measured in units of h

2.3 Sum over path:

As we have learnt in undergraduate level the integrationba

f(x) can bewritten as the limiting case of the sum of the rectangles defined by thegraph y=f(x) i.e b

a

f(x) = limh0

i

hf(xi) (2.14)


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Let us interpret K(b,a) using (2.12) and(2.13) as a integral.Let ta and tbbe divided into N parts such that

N = tb ta

x0 = xa = x(ta)

xN = xb = x(tb)

t0 = ta

tN = tb


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Now from (2.12) we have

K(b, a) = all paths

A[x(t)]

allpathexcludingx1

x1

(a x1 b)

allpathexcludingx1

x1

(a x1)(x1 b)

allpathexcludingx1

dx1(a x1)(x1 b)

allpathexcludingx1 andx2

x2

dx1(a x1)(x1 x2)(x2 b)

allpathexcludingx1 andx2

dx1 x2(a x1)(x1 x2)(x2 b)

. . .

allspace

dx1dx2 . . . d xN1[x(t)]

= A

. . .

all space

dx1dx2 . . . d xN1ehS[x(t)]

The value of A determined by the normalization condition.For the la-

grangian defined in 2.5 value of A is BN, where B=

2hm

12

K(b, a) =1

B

. . .

dx1B

dx2

B. . .

dxN1

BehS[a,b] (2.15)

where S[a, b] =tbta

L(x, x, t). In most general way we can write;

K(b, a) =

ba

ehS[a,b]Dx (2.16)

and this is called a path integral.We shall rarely encounter to the equa-tion (2.15).


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2.4 Quantum limits to classical motion:

From (2.12) and (2.13) we see that all the possible path contributes tothe total probablity amplitude.For, classical motion the action seems tobe small for the neighbouring path but as compared to the h it is veryhuge so the phase oscillates rapidly for the classical motion and canel theoverall contribution.But for the path in the neighbouhood of the stationaryaction is constant upto first order so they dont cancel eachother.So, thecontribution to the amplitude only comes thorugh the path for whichthe action is stationary for the classical motion.However,for the quantummotion the action is not so significiant as comparedt to h i.e S h,so thephase do not oscillate rapidly and dont cancel each other due to which allthe possible paths contribute to the probablity amplitude of the motion.

2.5 Event occuring in succession:

Let in between a and b positions introduce a position c then

S[a, b] = S[a, c] + S[c, b]


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Then; from (2.16)

K(b, a) =ba

DxehS[a,b]

=

ba

Dxeh

(S[a,c]+S[c,b])

=

ba

DxehS[a,c]e

hS[c,b]

The integration is done from a to c and then c to b but integration mustbe done for all points c.So,

K(b, a) =

ba

Dx

ba

dxcehS[a,c]e

hS[c,b]

=ba

dxcK(b, c)K(c, a)

If time scale is divided into three points c,d and e such that tc < td < tethen we can write

K(b, a) =

ba

ba

ba

dxcdxddxeK(b, d)K(d, c)K(c, a)

Note:

(a) K(c,d)=0 if tc < td

(b) K(i + 1, i) = 1B

ehLxi+1xi

,xi+1+xi

2 ,ti+1+ti

2

(c) For a system with many variables

K(xb, yb, zb, tb; xa, ya, za, ta) =

ba

e{hL(x,y,z,x,y,z,t)}Dx(t)Dy(t)Dz(t)

(d) For a separable system:S[x,X] = Sx[x] + SX[X] then

K(xb, Xb, tb; xa, Xa, ta) = Kx(xb, tb; xa, ta)KX(Xb, tb; Xa, ta)

(e) (x2, t2) =

K(x2, t2; x3, t3)(x3, t3)dx3


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Chapter 3

Schrodinger Equation:

In our undergraduate level we were familiar with the a differentialequation in quantum mechaincs known as Schrodinger Equation. Weused to set up the Schrodinger equation for a given problem and useboundary condition to find the energy eignen values and requiredwave function. Here we are deriving same differential equation viapath integral approach.

For any point c in between a and b we can write action as S[a, b] =S[a, c] + S[c, b] and this gave us possiblity to derive the schrodingerequation.

Here we will compare the wave function in the infinitisally closer timet and t+ taking only consideration up to first order of

We know that,

(x, t + ) =

K(x, t + ; y, t)(y, t)dy

And

K(x, t + ; y, t) =1

BehL(xy ,

x+y2 ,

2t+2 )

Let the lagrangian be:

L(x, x, t) =1

2mx2 V(x, t)

Combing above equation we can write;

(x, t + ) =

1B

e h L( x

y ,x+y2 , 2t+2 )(y, t)dy

(x, t + ) =1

B

em(xy)2

2h ehV( x+y2 ,

2t+2 )(y, t)dy

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CHAPTER 3. SCHRODINGER EQUATION: 15

Since (xy)2 is in the phase, only those value contribute for whic x-yis small else will oscillate rapidly and do not contribute to integral.

So, let us assume x = y + . 2t+2 = t for very small . Now aboveequation becomes

(x, t + ) =1

B

em2

2h ehV(x+ 2 ,t)(x + , t)d

Since the phase in first exponential changes by 1 for =

2hm

. So

expanding above equation upto first order of and second order of we get

(x, t)+(x, t)

t=

1

B

em2

2h

1

hV(x, t)

(x, t) +

(x, t)

x+ 2

2(x, t)

x2

d

(x, t)+(x, t)

t=

1

B

em2

2h

(x, t)

hV(x, t)(x, t) + 2

2(x, t)

x2

d

Using

eix2

xdx = 0

em2

2h d =

2h

m

12

em2

2h 2d =h

m

we get

B =

2h

m

12

h(x, t)

t=h2

2m

2

x2(x, t) + V(x, t)(x, t)

which is the required differential equation known as SchrodingerEquation.This procedure can be used to derive the normalization constant Band the Hamiltonian for any other complicated cases by comparingthe expression derived with the expression below

H = h

t


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Chapter 4

Perturbration theory:

All the quantum mechanical problems cannot be solved in the closedform. However, we can study the system approximately by the per-tubration expansion of the small pertubrating term. Let a particle ofmass m moves in a small potentil V(x,t) then its lagrangian becomes;

L =1

2mx2 V(x, t)

Now from 2.16 we get

Kv(b, a) =

eh

tbta

L(x,x,t)dtDx

= eh

tbta

12mx

2dteh

tbta

V(x(t),t)dtDx

Since V(x(t),t) is a pertubrating potential its very very small. Onexpanding the exponential of V(x,t) we get;

eh

tbta

V(x(t),t)dt = 1 +

h

tbta

V[x(s), s]ds

+1

2!

h

tbta

V[x(s), s]ds

2+ . . .

= 1

h

tbta

V[x(s), s]ds1

2!h2

tbta

tbta

V[x(u), u] V[x(s), s] duds + . . . . . .

(4.1)

Then we can write;

Kv(b, a) =

eh

tbta

12mx

2

1

h

tbta

V[x(s), s]ds1

2!h2

tbta

tbta

V[x(u), u] V[x(s), s] duds

16


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CHAPTER 4. PERTURBRATION THEORY: 17

which can be written as

Kv(b, a) = K0(b, a) + K(1)

v (b, a) + K(2)

v (b, a) + . . . . . . (4.2)where;

K0(b, a) =

eh

tbta

12mx

2dtDx, : free particle integral (4.3)

K(1)v (b, a) =

h

tbta

eh

tbta

12mx

2dtV[x(s), s]dsDx :First order pertubration

(4.4)

K(2)v (b, a) = 1

2!h2

tbta

tbta

eh

tbta

12mx

2dtV[x(u), u] V[x(s), s] dudsDx :Second order per

(4.5)

.

.....

.

.....

......

......

......

......

......

...

4.1 First order pertubration:

K(1)v (b, a) = h

tbta

eh

tbta

12mx

2dtV[x(s), s]dsDx This can be writ-ten as;

K(1)v (b, a) = htbta

F(s) ds

whereF(s) =

eh

tbta

12mx

2dtV[x(s), s]DxThe above integral represents the sum of the possible each weightedby the potential. So,

F(s) =

eh

tbta

12mx

2dtV[x(s), s]Dx

=

eh [

s

ta

12mx

2dt+tbs

12mx

2dt]V[x(s), s]Dx dx(s)

=

K0(b, s)V[x(s), s]K0(s, a)dx(s)

So;

K(1)v (b, a) =

h

tbta

K0(b, s)V[x(s), s]K0(s, a)dx(s)ds


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In a simple form it can be written as;V[x(s),s] V(s)

dx(s) dxsds dtsdxsdts ds.

K(1)v (b, a) =

h

K0(b, s)V(s)K0(s, a)ds (4.6)

4.1.1 First order pertubration interpretation:

From (4.6) we can say that the free particle moves from postition ato position s where it encounters a potential at s i.e V(s). After in-teracting with potential at position s it then moves from the positions to position b as a free particle. Since, postion s can be any point

where potential V =0, so to get the first order pertubration we mustintegrate over the point s both in space and time.

SV

b

a

Figure 4.1: Interpretion of the first order perturbration

4.2 Second Order pertubration:

By the similar process carried on above and with analogous to 4.6we can write

K(2)v =1

2!

h

2 tbta

tbta

K0(b, s)V(s)K(s, u)V(u)K0(u, a)dxsdxudtsdtu


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Now; using constraint K(u, v) = 0, for tu < tv

tbta

tbta

V(s)V(u)dsdu =tbta

tbs

V(s)V(u)duds+tbta

sta

V(s)V(u)duds

The first integral of the RHS satisfies above constarint and the secondintegral after interchanging role of s and u it becomes

tbta

tbu

V(u)V(s)dsdu

Performing u s tbta

tbs

V(u)V(s)dsdu

So finally the second order perturbration can be written as:

K(2)v =

h

2 tbta

tbts

K0(b, s)V(s)K(s, u)V(u)K0(u, a)dxsdxudtsdtu

Finally;

K(2)v =

h

2 K0(b, s)V(s)K(s, u)V(u)K0(u, a)dsdu (4.7)

Similarly n! is removed from the K(n)v .


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4.2.1 Interpretion of the second order perturba-

tion:

From (4.7) we can say that the free particle moves from postition ato position u where it encounters a potential at u i.e V(u). Afterinteracting with potential at position u it then moves from the po-sition u to position u as a free particle where it again encounters apotential at s where it is scattered.From the postion s to postion bit moves as a free particle. Since, postion u and s can be any pointwhere potential V =0, so to get the first order pertubration we mustintegrate over the point s and u such that the constraint K(a, b) = 0for ta < tb both in space and time.

u

V

b

a

v

Figure 4.2: Second order perturbration

4.3 Integral expression for Kv

From (4.2), (4.6),(4.7) we can expand Kv as

Kv(b, a) = K0(b, a)

h

K0(b, s)V(s)K0(s, a)ds

+

h 2

K0(b, s)V(s)K(s, u)V(u)K0(u, a)dsdu + . . . . . . . . .which can be written as

Kv(b, a) = K0(b, a)

h

K0(b, s)V(s)

K0(s, a)

h

K0(s, u)V(u)K0(u, a)du

ds


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Let us try to develop a method for nth order perturbration;

K0(u, a) K0(u, a) hK0(u, r)V(r)K0(r, a)dr (4.8)

And from the first order pertubration theory

Kv(b, a) = K0(b, a)

h

K0(b, s)V(s)K0(s, a)ds (4.9)

Doing the substitution 4.8 in 4.9 once we get second order perturba-tion.i.e

K(2)v (b, a) = K0(b, a)

h

K0(b, s)V(s)

K0(s, a)

h

K0(s, r)V(r)K0(r, a)dr

ds

(4.10)

Substitution 4.8 in 4.10 we get third order perturbration theory.i.e

K(3)v (b, a) =K0(b, a)

h

K0(b, s)V(s)

K0(s, a)

h

K0(s, r)V(r)

K0(r, a)

h

K0(r, p)V(p)K0(p, a)dp

second substitution to K0(r,a)

dr

First substition to K0(s, a)

For nth order perturbation we make (n1) times substitution of the

(4.8) to (4.9) and we get the expression for K(n)v (b, a)

4.4 Wave function expansion:

Let Kv(b, a) be the kernal for the particle attextbfa to move to particle at b. Let (a) be the inital wave functionat position at a.Then the wave function at b is given by

(b) =

Kv(b, a)(a)da

And this can be expanded as;

(b) =

[K0(b, a) h

K0(b, s)V(s)K0(s, a)ds

+

h

2 K0(b, s)V(s)K(s, u)V(u)K0(u, a)dsdu + . . . . . . . . .](a)da


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which can be written as;

(b) = (b) hK0(b, s)V(s)(s)ds

+

h

2 K0(b, s)V(s)K(s, u)V(u)(u)dsdu + . . . . . . . . .

(4.11)

The above expansion of is known as Born series expanison of.If only first two terms are considered in the expansion then it isknown as First Born approximation. If first three terms are con-sidered in the expansion then it is known as Second Born approx-imation.

Note:Here, till now we have considered V(x) as the pertubrating

potential and K0 the free particle kernel. If the total potential

is U(x)+V(x) and V(x) be pertubrating potential. Then we

can use the formula derived till now but we must replace free

particle kernel K0 by the partilce kernal in presence potential

U(x) denoted by KU

Path Integral Formulation of Qm

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