COMENIUS UNIVERSITY IN BRATISLAVAFACULTY OF MATHEMATICS, PHYSICS AND

INFORMATICS

QUANTUM DESCRIPTION OF THE UNIVERSE CLOSETO BIG BANG

Master’s thesis

2014 Fridrich Valach

COMENIUS UNIVERSITY IN BRATISLAVAFACULTY OF MATHEMATICS, PHYSICS AND

INFORMATICS

QUANTUM DESCRIPTION OF THE UNIVERSE CLOSETO BIG BANG

Master’s thesis

Study programme: Theoretical physicsBranch of study: 1160 PhysicsSupervisor: Vladimír Balek, doc. RNDr., CSc.

Bratislava, 2014 Fridrich Valach

14270975

Comenius University in BratislavaFaculty of Mathematics, Physics and Informatics

THESIS ASSIGNMENT

Name and Surname: Bc. Fridrich ValachStudy programme: Theoretical Physics (Single degree study, master II. deg., full

time form)Field of Study: 4.1.1. PhysicsType of Thesis: Diploma ThesisLanguage of Thesis: EnglishSecondary language: Slovak

Title: Quantum description of the universe close to Big Bang

Aim: To investigate solutions of Wheeler-DeWitt equation in minisuperspace,describing creation of the universe from ``nothing'', with logarithmicasymptotics for the radius of the universe approaching zero.

Literature: T. Banks, C. M. Bender, T. T. Wu: Coupled Anharmonic Oscillators: I. EqualMass Case, Phys. Rev. D8, 3346 (1973) A. Vilenkin: Boundary conditions inquantum cosmology, Phys. Rev. D33, 3560--3569 (1986) S. Coleman, F. deLuccia: Gravitational Effects On and Of Vacuum Decay, Phys. Rev. D21, 3305(1980)

Annotation: For a universe created from `nothing'' there should exist a one-parametric classof solutions, so far not considered in the literature, with the asymptotics for theradius approaching zero proportional to the logarithm of the radius. In the thesisproperties of these solutions should be investigated and subtleties of the methodused in their derivation should be discussed, first in the quantum-mechanicalproblem of a particle tunneling from a potential well in two dimensions and thenin the cosmological setting, as a tool for the description of the initial stage ofthe evolution of the universe.

Keywords: Wheeler-DeWitt equation, tunneling, creation of the universe from ``nothing''

Supervisor: doc. RNDr. Vladimír Balek, CSc.Department: FMFI.KTFDF - Department of Theoretical Physics and Didactics of

PhysicsHead ofdepartment:

doc. RNDr. Tomáš Blažek, PhD.

Assigned: 27.11.2012

Approved: 28.11.2012 prof. RNDr. Melánia Babincová, CSc.Guarantor of Study Programme

Student Supervisor

Acknowledgement

I would like to thank my supervisor doc. RNDr. Vladimír Balek, CSc. forhis guidance, help, kindness, patience and also for the choice of this interesting topic.I am also deeply indebted to my family for their invaluable support.

4

Abstract

In the thesis we study WKB solutions of the Wheeler–DeWitt equation describingcreation of the universe from “nothing”. This was studied by Vilenkin in 1986. Mo-tivated by his paper, we consider the case of a two-dimensional minisuperspace fora closed FLRW universe containing scalar field with quadratic potential. There is aclose resemblance between our problem and the problem of tunnelling into a potentialwell in the two-dimensional quantum mechanics. The tunnelling from a potential wellwas investigated by Banks, Bender and Wu in 1973. We first adjust their techniqueto our problem and then use the resemblance of the quantum mechanical and quan-tum cosmological problems to explore the latter, as proposed by Balek in 2014. Inparticular, we construct a class of solutions so far not considered in the literature.These solutions arise when we require an asymptotic behaviour different from that ofVilenkin.

Keywords: 2D tunnelling, Wheeler–DeWitt equation, scalar field, minisuperspace,creation of the universe from “nothing”

5

Abstrakt

V práci skúmame WKB riešenia Wheelerovej–DeWittovej rovnice popisujúce vznikvesmíru z „ničoho“. Touto problematikou sa zaoberal Vilenkin v roku 1986. Motivo-vaní jeho článkom sa zameriame na prípad dvojrozmerného minisuperpriestoru preuzavretý FLRW vesmír obsahujúci skalárne pole s kvadratickým potenciálom. Našaúloha sa veľmi podobá tunelovaniu dovnútra potenciálovej jamy v kvantovej mecha-nike. Prípad tunelovania smerom von bol preskúmaný Banksom, Benderom a Wu vroku 1983. Najskôr upravíme ich metódu a potom využijeme spomínanú podobnosťproblémov na preskúmanie kvantovokozmologickej úlohy, ako navrhuje Balek v článkuz roku 2014. V rámci toho skonštruujeme triedu riešení, ktorá sa doteraz neuvažovalav literatúre. Tieto riešenia dostaneme, ak požadujeme iné asymptotické správanie akoVilenkin.

Kľúčové slová: tunelovanie v dvoch rozmeroch, Wheelerova–DeWittova rovnica, ska-lárne pole, minisuperpriestor, vznik vesmíru z „ničoho“

6

Preface

The thesis is devoted to quantum cosmology. I have chosen the topic, because itbelongs to the field of quantum gravity, the centre of attention of many theoreticalphysicist today. This theory is not yet complete, which allows anybody working in itto develop his own ideas and approaches. On the other hand, it is generally believedthat finding a satisfactory and well-defined theory of quantum gravity would be amajor leap forward in theoretical physics.

Understanding of the thesis requires the knowledge of quantum mechanics.In addition, in section 4.1 we use some basic concepts from general relativity as wellas the Lagrangian and Hamiltonian formalism.

7

Contents

1 Quantum-mechanical tunnelling 11

1.1 WKB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Matching the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 2D tunnelling outwards . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Modification of the problem 27

2.1 2D tunnelling inwards . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Outside the barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Some pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 In polar coordinates 40

4 Tunnelling in cosmology 45

4.1 Wheeler–DeWitt equation . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Vilenkin’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Alternative solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A Associated Legendre functions 58

8

Introduction

From the birth of quantum theory and general theory of relativity there have beenmany attempts to unify them, or to find a more general theory from which they wouldarise as respective limits. Over years most efforts have formed several “streams” [12].One significant line of research, often called covariant approach, is covered by stringtheory. In this work we focus on another approach, called canonical. It uses theprocedure of canonical quantisation, which is applied to Hamiltonian formulation ofgeneral relativity [7] (for that formulation see also [11]). The result is the Wheeler–DeWitt equation, which is in general a (functional) differential equation acting ona functional defined on the space of all possible three-geometries. This functional isreffered to as the wavefunction of the universe.

It turned out, however, that the Wheeler-DeWitt equation is ill-defined inthe general case. A resolution of the problem came only with loop quantum gravity,20 years after the moment the equation was first written down [12]. Nevertheless,different versions of the equation, restricted to some special cases of three-geometries,are consistent and have been intensively studied (see for example [7], [13], [14], [15];for more recent see [3], [5]). This will be also the focus of our thesis. More specifically,we will investigate the case of a closed FLRW universe with spatially homogeneousscalar field with quadratic potential. The classical state of the universe is at anymoment given by two parameters – radius of the universe and value of the scalarfield. Hence, the wavefunctional is reduced to a wavefunction dependent on just twovariables. The corresponding Wheeler–DeWitt equation has a form very similar tothe time-independent Schrödinger equation in two dimensions. Because of this, wecan employ techniques known from ordinary quantum mechanics – in the first partof our thesis we describe in detail the methods adopted from this theory which wewill later use in the quantum cosmological problem. Let us emphasise that all thework will be done in the WKB approximation, which is a reasonable framework fordiscussing the physical implications of solutions of the Wheeler–DeWitt equation (seefor example [6]).

We start with a brief outline of the basic principles and results of the WKBapproximation in quantum mechanics [10], [9] and then we proceed to describe anapproach to the problem of a particle tunnelling outwards from a non-rotationally

9

symmetric potential well in two dimensions, as proposed in [4] (see also [5] and [3]). Inthe second chapter, we modify the method so that instead of a particle escaping fromthe well, we examine an incoming beam of particles tunnelling through the barrierto the inside of the well [3]. We describe in detail the known procedures, at somepoint including our own arguments. The latter is done mainly at the point where wedeal with the separation constant (analysis around formula (1.62) and similar onesin the following chapters), when constructing the incoming wave corresponding tothe tunnelling solution (section 2.2), and when finding a neccessary condition for thesolution to be well-behaved (section 2.3). As a result, we arrive at a one-parametricclass of solutions for the problem considered.

In the short third chapter we obtain solutions of the same problem afterperforming a similar procedure as in previous chapters, but in polar coordinates (itis again done along the lines of [3]). This is an important midpoint on the way toquantum cosmology, because the Wheeler–DeWitt equation has almost the same formas Schrödinger equation in polar coordinates, the differences being only the reversesign of the kinetic term for scalar field and the value of energy.

In the last chapter, we turn our attention to the Wheeler–DeWitt equationitself. After constructing it, we describe the approach of Vilenkin [14], who examinedthe case of a universe tunnelling from zero radius into the domain of finite-sizeduniverses. This is called “creation of the universe from nothing”. It is important forus that Vilenkin in [14] uses a condition that the dependence of the wavefunctionon the value of the scalar field vanishes as the radius of the universe approacheszero. Motivated by [8], in the last section of the chapter 4 we restrict ourselvesto a particular choice of the ordering parameter1, and we formulate our alternativecondition, which together with the techniques from previous chapters yields a newone-parametric class of solutions2.

Finally, we devoted the Appendix to formulae concerning the associatedLegendre functions, as we need these throughout the computations.

1This parameter represents the ambiguity in the ordering of non-commuting operators, arisingduring the quantisation process.

2The reverse sign in the Wheeler–DeWitt equation implies that despite the fact that we probethe case of tunnelling outwards, we obtain the whole class of solutions as in the quantum mechanicaltunnelling inwards.

10

Chapter 1

Quantum-mechanical tunnelling

1.1 WKB approximation

One of the most widely used approaches to semiclassical approximation of quantummechanics is based on works of Wentzel, Kramers and Brillouin (thus the abbreviationWKB) and goes as follows [10, p. 232]:

The main idea is that a situation can be called semiclassical when thecorresponding de Broglie wavelength is much shorter than a characteristic lengthassociated with the problem, i.e. a typical scale on which the potential changes.Moreover, the de Broglie wavelength itself varies on this scale and thus we are leadto another equivalent formulation of the condition of semiclassicity

∆λ|λ λ, (1.1)

where ∆λ|λ is the change of λ on the distance of λ.

We now rewrite the condition in terms of “more useful” variables. From

λ = h

p= h√

2m(E − V (x)), (1.2)

where h is the Planck constant, p the momentum of the particle, m its mass, E itsenergy and V the potential, we have

∣∣∣∣∆λ|λ∣∣∣∣ ≈

∣∣∣∣∣dλ

dx

∣∣∣∣∣λ =∣∣∣∣∣∣

h√8m(E − V )3

dV

dx

∣∣∣∣∣∣λ = hm

p3

∣∣∣∣∣dV

dx

∣∣∣∣∣λ. (1.3)

Thus, the equivalent formulation of the condition is

hm

p3

∣∣∣∣∣dV

dx

∣∣∣∣∣ 1. (1.4)

11

Now we turn our attention to the one dimensional time-independent Schrö-dinger equation and to its solution, the wavefunction Ψ(x). For a free particle, wehave Ψ = Aeipx/~, where ~ is the reduced Planck constant. Because the potential inour case varies very slowly, the solution should resemble the one of a free particle.Therefore we write

Ψ = AeiS(x)/~ (1.5)and expect the coefficient A to be almost constant. After passing to exactly constantA and inserting this into the Schrödinger equation we obtain

−i~d2S

dx2 +(dS

dx

)2

= p2. (1.6)

Now we perform the crucial step: we expand the S function into the Tay-lor series in ~. One motivation is that the (reduced) Planck constant in some waycharacterises the quantum world (e.g. through the uncertainty principle) and “thesmaller the constant, the smaller the quantum effects”. Considering this, it seemsnatural that when investigating the semiclassical effects, we write such an expansionand then consider only first few terms in it. However, there is an alternative point ofview, which we adopt here. In this approach, we write the expansion (which shouldof course exist for every well behaved function) and then we inspect the terms one byone and see whether we can use the condition (1.4) to cut off the series.

When we put

S(x) = S0(x) + ~S1(x) + ~2

2 S2(x) + . . . (1.7)

into (1.6) and collect together the terms of the same order we have equations(dS0

dx

)2

= p2 (1.8)

dS0

dx

dS1

dx− i

2d2S0

dx2 = 0 (1.9)

dS0

dx

dS2

dx+(dS1

dx

)2

− id2S1

dx2 = 0 (1.10)

...

From (1.8) we immediately have

S0(x) = ±∫p(x)dx. (1.11)

The equation (1.9) impliesdS1

dx= i

2pdp

dx. (1.12)

12

After eliminating dx and integrating both hand sides we obtain

S1(x) = i

2 log(C1p). (1.13)

Note that we have constants similar to C1 in the formulae for S0 and S2 as well, butwe have hidden them into the integrals. Eventually, we get rid of all these constantsby absorbing them into the preexponential factor.

The third equation (1.10) gives us

S2(x) =∫ 1

dS0dx

id2S1

dx2 −(dS1

dx

)2 dx =

= ±∫ 1p

−1

2

(p′

p

)′+ 1

4

(p′

p

)2 dx =

= ∓14

∫ (2p′′p2 − 3p

′2

p3

)dx = (1.14)

= ∓(p′

2p2 + 14

∫ p′2

p3 dx

)=

= ±(mV ′

2p3 −m2

4

∫ V ′2

p5 dx

)

where we denoted d/dx by a prime and used integration by parts as well as the factthat p =

√2m(E − V (x)). Putting all together we have

S(x) = ±∫pdx+ i~

2 log(C1p)±~2

2

(mV ′

2p3 −m2

4

∫ V ′2

p5 dx

)+ . . . (1.15)

Considering the condition (1.4) we now see that the term proportional to~2 can be neglected. So can the higher terms, which is a fact that we will not provehere (it seems rather intuitive, though).

Finally we have obtained the expression

Ψ(x) ≈ A

k1/2 exp(i∫kdx

)+ B

k1/2 exp(−i∫kdx

), (1.16)

where we used k = p/~.

At this point a discussion is needed. In (1.3) we can see that near theturning point (i.e. point, where E = V (x), see figure 1.1) our approximation is nolonger valid. It means that we can use the formula (1.16) only in the regions farfrom these points (in our case far left and far right of the point b, respectively).A question arises concerning the nature of the solutions in the regions excluded orat least about the relation of the coefficients in the respective domains (we meanpreexponential coefficients in (1.16)). Let us now briefly describe how to (partiallybut rather satisfactorily) resolve these issues.

13

V(x)

xb

E

Figure 1.1: Situation near the turning point

1.2 Matching the solutions

A “standard” approach [10, p. 236] is to linearise the potential in the region nearthe turning point and presume that the domain of validity of this approximationoverlaps with the domains where we have used the WKB approximation. By solvingthe Schrödinger equation in the linearised case we obtain Airy functions, which wethen use to relate the WKB coefficients from the “left” and from the “right”. There is,however, another method, described in [9, p. 164], which uses analytic continuationof the solution to bypass in the complex plane the “dangerous” turning point. Ofcourse, both ways we come to the same result. However, the latter method is ratherelegant and we can easily generalise it to our case of tunnelling in 2D. We thereforeturn our attention to it.

Consider a potential similar to that depicted in figure 1.1. To avoid confu-sion when using square roots we define the function

w(x) = 2m~2 (E − V (x)) . (1.17)

In terms of w the solution in the left region is

ΨL = C1

w1/4 exp(i∫ x

b

√wdx

)+ C2

w1/4 exp(−i∫ x

b

√wdx

). (1.18)

For the wavefunction on the right we have

ΨR = C

2(−w)1/4 exp(−∫ x

b

√−wdx

)(1.19)

(in the latter formula we have omitted the solution increasing to infinity when x →∞).

14

We now claim that it is possible to solve the time independent Schrödingerequation in the complex plane and that the left and right solutions will be smoothlyconnected by a (smooth) solution in the complex region between the two segmentsof the real axis to which the solutions refer (see figure 1.2; the grey colour marksthe region where we are not allowed to use the WKB approximation - we have to“go” around). Thus, both ΨL and ΨR being parts of the same solution, we can easilyunveil the relation between the coefficients appearing in them.

b

Im

Re

x

x

Figure 1.2: Region where the WKB approximation is not valid

From now on, we set the cut of the (complex) square root function on thenegative real axis. In other words, we have

√eiϕ = e

i2ϕ for ϕ ∈ (−π, π). (1.20)

We start from the solution on the right and analytically continue it first to the uppercomplex half plane and then to the lower complex half plane. We must also analyt-ically continue the potential V (x). For this purpose, we get along with the linearapproximation like in the method of Airy functions.

The expression (1.19) is valid not only on the real axis for x b, but in thewhole upper region far from B up to the real axis on “the other side” (i.e. for x b)1.After we choose small ε, set x = b + rei(π−ε) (figure 1.3, in this case ϕ = π − ε), anddenote α = V ′(b), we have

√−w(x) ≈

√2m~2 αre

i(π−ε) ≈ i

√2m~2 αr ≈ i

√w(x), 2 (1.21)

1On the axis itself (left to b) the square root in solution (1.19) has a cut, which is why we muststop our analytic continuation before reaching the axis. However, this is not a problem, becausewithout any further trouble we can continue (1.18) so that it meets the solution just next to the realaxis.

2When we consider only the beginning and the end, the complexity of the expression may look a

15

and the solution carried over from the right has the form

C

2√i(w)1/4

exp(−i∫ x

b

√wdx

)= Ce−iπ/4

2w1/4 exp(−i∫ x

b

√wdx

), (1.22)

which is exactly the second term in (1.18). We would be perfectly satisfied were itnot the fact that we reconstructed only half of the solution. What is the reason?

One good idea is to look at the behaviour of the terms in (1.18) whenanalytically continuing ΨL. When continuing the left solution towards the rightregion in the upper plane, the first term becomes negligible compared to the second.To see this, consider an upper semicircle around b with the radius R. For x = b+Reiϕ,x = b+ reiϕ, ϕ ∈ (0, π], we have

∫ x

b

√wdx = α

√2m~2

∫ R

0

√−reiϕeiϕdr = α

√2m~2

∫ R

0

√rei(ϕ−π)eiϕdr =

= 23α√

2m~2 R

32 ei

ϕ−π2 +iϕ = −i23α

√2m~2 R

32 ei

32ϕ,

(1.23)

where we integrated along the path of constant ϕ. Notice that we have used−1 = e−iπ

b

Im

Re

x

x

r

φ

Figure 1.3: Coordinates description

instead of (perhaps more popular) −1 = eiπ. This is in order to satisfy the conditionthat the phase of the exponential lies in the interval (−π, π) so that we can use (1.20).Now we can see that the module

∣∣∣∣exp(i∫ x

b

√wdx

)∣∣∣∣ = exp2

3α√

2m~2 R

32 cos

(32ϕ) (1.24)

bit absurd at first sight. But it is actually very important to distinguish whether we obtain√−w =

i√w or

√−w = −i√w, which we achieve by realising that π − ε < π implies√ei(π−ε) ≈ eiπ/2 = i

due to (1.20).

16

decreases exponentially when ϕ decreases from π, while the module of the secondterm in (1.18) increases exponentially. The implication is that we cannot expect thefirst term in (1.18) to appear in the analytic continuation of ΨR, because it couldhave been produced only by a term too small to survive the WKB approximation3.

Let us return to our result. We obtained the relation

C2 = C

2 e−iπ4 . (1.25)

When considering analytic continuation to the lower half plane and performing asimilar analysis (the crucial formula now being

√−w = −i√w), we arrive at thecondition

C1 = C

2 eiπ4 . (1.26)

As a result, we can write

ΨL = C

w1/4 cos(∫ x

b

√wdx+ π

4

). (1.27)

Putting all together we have

ΨL = C√k

cos(∫ x

bkdx+ π

4

), ΨR = C

2√|k|

exp(−∫ x

b|k|dx

)4. (1.28)

We now derive a rather famous consequence of these formulae (see [9,p. 170]). Imagine a situation similar to the one depicted on 1.4. The matchingformulae for the turning point b are given by (1.28), so for the internal solution wehave

Ψinside = C√k

cos(∫ x

bkdx+ π

4

). (1.29)

After performing another analysis, this time of the left turning point (the solution tothe far left of a is C ′/(2

√|k|) exp (

∫ xa |k|dx)), we arrive at

Ψinside = C ′√k

cos(∫ x

akdx− π

4

). (1.30)

We require this to be equal to (1.29), and as a result, we arrive at the condition∫ b

akdx− π

2 = nπ, (1.31)

3In other words, in our approximation, the function that gives birth only to the second term in(1.18) is in the region (relatively) far above the real axis indistinguishable from the function thatproduces both terms. Because of this, we can say that the continued ΨR function smoothly becomesthe second term in (1.18) modulo any function with the behaviour of the first term from (1.18).

4In this context from now on we will use the following (perhaps barbaric) notation: by |p| wewill mean

√|w|. (The barbarism is in the fact that if V > E, then p lies precisely on the cut of the

square root function.)

17

V(x)

x

b

E

a

Figure 1.4: A generic potential well

which is better known in its equivalent form1

2π

∮kdx = n+ 1

2 (1.32)

as the Bohr-Sommerfeld quantisation rule (by∮

we now mean simply going from a

to b and back again).

This condition answers a natural question concerning the nature of the re-lation between the WKB formula (1.16) and the time-independent Schrödinger equa-tion: The Schrödinger equation gives us stationary states as well as values of energybelonging to them. Energy spectrum is simply the spectrum of the Hamiltonian oper-ator present in the equation. On the other hand, the formula (1.16) allows us to com-pute (approximately) the wavefunctions of the stationary states, but does not itselfdescribe which energies actually belong to the spectrum. It is the Bohr-Sommerfeldrule (or possibly anything else describing the relations between the solutions fromdifferent regions) which gives us the answer to the possible energy values and thustells us which energies we are allowed to use (through k) when working with (1.16).

1.3 2D tunnelling outwards

We now describe the technique developed by Banks, Bender and Wu [4] (for a detaileddescription see also [5]) for treating multidimensional QM tunnelling out of a non-rotationally symmetric potential well. We focus on the case of two dimensions andthe potential of the form

V (x, y) = 14(x2 + y2)− 1

4ε(x4 + y4 + 2cx2y2), (1.33)

18

where 0 < ε 1 and |c| < 1 (for a typical visage of the potential, see figure 1.5). Weare interested in the situation when a particle in the ground state is put inside thewell and then tunnels out of it. We may visualise this for example by considering a2D quadratic potential with a particle in the ground state and then quickly changingthe potential to the one in (1.33), which causes the particle to slowly escape from thewell.

Figure 1.5: Potential V (x, y) for c = 3/8, ε = 1/100

The most penetrable parts of the barrier are the domains close to the xand y axes. These are called MPEP - the most probable escape paths. We will focuson the domain near the positive x axis. It is useful in this case to set a conditionthat the wavefunction behaves like an outgoing wave when x → ∞. This boundarycondition causes the hamiltonian to be non-hermitian5. As a consequence, the eigen-states, which we call quasistationary states, have an imaginary part in their energy6.The claim is that we can describe the situation of the particle tunnelling out by aquasistationary state with energy close to the one of the ground state in the quadraticpotential.

Near the x-axis the potential can be also expressed in a rather simple form

V (x, y) = V0(x) + 14k(x)y2, k = 1− 2εcx2, V0 = 1

4x2(1− εx2), (1.34)

where we have neglected the y4 term.

In our case with ε 1 we are allowed to use the WKB approximation. Thereason is that as a characteristic length of our potential we may choose the distance

5When “shifting” the hamiltonian operator in∫

(HΨ)∗ΦdV from Ψ to Φ by means of integrationby parts, the boundary term does not vanish.

6Among other things this causes ddt

∫|Ψ(t)|2dV 6= 0, where we have integrated over some compact

region centered around the well. This is very convenient for our purpose.

19

of the maximum of the potential (on the x-axis) from the centre, which is 1/√

2ε andthis is much larger than the mean squared displacement of the particle in the groundstate of the one-dimensional quadratic potential, (∆x)2 = 1, which is a quantity wecan in our case take instead of the de Broglie wavelength.

Now we set ~ = 1 to simplify the expressions. We will investigate theSchrödinger equation for a particle of mass m = 1/2

[−∂2x − ∂2

y + 14(x2 + y2)− 1

4ε(x4 + y4 + 2cx2y2)− E]Ψ(x, y) = 0. (1.35)

We begin the calculations with an observation that the ground state of theparabolic approximation of our potential (the ground state of a 2D QM oscillator;it has energy E = 1) is equivalent to two 1D QM oscillators. Thus, we can, whenconstructing our solution, assign energy 1/2 to the x and y directions respectively.Using this fact along with the WKB approximation we write the wavefunction in thefollowing form:

Ψ(x, y) = A(x, y)p− 12 exp

(−∫pdx− 1

4f(x)y2), (1.36)

where

p =√

V0 −E

2 =√

14x

2(1− εx2)− 12 .

7 (1.37)

We have introduced two new functions f(x) and A(x, y). Our plan is to put thisexpression into (1.35), use again the WKB approximation to neglect some terms andchoose f(x) so that it eliminates another group of “unpleasant” terms.

After inserting the expression into the Schrödinger equation, neglecting they4 term and imposing a condition that the terms proportional to y2 vanish8, we obtainequations

−f 2A+ 2fxAx − fxAp−1px − 2fxAp+ fxxA+ kA = 0 (1.38)and

−Ayy + fyAy + 12fA− Axx + Axp

−1px + 2Axp−34Ap

−2p2x+

+12Ap

−1pxx − Ap2 + 14x

2(1− εx2)A− A = 0,(1.39)

where we have used subscripts to denote the derivatives with respect to x and y.7We factored out of the function Ψ(x, y) a 1D WKB function. Obviously, this function must cor-

respond to the energy 1/2 in order that it matches properly with the ball-shaped function describingthe motion of the particle in the x-direction inside the well.

8This gives us the condition for f .

20

Thanks to our approximation, we can neglect Ax, px, fx, Axx, pxx and fxxcompared to Ap, p2, fp, Axp, pxp and fxp, respectively. The reason is that we canassert for example Ax ∼ A/l A/λ ∼ Ap, where l is the “radius” of the potential.Due to (1.37) we also have

−Ap2 + 14x

2(1− εx2)A− A = −12A. (1.40)

Now we introduce new variables: x→ ε−1/2x, p→ ε−1/2p, which we are going to usefrom now on (we have for example k = 1− 2cx2). We can also write

p =√

14x

2(1− x2)− ε

2 ≈12x(1− x2) 1

2 . (1.41)

Putting all together, we obtain

−2pfx = f 2 − k (1.42)

−Ayy + fyAy + 12fA+ 2Axp−

12A = 0. (1.43)

Let us now define w =√

1− x2. We have −w∂x = x∂w and therefore

2p∂x = xw∂x = −x2∂w, (1.44)

so we can write equations (1.42) and (1.43) in the form

x2fw = f 2 − k (1.45)

−Ayy + fyAy + 12fA− x

2Aw −12A = 0. (1.46)

The equation (1.45) is known as the Riccati equation. The standard sub-stitution for its linearisation is

f = −x2uwu, (1.47)

where we have introduced a new function u(w). Inserting the substitution into (1.45)we get the well known associated Legendre equation (see Appendix A)

(1− w2)u′′ − 2wu′ +(

2c− 11− w2

)u = 0 (1.48)

(in this case we have denoted the derivative with respect to w with prime to make theequation look more familiar). The solutions are the associated Legendre functions P µ

ν

and Qµν with their indices given by

ν(ν + 1) = 2c, µ2 = 1. (1.49)

21

It can be shown thatP 1ν ∝ P−1

ν , Q1ν ∝ Q−1

ν , (1.50)and in our calculations we are also free to choose and fix one of the two roots ofν(ν + 1) = 2c. We therefore take

u(w) = a1P−1ν (w) + a2Q

1ν(w) (1.51)

with ν being the larger root of ν(ν + 1) = 2c and a1, a2 being some fixed coefficients.

We now return to the equation (1.46). Instead of variables w and y we takew and s = y

u(w) . The equation will then have the form

Ass = −u2(1− w2)Aw + 12u

2A(f − 1). (1.52)

Using the method of separation we set A(w, s) = W (w)S(s) and obtain the equations

Sss = CS, (1.53)

u2(1− w2)Ww =(1

2u2(f − 1)− C

)W. (1.54)

The solutions for S are

S ∝

cos(√−Cs) for C ≤ 0

cosh(√Cs) for C > 0 (1.55)

and for W we have∫ dW

W=∫ [

f − 12(1− w2) −

C

u2(1− w2)

]dw, (1.56)

log |W | = −∫ C

u2(1− w2)dw −∫ uw

2udw −∫ dw

2(1− w2) =

= −∫ C

u2(1− w2)dw −12 log |u|+ 1

4 log∣∣∣∣1− w1 + w

∣∣∣∣ ,(1.57)

and finally

W ∝ |u|− 12

(1− w1 + w

) 14

exp[−C

∫ w

w0

dw

u2(1− w2)

]. (1.58)

The purpose of f from (1.36) is to make our calculations easier. Thus,once we have learned that it should be of the form (1.51), we are free to choose thecoefficients a1 and a2. Either way it eliminates the unpleasant terms. However, notall choices lead to “nice” formulas, as we will now show.

22

Let us start with considering the asymptotical behaviour of the functionsP−1ν and Q1

ν for small x:P−1ν ∼

x

2 , Q1ν ∼ −

1x. (1.59)

In (1.58) we choose the lower bound of integration w0 to lie in the region where thisapproximation is valid.

Now we examine the situation when u is chosen so that it contains Q1ν , i.e.

u(w) = ηP−1ν +Q1

ν , (1.60)

η being some coefficient9. For the f function we have f ≈ −1 for small x, the factor(1−w1+w

)1/4from (1.58) equals approximately

√x/2 in the same limit and for the integral

in (1.58) we obtain∫ w

w0

dw

u2(1− w2) =∫ x0

x

dx

x√

1− x2u2=∫ x0

xxdx = K − x2

2 (1.61)

where K is a constant. Imagine now a linear superposition of functions belongingto different separation constants. We will inspect whether it is possible to obtain aplausible solution by means of such a superposition. More specifically, we will try toconstruct a function with the asymptotic form proportional (up to some function ofx) to exp(−y2/4) for very small x10. In other words we search for C(α), D(α) suchthat

∫ ∞

0dα|u|− 1

2p−12

(1− w1 + w

) 14

exp(−1ε

∫pdx− 1

4fy2)·

·C(α) cosh

(αy

u

)exp

[−α2

∫ w

w0

dw

u2(1− w2)

]+ (1.62)

+D(α) cos(αy

u

)exp

[α2∫ w

w0

dw

u2(1− w2)

]∼ e−

y24 ,

where by ∼ we mean that the two expressions match asymptotically up to somefunction of x multiplying one or another hand side11.

Changing our integration path from the real axis to the imaginary one (bymeans of the standard method of analytic continuation and using

∮Φ(z)dz = 0 for

9As can be seen from the equations, the result does not depend on the “global” constant in u(w)(both u(w) and au(w) produce the same outcome). This is why we set the constant in front of Q1

ν

to 1.10This is the behaviour we require in order to match our solution with the internal bell-shaped

function correctly. (We presume that there is a region, where both the tunelling and the internalsolution are valid simultaneously.)

11The factor ε−1 in the first exponent is due to the rescaling of variables.

23

analytic Φ) for the integral containing the hyperbolic cosine we obtain precisely theform of the other term containing the (ordinary) cosine, because

cosh(iαy

u

)exp

[−(iα)2

∫ w

w0

dw

u2(1− w2)

]=

= cos(αy

u

)exp

[α2∫ w

w0

dw

u2(1− w2)

].

(1.63)

Thus, we can consider only one function F instead of C, D and the expression sim-plifies to

∫ ∞

0dα|u|− 1

2p−12

(1− w1 + w

) 14

exp(−1ε

∫pdx− 1

4fy2)·

·F (α) cos(αy

u

)exp

[α2∫ w

w0

dw

u2(1− w2)

]∼ e−

y24

(1.64)

orey24

∫ ∞

0dαF (α) cos(αxy)eα2(K−x2/2) ∼ e−

y24 , (1.65)

when the asymptotic behaviour is taken into account. The function

cos(αxy) exp(α2(K − x2/2)

)(1.66)

is even in α, which leads us to define

F (−α) = F (α) for α > 0 (1.67)

and write the equation (1.65) as∫ ∞

−∞dαF (α)eiαxyeα2(K−x2/2) ∼ e−

y22 . (1.68)

Multiplying both hand sides by exp(−iβxy), integrating with respect to y, and using∫ ∞

−∞ei(α−β)xydy = 2πδ(x(α− β)), (1.69)

∫ ∞

−∞e−y

2/2−iβxydy =√

2πe−β2x2/2, (1.70)

δ(x(α− β)) = x−1δ(α− β) (when computing∫dα) (1.71)

we obtain approximatelyF (β) ∝ e−Kβ

2. (1.72)

According to this result we are allowed to set

u(w) = ηP−1ν +Q1

ν , (1.73)

24

but then we have to use the whole superposition of solutions with the weight F (β) ∝e−Kβ

2 .12 This seems rather inconvenient for any further calculations. Fortunately,the case u = P−1

ν will provide us with a (much) more suitable alternative. Let ustherefore turn our attention to this option.

For u = P−1ν we have (for small x) u ≈ x/2, f ≈ 1, and for the integral in

(1.58) we write∫ w

w0

dw

u2(1− w2) =∫ x0

x

dx

x√

1− x2u2≈∫ x0

x

4dxx3 ≈

2x2 . (1.74)

In the latter we have used the fact that we are interested in x which is much closerto 0 than x0. Putting all together and requiring the plausible solution (i.e. solutionwith the same correct asymptotic behaviour as we tried to construct in the previouscase) we arrive at the condition (compare with (1.65))

e−y24

∫ ∞

0dαF (α) cos(2αy/x)e2α2/x2 ∼ e−

y24 . (1.75)

Performing the next step as in the previous case we obtain the form∫ ∞

−∞dαF (α)e2iαy/xe2α2/x2 ∼ constant in y. (1.76)

After multiplying both hand sides by exp(−2iβy/x) and integrating with respect toy we have approximately

F (β)e2β2x2 ∝ δ (β) (1.77)

or equivalentlyF (β) ∝ δ(β), (1.78)

which is really good news for us, because it means that we have to set the separationconstant from (1.53) and (1.54) to zero. Thanks to this, we (gladly) take u = P−1

ν .13

The solution is

Ψ(x, y) ≈ A|u|−1/2(1− w

1 + w

)1/4p−1/2 exp

(−1ε

∫pdx+ x2

4uwuy2), (1.79)

12Here it is appropriate to note that we have actually proven the other options (other weights)to be “bad behaved”. We have not rigorously proven, nor we ever will in this text, that the optionF (β) ∝ e−Kβ2 is correct in all conceivable aspects. For example it is possible that the respectivewavefunction will be inacceptable due to its behaviour in some region where our approximation isno longer valid. It is therefore impossible to prove any further results in this direction withoutexceeding completely the (WKB) framework of this thesis.

13One might argue that this analysis was perhaps unnecessarily long and complicated. Somewould probably have said from the beginning that from the two functions P−1

ν and Q1ν the first is

much more convenient (because of its nice behaviour) than the latter. This might indeed be true.Nevertheless, our analysis will be important for us in the next section, when it will turn out thatthe situation (i.e. the correct asymptotic behaviour) requires the presence of the Q1

ν function.

25

A being a constant, w =√

1− x2, and u = P−1ν . We can also take |u| = u due to the

fact that P−1ν (w) is positive on w ∈ (0, 1), ν ∈

(−1

2 , 1)(c ∈

(−1

8 , 1)), as can be seen

from numerical computations.

Finally, we show that our solution matches the internal solution not only inthe y, but also in the x direction. The internal solution, when written in our rescaledvariables, is

Ψinternal ∝ exp[−1

4

(1εx2 + y2

)]. (1.80)

In (1.79) we now consider x to be much smaller than 1 (so that it refers to positionsnear the centre), but sufficiently far (to the right) from the inner turning point (sothat the WKB approximation is still valid). We have

|u|−1/2 ≈√

2x,

(1− w1 + w

)1/4≈√x

2 , p−1/2 ≈√

2x, f ≈ 1. (1.81)

For the integral in the exponent we need also the next-to-leading order,∫ x

√2εpdx =

∫ x

√2ε

12√x2 − 2ε dx = x2

4 −ε

2 log(2xε

)− ε

4 +O( 1x2

)(1.82)

where we have used (1.41) and integrated (approximately) from the inner turningpoint14. Putting all together we obtain

Ψ ≈ constant · exp[−1

4

(x2

ε+ y2

)], (1.83)

which exactly matches (1.80).

14One might argue that we have another term in the exponential, 14fy

2, and this we should alsoexpand up to the next-to-leading order. However, we would obtain an expression proportional tox2y2 without any 1

ε factor and this can be neglected compared to the term 14y

2 (because the rescaledx variable is very small in this sector).

26

Chapter 2

Modification of the problem

2.1 2D tunnelling inwards

Now we turn our attention to the inverse problem (see [3]1). Imagine a beam ofparticles with E = 1 coming from both the plus and minus infinity in the x directionand heading towards the central potential well2. The wavefunction will have thefollowing form: In the region outside the barrier it will be a superposition of theincoming and outcoming wave (due to reflection) while in the barrier its module willdecrease in the x direction towards the centre. Our task is to determine the form ofthe wavefunction in this region.

Near the centre itself we assume that we have Ψ ∼ e−y2/4, where by ∼ we

again mean that the two functions match assymptotically each other, up to somefunction of x. This can happen in at least two different cases:

• The incoming wave is strongly focused around the x axis so that the module ofthe function near the upper or lower “pass” is smaller than the module insidethe well. As a consequence, it is reasonable to assume Ψinside → 0 for large y.

• The potential has some form that is more complicated than V (x, y) = 14(x2 +

y2)− 14ε(x

4 + y4 + 2cx2y2), and the form which we use is only approximate nearthe x axis. If the complete potential does not contain any other “passes”, wededuce again Ψinside → 0 for large y.

1In the paper [3] the author deals with both problems simultaneously.2Note that this problem allows us to choose the value of energy from a continuous spectrum as

opposed to the previous case, where the wavefuction originated from a state with energy from thediscrete spectrum of parabolic potential. We have chosen the value 1 just to demonstrate clearlythe correspondence with the previous case.

27

When solving the Schrödinger equation in the well (after neglecting the terms ofhigher order in x and y) we can use the method of separation of variables and forΨinternal(x, y) = X(x)Y (y) we arrive at two equations for a quantum harmonic oscil-lator. Both options mentioned imply that Ey = 1/2 + n for some n ∈ N. The choiceEy = 1/2 (and thus Ψ ∼ e−y

2/4) corresponds to the previous problem and thereforewe choose to study this case.

From Ey = 1/2 we immediately have Ex = 1/2. For the X function wethen obtain

−Xxx(x) +(1

4x2 − 1

2

)X(x) = 0, (2.1)

X(x) ∝ e−x2/4(

1 + a∫ x

0ex

2/2dx), (2.2)

a being an integration constant. The value of∫ x

0ex

2/2dx is for large x approximately1xex

2/2. Putting all together, we write (for large x)

Ψinternal ≈ constant · e− 14 (x2+y2) 1

xex22 = constant · 1

xe

14 (x2−y2). (2.3)

This result will be needed later (during the matching process).

Returning to our search for the tunnelling solution, we now write the wave-function as

Ψ(x, y) = A(x, y)p− 12 exp

(∫pdx− 1

4f(x)y2), (2.4)

where

p =√

V0 −E

2 =√

14x

2(1− εx2)− 12 . (2.5)

Note that the only difference between these formulae and (1.36), (1.37) is the oppositesign in front of

∫pdx. After inserting the expression into the Schrödinger equation,

neglecting the same terms as in the previous case and rescaling x and p, we obtain3

2pfx = f 2 − k (2.6)

−Ayy + fyAy + 12fA+ 2Axp−

12A = 0. (2.7)

This time we choose our substitution with the sign plus

f = x2uwu

(2.8)

to arrive again at the associated Legendre equation

(1− w2)u′′ − 2wu′ +(

2c− 11− w2

)u = 0. (2.9)

3It may be useful to compare the following expressions to their counterparts in section 1.3.

28

We choose the same form of the solution as before,

u(w) = a1P−1ν (w) + a2Q

1ν(w), (2.10)

ν being the larger root of ν(ν + 1) = 2c and a1, a2 being some fixed coefficients.

After changing the variables in (2.10) from w, y to w, s = yu(w) and using

the method of separation of variables, we have

Sss = CS, (2.11)

−u2(1− w2)Ww =[12u

2(f − 1)− C]W. (2.12)

This time the solutions are

S ∝

cos(√−Cs) for C ≤ 0

cosh(√Cs) for C > 0 (2.13)

andW ∝ |u|− 1

2

(1− w1 + w

)− 14

exp[C∫ w

w0

dw

u2(1− w2)

]. (2.14)

To simplify the analysis, we again choose w0 to lie in the region where the leadingterm approximation to P−1

ν and Q1ν is valid.

Let us now consider different linear combinations of P−1ν and Q1

ν and per-form a similar analysis as in the previous case. We start with

u(w) = P−1ν . (2.15)

For small x we have u ≈ x/2, f ≈ −1, and the integral in (2.14) equals approximately2/x2 (see (1.74)). We again want to find out whether it is possible to form a linearcombination of solutions with different separation constants C such that the resulthas the correct asymptotic behaviour (for small x)4. After repeating the steps in thesection 1.3 we arrive at

∫ ∞

−∞dαF (α)e2iαy/xe−2α2/x2 ∼ e−y

2/4, (2.16)

and after multiplying by exp(−2iβy/x), integrating with respect to y, and using∫ ∞

−∞e−y

2−2iβy/xdy =√

2πe−2β2/x2, (2.17)

we have approximatelyF (β)e−

2β2x2 ∝ e−

2β2x2 (2.18)

4i.e. we can possibly match it with the internal solution (we again presume that there is a regionwhere both the internal and tunnelling solution are valid simultaneously)

29

or equivalentlyF (β) = constant. (2.19)

The result is that u = P−1ν can be used, but in order to satisfy the conditions of good

asymptotic behaviour, we have to use solutions with all C’s in the superposition.However, we still have the option u = ηP−1

ν + Q1ν . This, as we will now show, leads

to much more elegant formulae.

In this case we have u ≈ −1/x, f ≈ 1 and the integral in (2.14) equalsapproximately K − x2/2 (see (1.61)). We repeat the same steps and obtain

∫ ∞

−∞dαF (α)eiαxye−α2(K−x2/2) ∼ constant, (2.20)

from which we deduce (after multiplying by eiβxy and integrating with respect to y)

F (β)e−β2(K−x2/2) ∼ δ(β) (2.21)

or equivalentlyF (β) ∝ δ(β). (2.22)

We therefore choose and will examine the case of

u = ηP−1ν +Q1

ν , (2.23)

when we have

Ψ(x, y) ≈ A|u|−1/2(1− w

1 + w

)−1/4p−1/2 exp

(1ε

∫pdx− x2

4uwuy2). (2.24)

It is easy to show that for all values of η function (2.24) matches (2.3).Using

|u|−1/2 ≈ √x,(1− w

1 + w

)−1/4≈√

2x, p−1/2 ≈

√2x, f ≈ 1 (2.25)

and (1.82) for the integral in the exponent, we have (in the same limit as in theprevious case)

Ψ ≈ constant · 1x

exp[14

(1εx2 − y2

)], (2.26)

which corresponds to (2.3).

Now we must emphasise that the setting with a particle (or a beam ofparticles) tunnelling inwards is in some sense much more complicated than the caseof tunnelling outwards. The complication arises from the fact that it was natural toexpect that the problem of particle escaping from the well has a unique solution. Thisreadily follows from the formulation of the problem: One has a particle in a well in the

30

ground state and the particle slowly escapes; what will be the probability amplitude?In the case of a beam of incoming particles we can no longer expect that the solutionwill be unique, because we have substantial freedom in choosing the exact form of theincoming wave. This is reflected in different wavefunctions in the tunnelling region.As a consequence, it is plausible that different values of the η coefficient in (2.23)will result in different Ψ’s. This is indeed the case. Before demonstrating all this, wenow show how to relate the wavefunction in the tunnelling region to the wavefunctionoutside the barrier.

2.2 Outside the barrier

We use the technique decribed in 1.2. Again it is reasonable to expect that the WKBapproximation is valid not only on the real axis, but also throughout the complexplane, as long as we stay far away from the turning point. Thus, we can construct theanalytic continuation of our solution, bypass the turning point by going in the upperhalf-plane, arrive again at the real axis on the other side of “the dangerous region”and proclaim that the solution obtained is “half” of the correct solution. The other“half” we get when we bypass the turning point in the opposite half-plane.

We now presume that the function u does not change the sign for x ∈ (0, 1),from which it immediately follows that |u| = −u due to (1.59). This is a reasonableassumption, as we will show later5.

The formula

Ψ(x, y) ≈ Au−1/2(1− w

1 + w

)−1/4p−1/2 exp

(1ε

∫pdx− x2

4uwuy2)

(2.27)

is valid in the complex plane up to the real axis on the opposite side of the point P(see figure 2.1). In the regions A and B (both of them being very close to the realaxis), the functions w, p and √p in (2.24) have the form

A (up) B (down)wu(x) = −i

√x2 − 1 wd(x) = i

√x2 − 1 (2.28)

pu(x) = − i2x√x2 − 1 pd(x) = i

2x√x2 − 1 (2.29)

√pu(x) = e−iπ/4

√1/2x

√x2 − 1

√pd(x) = eiπ/4

√1/2x

√x2 − 1, (2.30)

5The basic idea will be that when u passes 0, the wavefunction behaves non-physically (see u inthe denominator of (2.24)).

31

Im

Re

x

x

A

B

P

Figure 2.1: Complex plane (in x) with turning point P

where we have chosen the indices u and d so that it is obvious to which region dothe functions refer. Contrary to this, let us write u without indices, i.e. u(w) =ηP−1

ν (w) + Q1ν(w) and again use the subscript to denote the derivative with respect

to x. Then we can write for x > 1

Ψ ≈ C

[p−1/2u (u(wu))−1/2

(1 + wu1− wu

)1/4exp

(1ε

∫pudx+ xwu

4ux(wu)u(wu)

y2)

+

+p−1/2d (u(wd))−1/2

(1 + wd1− wd

)1/4exp

(1ε

∫pddx+ xwd

4ux(wd)u(wd)

y2)]

.

(2.31)

In Appendix A ((A.12), (A.13)) we prove the following equalities for η ∈ R:

u(w) = u(w), u′(w) = u′(w), (2.32)

where z denotes the complex conjugate of z and the prime denotes derivative withrespect to the argument. As a consequence, we have

u(wu) = u(wd) (2.33)

and

ux(wu) = ux(−i√x2 − 1) = −iu′(wu)(

√x2 − 1)x =

= −iu′(wd)(√x2 − 1)x = iu′(wd)(

√x2 − 1)x = (2.34)

= ux(i√x2 − 1) = ux(wd).

Putting all together, it can easily be seen that the second term in (2.31) is

32

a complex conjugate of the first term. Thus, we finally have

Ψ ≈ 2CRe

p−1/2d (u(wd))−1/2

(1 + wd1− wd

)1/4·

· exp(

1ε

∫pddx+ xwd

4ux(wd)u(wd)

y2)

,

(2.35)

where Re z is the real part of z ∈ C.

2.3 Some pictures

Now we show how the results for the case of tunnelling inwards look like, both in-side and outside the barrier. Figures 2.2 to 2.8 show |Ψ|2 in the region x ∈ (0, 1)near the x-axis (small y). The parameters have values ε = 0.1, ν = 0.5, η ∈−4,−2,−1, 0, 1, 2, 4.

As can be seen from the graphs, the function Ψ behaves unacceptable forlarge η, because the module goes to infinity for some x smaller than 1. We musttherefore abandon such solutions6. Similarly, for η much smaller than 0 the wave-function starts to increase in the transversal direction for some fixed x < 1, which isinacceptable for the particle in the barrier. These solutions are therefore “bad”, too.

The first problem (singularity in Ψ) is caused by the fact that u(x) crosses 0for some x ∈ (0, 1)7. A generic example of such bad-behaved function u(x) is depictedon 2.9. From the assymptotic form we also know that for every η 6= 0 the function u isnegative near x = 0. It is therefore tempting to try to find out for which η and ν hasthe function negative value also in x = 1. Clearly the answer gives us a neccessarycondition for determining whether the solution is well-behaved.

From (A.7), (A.15), (A.16), and

Γ(x)Γ(1− x) = π

sin(πx) (2.36)

we have

u(x = 1) = ηP−1ν (w = 0) +Q1

ν(w = 0) =

= − η

ν(ν + 1)P1ν (w = 0) +Q1

ν(w = 0) =

= −√π

Γ(−ν

2

)Γ(

12 + ν

2

)[

2ην(1 + ν) + π tan π(ν + 1)

2

]=

6This is not something new, we have already abandoned solutions with other separations con-stants/weight functions due to their wrong (assymptotical) behaviour.

7Another implication is divergence of f .

33

Figure 2.2: ν = 0.5, η = −4 Figure 2.3: ν = 0.5, η = −2

Figure 2.4: ν = 0.5, η = −1 Figure 2.5: ν = 0.5, η = 0

=√π

sin πν2

Γ(1 + ν

2

)

Γ(

12 + ν

2

)[

2ην(1 + ν) − π cot πν2

]. (2.37)

Recall that we chose ν to be the larger root of ν(ν + 1) = 2c. Thus

ν = −1 +√

1 + 8c2 . (2.38)

If we restrict ourselves to c ∈(−1

8 , 1)so that both roots are real, we have

ν ∈(−1

2 , 1). (2.39)

Moreover, for real positive arguments we have Γ(x) > 0. Putting all together we seethat u(x = 1) > 0 is equivalent to

1sin πν

2

[2η

ν(ν + 1) − π cot πµ2

]< 0, (2.40)

34

Figure 2.6: ν = 0.5, η = 1 Figure 2.7: ν = 0.5, η = 2

Figure 2.8: ν = 0.5, η = 4 Figure 2.9: u(x) (ν = 0.5, η = 4)

which can be simplified to

η < ηcrit, ηcrit = π

2 ν(ν + 1) cot πν2 . (2.41)

This is the neccessary condition for the solution to be well-behaved.

The second problem (the increase in transversal direction) is caused bynegative f(x). Again, from the assymptotic form we have that for small x the ffunction is positive,

f(x) = x2uwu

= −xwuxu≈ −x

1x2

− 1x

= 1. (2.42)

One more time, we can formulate the neccessary condition for the function to be“acceptable”:

f(x = 1) > 0. (2.43)

35

We first find for which η we have uw < 0. With the help of (A.7), (A.15), (A.16),(A.17), (A.18), and

xΓ(x) = Γ(x+ 1) (2.44)we have

uw(x = 1) =( 1w2 − 1

η[wνP−1

ν − (−1 + ν)P−1ν

]+

+[wνQ1

ν − (1 + ν)Q1ν−1

])∣∣∣x=1

== η(ν − 1)P−1

ν−1(w = 0) + (ν + 1)Q1ν−1(w = 0) =

= −ηνP 1ν−1(0) + (ν + 1)Q1

ν−1(0) =

=√π

Γ(ν2

)Γ(

12 − ν

2

)[−2ην− π(ν + 1) tan πν2

]=

= −√π

2Γ(1 + ν

2

)Γ(

12 − ν

2

)[2η + πν(ν + 1) tan πν2

](2.45)

As a result, when using ν ∈ (−12 , 1), the inequality uw(x = 1) < 0 is equivalent to

2η + πν(ν + 1) tan πν2 > 0, (2.46)

which gives usη > η′crit, η′crit = −π2 ν(ν + 1) tan πν2 . (2.47)

Consequently, for η < ηcrit (the first condition) the value f(x = 1) is negative if andonly if (2.47) is satisfied. We thus have another neccessary condition.

Putting all together, we have

−π2 ν(ν + 1) tan πν2 < η <π

2 ν(ν + 1) cot πν2 , (2.48)

which defines a region in the η, ν plane (see (2.10)). Although we have only proventhat the conditions are neccessary, numeric computation suggests that they mightbe also sufficient (or perhaps sufficient everywhere except for some small regions).However, a rigorous proof (if the reversed implication indeed holds) would probablybe much more complicated than the one we gave in this text.

Finally, in figures 2.11 to 2.24 we provide graphs of the |Ψ|2 function behindthe barrier, i.e. for x > 1 (again we focus only on the case of tunnelling inwards).The parameters have values ε = 0.1, ν ∈ 0.5, 0.9, η ∈ −2,−1, −0.5, 0, 0.5, 1, 28.

8We included also several graphs corresponding to values of ν and η lying outside the grey area infigure 2.10. We can see that the incoming waves constructed in the same way as for the well-behavedsolutions have qualitatively the same form.

36

Figure 2.10: Region where the two neccessary conditions are satisfied

We can now confirm that various η values indeed correspond to different incomingwaves.

As a curiosity, observe that while for x < 1 was the behaviour of |Ψ|2 forfixed x always of the form e−

14f(x)y2 (see (1.79)), behind the barrier it is no longer so.

The reason is that we have used only the real part of the function (see (2.35)).

Figure 2.11: ν = 0.5, η = −2 Figure 2.12: ν = 0.5, η = −1

37

Figure 2.13: ν = 0.5, η = −0.5 Figure 2.14: ν = 0.5, η = 0

Figure 2.15: ν = 0.5, η = 0.5 Figure 2.16: ν = 0.5, η = 1

Figure 2.17: ν = 0.5, η = 2 Figure 2.18: ν = 0.9, η = −2

38

Figure 2.19: ν = 0.9, η = −1 Figure 2.20: ν = 0.9, η = −0.5

Figure 2.21: ν = 0.9, η = 0 Figure 2.22: ν = 0.9, η = 0.5

Figure 2.23: ν = 0.9, η = 1 Figure 2.24: ν = 0.9, η = 2

39

Chapter 3

In polar coordinates

This short chapter deals with the same problem, but shows how to obtain solutionswhen working in polar coordinates (again see [3]). It will be useful in the quantum-cosmological problem, because, as we will show in the next chapter, the Wheeler-DeWitt equation has (in our case) a form very similar to Schrödinger equation in polarcoordinates. However, as it is unneccessary to provide all details of the calculation(the ideas are practically the same as in the sections 1.3 and 2.1), we will focus onlyon differences from the previous calculations and main results. We again consider thecase E = 1 (although the wave can have any energy from a continuous spectrum)and this time we compute the wavefunctions for tunnelling inwards and outwardssimultaneously.

Let us begin with writing equation (1.35) in the r, ϕ variables:[−1r∂rr∂r −

1r2∂

2ϕ + 1

4r2−

− 14εr

4(cos4 ϕ+ sin4 ϕ+ 2c cos2 ϕ sin2 ϕ

)− 1

]Ψ(r, ϕ) = 0.

(3.1)

After restricting ourselves to ϕ 1 and defining Ψ =√rΨ, for which

1r∂rr∂rΨ = 1√

r

(∂2r Ψ + 1

4r2 Ψ)≈ 1√

r∂2r Ψ, 1 (3.2)

we arrive at[−∂2

r −1r2∂

2ϕ + 1

4r2(1− εr2 + 2εγr2ϕ2)− 1

]Ψ(r, ϕ) = 0, (3.3)

where γ = 1− c.1The last is due to the WKB approximation and the fact that we are interested in large values

of r.

40

We look for the solution in the form

Ψ(r, ϕ) = B(r, ϕ)q− 12 exp

(∓∫qdr − 1

4g(r)ϕ2), (3.4)

where

q =√

14r

2(1− εr2)− E =√

14r

2(1− εr2)− 1 (3.5)

and the − and + signs refer to tunnelling outwards and inwards, respectively. Weintroduced two new functions, B(r, ϕ) and g(r). Again, the purpose of the latter is toeliminate a group of unpleasant terms (this time the terms with the common factorϕ2). After inserting (3.4) into (3.3), using the WKB approximation to neglect someterms, redefining the variables and functions according to

r → ε−1/2r, ϕ→ ε1/2ϕ, q → ε−1/2q, g → ε−1g, (3.6)

and definingκ = 2γr4, ξ =

√1− r2, (3.7)

we obtain

±r2gξ = 1r2 g

2 − κ, (3.8)

±2r3

ξqBξ +Bϕϕ −

12gB − gϕBϕ = 0, (3.9)

q ≈ 12r√

1− r2, (3.10)

where we have again denoted the respective derivatives with subscript.

To solve (3.8) we make a substitution (as proposed by Genzor in [5])

g = ∓r4vξv

(3.11)

and obtain the Gegenbauer equation (see (A.31))

(1− ξ2)v′′ − 4ξv′ − 2γv = 0, (3.12)

where we have again used vξ = v′. According to (A.34) the solution is

v(ξ) = 1r

[a1P

−1ν (ξ) + a2Q

1ν(ξ)

], (3.13)

where again ν is the larger root of ν(ν + 1) = 2c and a1, a2 are arbitrary constants.

In (3.9) we change the variables ξ and ϕ to ξ and ψ = ϕv(ξ) . Now the

equation assumes the form

∓r4(v2Bξ + 1

2vvξB)

= Bψψ. (3.14)

41

We again use the method of separation of variables, and after setting

B(ξ, ψ) = Ξ(ξ)Ψ(ψ)2 (3.15)

we obtain

∓r4(v2Ξξ + 12vvξΞ) = CΞ, (3.16)

Ψψψ = CΨ. (3.17)

The solutions for Ψ are

Ψ ∝

cos(√−Cψ) for C ≤ 0

cosh(√Cψ) for C > 0 (3.18)

and from (3.16) we have∫ dΞ

Ξ = −12

∫ dv

v∓ C

∫ dξ

r4v2 , (3.19)

Ξ ∝ |v|−1/2 exp[∓C

∫ ξ

ξ0

dξ

(1− ξ2)2v2

]. (3.20)

Once again, we choose ξ0 to lie in the region where our approximation of associatedLegendre functions is valid.

We now try to construct a superposition of solutions with different separa-tion constants C so that it has the correct assymptotic behaviour. First we deal withthe particle tunnelling outwards. We expect that near the centre, the wavefunctionwill not depend on ϕ. In other words, we would like to find a function G(α)3 suchthat

∫ ∞

−∞dα|v|− 1

2 q−12 exp

(−1ε

∫qdr − 1

4gϕ2)·

·G(α) exp(iαϕ

v

)exp

[α2∫ ξ

ξ0

dξ

(1− ξ2)2v2

]∼ constant in ϕ,

(3.21)

where by ∼ we mean that the two expressions assymptotically match (for small r)up to some function of r.

Consider the case v = 1rP−1ν (ξ). Then, for small r, we have v ≈ 1/2,

g ≈ 2r4v′′(r = 0),4 and the integral in the third exponential in (3.21) can be writtenas ∫ ξ

ξ0

dξ

(1− ξ2)2v2 =∫ r0

r

dr

r3√

1− r2v2≈ 4

∫ r0

r

dr

r3 ≈2r2 . (3.22)

2We denote the function of ψ by tilde to avoid confusion with the wavefunction.3We skip the steps which are similar to those in the previous chapters.4The second derivative is due to absence of a linear term in v (see (A.19)).

42

From (3.21) we have∫ ∞

−∞dαG(α)e2iαϕe

2α2r2 ∼ constant in ϕ (3.23)

and finally

G(β)e2β2r2 ∝ δ(β), (3.24)

G(β) ∝ δ(β). (3.25)

In the case v = 1r(ηP−1

ν +Q1ν) we have v ≈ −1/r2, g ≈ −2r2 and the integral

is∫ ξ

ξ0

dξ

(1− ξ2)2v2 =∫ r0

r

dr

r3√

1− r2v2≈∫ r0

rrdr = K − r2

2 . (3.26)

Condition (3.21) is equivalent to∫ ∞

−∞dαG(α)e−iαr2ϕe

α2(K− r2

2

)∼ e−

12 r

2ϕ2, (3.27)

G(β)eβ2(K− r2

2

)∝ e−

12 r

2β2, (3.28)

G(β) ∝ e−Kβ2. (3.29)

As a result, we choose the first option v = 1rP−1ν , because the expressions will be

much simpler. The wavefunction then has the form

Ψ(r, ϕ) ≈ B|v|−1/2q−1/2r−1/2 exp(−1ε

∫qdr + r4

4vξvϕ2), (3.30)

B being a constant.

For tunnelling inwards the task is again more complicated. We cannotexpect that the wavefunction in the centre will be independent of ϕ. However, theprecise dependence on ϕ can be easily determined when we (once again) presume thatthe wavefunction in the centre is supressed in the y direction in the same manner asbefore. From (2.3) we then have (in the rescaled variables and again restrictingourselves to ϕ 1)

Ψinternal ≈ constant · 1r cos (

√εϕ) exp

(−r

2

4ε

)exp

[r2

2ε cos2(√

εϕ)]≈ (3.31)

≈ constant · 1r

exp(−r

2

4ε

)exp

[r2

2ε(1− εϕ2

)]≈ (3.32)

≈ constant · 1r

exp(r2

4ε

)exp

(−1

2r2ϕ2

). (3.33)

43

We therefore want the tunnelling solution to satisfy Ψ ∼ e−12 r

2ϕ2 instead of Ψ ∼constant. Next, we search for a correct weight function G(α).

For v = 1rP−1ν we arrive at

∫ ∞

−∞G(α)e2iαϕe−

2α2r2 dα ∼ e−

12 r

2ϕ2, (3.34)

from which we deduceG(β) = constant. (3.35)

For v = 1r(ηP−1

ν +Q1ν) we have

e−12 r

2ϕ2∫ ∞

−∞G(α)e−iαr2ϕe

−α2(K− r2

2

)dα ∼ e−

12 r

2ϕ2, (3.36)

which impliesG(β) ∝ δ(β). (3.37)

Consequently, we choose v = 1r(ηP−1

ν + Q1ν). Again, we obtained a one-parametric

class of solutions. The wavefunction has the same form as in (3.30), difference beingonly in the choice of v.

Finally, we check that the results indeed match the respective internal so-lutions correctly even in the r direction. For tunnelling outwards we have

|v|−1/2 ≈√

2, q−1/2 ≈√

2r, g ≈ 2r4v′′(r = 0), (3.38)

∫ r

√4εqdr =

∫ r

√4ε

12√r2 − 4ε dr = r2

4 − ε log(r

ε

)− ε

2 +O( 1r2

), (3.39)

Ψ ≈ constant · exp(−r

2

4ε

), (3.40)

which corresponds to the ground state of the parabolic potential.

The case of tunnelling inwards differs in

|v|−1/2 ≈ r, g ≈ 2r2, (3.41)

which produces

Ψ ≈ constant · 1r

exp(r2

4ε

)exp

(−1

2r2ϕ2

). (3.42)

This is in accordance with (3.33).

44

Chapter 4

Tunnelling in cosmology

In this chapter we consider the quantum cosmological problem of closed, homogeneousand isotropic universe with spatially homogeneous scalar field Φ with potential V (Φ).At the beginning we describe the equation that governs quantum cosmology, both inits most general form and in its form restricted to our case.

4.1 Wheeler–DeWitt equation

The Wheeler–DeWitt equation arises in the “canonical” approach to quantisation ofthe general theory of relativity. Let us briefly describe its derivation. The procedureof rewriting the general theory of relativity into Hamiltonian formalism is due toArnowitt, Deser and Misner (see [11]). The quantisation process is done accordingto [7]. We use the metric tensor of signature (−,+,+,+).

In the general case, we define functions N and Ni (lapse and shift functions)which describe some particular choice of coordinates. We also need the metric hijinducted on the hypersurfaces of constant time,

hij = gij, i, j = 1, 2, 3. (4.1)

Using these functions we rewrite the Lagrangian density

L = − 12κE

(R + 2Λ)√−g, (4.2)

R being the scalar curvature, Λ the cosmological constant, κE the Einstein constant(κE = 8πG) and g the determinant of gµν (this is the L, from which we can derivethe left hand side of Einstein equations by means of the principle of least action). Wethen construct the Hamiltonian density H0, which turns out to have an elegant form

H0 = NiHi +NH, (4.3)

45

whereH andHi are functions of hij’s and their conjugate momenta πij’s only. Finally,using the procedure of canonical quantisation, we define Ψ to be a functional fromthe space of all hij’s to C. The only restriction imposed on this functional is that itis anihilated by H and Hi, i.e. by the operators formed by the standard procedure

πij → −iδ

δhij. (4.4)

It is easy to show that the equations HiΨ = 0 are equivalent to the statement thatthe Ψ functional in fact depends only on the three-geometries 3G, i.e. 3D manifoldsequipped with a metric tensor. In other words, Ψ is the same for different coordinaterepresentations of the same geometry. As a result, we are left with only one condition

HΨ = 0, (4.5)

which is called the Wheeler–DeWitt equation. We refer to the space of all three-geometries as superspace and to the functional Ψ[3G] as the wavefunction (wavefunc-tional) of the universe.

In our case we are interested in a closed FLRW (Friedmann–Lemaître–Robertson–Walker) universe, i.e. a universe with the metric of the form

ds2 = −dt2 + a2(t)(dχ2 + sin2 χdΩ2), (4.6)

where dΩ2 = dθ2 +sin2 θdϕ2. In this universe there “resides” a spatially homogeneousscalar field Φ with potential V (Φ). Its Lagrangian density is

LΦ =[−1

2(∂Φ)2 − V (Φ)]√−g. (4.7)

For every fixed moment t we therefore have a universe fully characterised by twonumbers: a, Φ. In quantum cosmology, the wavefunction Ψ is a function of these twovariables. We now construct the corresponding Wheeler–DeWitt equation.

Due to the fact that we restricted ourselves to three-geometries of a specialform, the only freedom left is reparametrisation of the t variable. We therefore choose

ds2 = −N2dt2 + a2(t)(dχ2 + sin2 χdΩ2) (4.8)

as our starting point. It contains all possible spacetime metrics of the required formand their parametrisations (when we glue together the three-spheres in the naturalway). The scalar curvature corresponding to (4.8) is

R = −6a−2[aN−1(N−1a)· +N−2a2 + 1].1 (4.9)1A quick derivation consists in the substitution dt → Ndt in the scalar curvature for the closed

FLRW universe.

46

Similarly, the scalar field Lagrangian density has the form

LΦ =(1

2N−2Φ2 − V (Φ)

)√−g.2 (4.10)

Putting all together, we obtain action

S =∫Lgravd

4x+∫LΦd

4x = (4.11)

=∫ 3

κE

[N−1a(N−1a)· +N−2a2 + 1

]a−2 − (4.12)

− 1κE

Λ +(1

2N−2Φ2 − V (Φ)

)Na32π2dt (4.13)

Now we set 12π2

κEto 1, define λ = Λ

3 , rescale the scalar field, Φ→ 1√2πΦ, and

skip the total derivatives. As a result, for the Lagrangian (i.e. the integrand whenintegrating with respect to t) we have

L = 12(−N−1aa2 +Na−Nλa3 +N−1a3Φ2 − 2NV (Φ)a3). (4.14)

Using the canonical momenta corresponding to a and Φ,

πa = −N−1aa, πΦ = N−1a3Φ, (4.15)

we construct the Hamiltonian,

H = πaa+ πΦΦ− L = (4.16)

= −N2a[π2a − a−2π2

Φ + a2(1− λa2 − 2a2V (Φ))]. (4.17)

In the quantum theory, we define the wavefunction of the universe Ψ(a,Φ).As was mentioned earlier, this is a function on the space of a’s and Φ’s3. Ψ mustsatisfy the Wheeler–DeWitt equation

[−a−p∂aap∂a + a−2∂2

Φ + a2(1− λa2 − 2a2V (Φ))]

Ψ(a,Φ) = 0, (4.18)

where p is a constant. The complicated form of the first term arises due to theoperator ordering problem – in the Hamiltonian we have a product of two operatorsa−1 and πa, which do not commute in the quantum theory. In general, it should beof the form aα∂aa

β∂aaγ, α + β + γ = 0. However, similarly as in [14], we restrict

ourselves to the case γ = 0. (The only difference between the form in (4.18) and theone containing α, β, γ, is in some multiplicative factor aθ in the solution.)

2This is due to (∂Φ)2 = Φ;µΦ;µ = gµνΦ;µΦ;ν = gµνΦ,µΦ,ν = g00Φ2 = − 1N2 Φ2.

3We call it the minisuperspace.

47

4.2 Vilenkin’s solution

In the paper [14] the author investigated equation (4.18) for the case of a universe“created from nothing” and found a tunnelling solution. Let us now describe hisapproach. We consider a scalar field with a potential that is bounded from above.We presume it to be of the form

V (Φ) = ρv −12µ

2Φ2 (4.19)

near the global maximum at Φ = 0.4 Moreover, we set ρv = 0, because the term isexactly the same as the one containing λ.5 Such a potential corresponds to a scalarfield with imaginary mass m = iµ. We also define U(a,Φ) = a2(1− λa2 − 2a2V (Φ))and η = µ2/λ. The equation can be written as

[−a−p∂aap∂a + a−2∂2

Φ + U(a,Φ)]

Ψ(a,Φ) = 0, (4.20)

withU(a,Φ) = a2

[1− λa2(1− ηΦ2)

]. (4.21)

Vilenkin divides the half-plane a ∈ [0,∞), Φ ∈ (−∞,∞) into two domains:so called Euclidean region, where U > 0, and Lorentzian region, where U < 0.These roughly correspond to classically forbidden and allowed region, respectively.In general, the border between the two domains is defined by

a2 = 1λ+ 2V (Φ) . (4.22)

We search for a solution that decreases exponentially for increasing a. Condition(4.22) together with (4.19) implies that the highest tunnelling probability will bealong the a-axis, i.e. for small Φ, as the barrier is most penetrable at this point.

In the computations, we restrict ourselves to the case η 1, λ 1. Ac-cording to Vilenkin, the first is required when we want to study a cosmologicallyinteresting inflationary scenario6 and the second must be satisfied if we want to usethe WKB approximation7. Vilenkin also keeps the value of p unspecified (his compu-tation is therefore valid for all p’s).

4In most cases we can find a global maximum and expand the potential near it to obtain aparabolic approximation. All we have to do is to shift the scalar field and we arrive at the potentialof the form (4.19).

5We can redefine λ→ λ+ 2ρv.6We want the lifetime of the false vacuum, i.e. of the state Φ = 0, to be much greater than the

expansion rate of a de Sitter space (which is a classical analogy of our problem).7The condition is equivalent to the statement that the energy is much smaller than the Planck

energy.

48

For small a we can simplify the form of the Wheeler–DeWitt equation(4.20)8 as (

−a−p∂aap∂a + a−2∂2Φ + a2

)Ψ(a,Φ) = 0, (4.23)

or equivalently (∂2a + p

a∂a − a−2∂2

Φ − a2)

Ψ(a,Φ) = 0. (4.24)

Vilenkin then searches for a solution that does not depend on Φ in this region. Thisis an important presumption, which distinguishes his solution from the solution thatwe will obtain in section 4.3. In his case the equation takes the form

(∂2a + p

a∂a − a2

)Ψ(a). (4.25)

Changing the independent variable from a to b = a2/2 and the function from Ψ to

Ω = b1−p

4 Ψ, (4.26)

we can rewrite (4.25) as

b2Ω′′ + bΩ′ −[(1− p

4

)2+ b2

]Ω = 0, (4.27)

where we have denoted the derivative with respect to b by a prime. This is themodified Bessel differential equation, which has the general solution

Ω(b) = a1I 1−p4

(b) + a2K 1−p4

(b), (4.28)

a1, a2 being some coefficients and I, K being the modified Bessel functions of the firstand second kind, respectively. Due to the fact that we describe tunnelling “outwards”,we want the modul of the solution to decrease for a → ∞. We therefore take Ω =K 1−p

4. Returning to a and Φ we have

Ψ(a) ∝ a1−p

2 K 1−p4

(a2

2

). (4.29)

For large a we obtainΨ(a) ≈ constant · a− p+1

2 e−a22 . (4.30)

We now return to the “full” Wheeler–DeWitt equation and construct atunnelling solution that matches (4.30). From (4.22) we see that the tunnelling region

8Note that we do not need to consider only small values of Φ. However, Φ must not leave thedomain of validity of our parabolic approximation of the potential.

49

ends at a = λ−1/2. This motivates the change of variable a→ λ−1/2a. We first rewrite(4.30) and obtain (for a 1)

Ψ(a) ≈ constant · a− p+12 e−

a22λ . (4.31)

The Wheeler–DeWitt equation has the form−a−p∂aap∂a + a−2∂2

Φ + a2

λ2

[1− a2(1− ηΦ2)

]Ψ(a,Φ) = 0, (4.32)

Vilenkin then proposes an approach to WKB approximation which uses an expansionin λ, i.e.

Ψ(a,Φ) = eiλS(a,Φ), S = S0 + λS1 + λ2S2 + . . . 9 (4.33)

Inserting this expression into (4.32), collecting the terms with the same order of λand defining v(a,Φ) = 1− a2 + ηa2Φ2, we arrive at

(∂S0

∂a

)2

− 1a2

(∂S0

∂Φ

)2

+ a2v(a,Φ) = 0, (4.34)

−2∂S0

∂a

∂S1

∂a+ 2a2∂S0

∂Φ∂S1

∂Φ + i∂2S0

∂a2 + ip

a

∂S0

∂a− i

a2∂2S0

∂Φ2 = 0. (4.35)

Since the wavefunction does not depend on Φ for small a (we have chosenit so) and Φ is in U accompanied by the small parameter η, Vilenkin concludes thatthe terms having derivatives with respect to Φ can be neglected. As a result, we have

(∂S0

∂a

)2

+ a2v(a,Φ) = 0, (4.36)

2∂S0

∂a

∂S1

∂a− i∂

2S0

∂a2 −ip

a

∂S0

∂a= 0. (4.37)

The first equation implies

S0 = ±ia∫ √

1− a2(1− ηΦ2)da =

= ∓ i

3(1− ηΦ2)

[1− a2(1− ηΦ2)]3/2 +H0(Φ),

(4.38)

where H0 is some function of Φ. We want the solution to decrease as a → ∞,which means that we use the upper sign. The function H0 can be determined usingthe requirement that the expontential factor in (4.31) matches correctly with thatappearing in the function (4.33). In other words, we want to satisfy

ia2

2λ = − i

3λ(1− ηΦ2)

[1− 3

2a2(1− ηΦ2) +H0

], (4.39)

9The leading term contains λ−1 so that we can reproduce (4.31) in the limit a→ 0.

50

which implies H0 = −1.

From equation (4.37) we have

∂S1

∂a= i

2

∂2S0∂a2∂S0∂a

+ ip

2a, (4.40)

S1 = i

2 log ∂S0

∂a+ ip

2 log a+ H1(Φ) (4.41)

S1 = i

2 log(ap+1

√v(a,Φ)H1(Φ)

), (4.42)

where H1 and H1 are some functions of Φ. Requiring that the expression asymptoti-cally corresponds to the preexponential factor in (4.31), we arrive at

H1 = constant. (4.43)

Putting all together, we obtain

Ψ ≈ constant · a− p+12 v−1/4 exp

[− 1− v3/2

3λ(1− ηΦ2)

]. (4.44)

In [14] Vilenkin also mentions the resemblence between his solution and thewavefunctions of Banks, Bender and Wu [4]. However, Vilenkin’s approach seems tobe less precise due to the larger number of neglected terms during the computation.It is therefore interesting to obtain the Vilenkin-type wavefunction using the methodsof Banks, Bender and Wu, and compare the two. This will be done as a byproductin the following section.

4.3 Alternative solutions

We now show how to obtain another set of solutions using the techniques developedin previous chapters. The difference from the Vilenkin’s solution lies in a requirementthat our function matches a different internal solution, this time dependent on Φ.However, we restrict ourselves to p = 1. This is a choice motivated by Hawking andPage [8] (mentioned also in [14]), who proposed that the differential operator in theWheeler–DeWitt equation should be the Laplace operator in the canonical metricof the superspace. We also choose to study the case µ 1, λ 1, the potentialis assumed to be of the same form as in the previous case and we use the notationη = µ2/λ.

We again presume that the most probable escape path leads through theregion close to the a-(half-)axis, i.e. where Φ is small, and we will investigate this

51

domain. After introducing new variables10

X = a cosh Φ, (4.45)Y = a sinh Φ, (4.46)

we use them to rewrite the differential operator as

a−1∂aa∂a − a−2∂2Φ = ∂2

X − ∂2Y .

11 (4.47)

For small a we therefore have the following equivalent form of equation (4.18):(−∂2

X +X2)

Ψ(X, Y ) =(−∂2

Y + Y 2)

Ψ(X, Y ). (4.48)

We have used a2[1−λa2(1−ηΦ2)] ≈ a2 = X2−Y 2. Motivated by a similar conditionin the chapter 3, we now require that the wavefunction is supressed in the Y directionand from all possible options we choose the one corresponding to the lowest energy. Inother words, we want that after the separation of variables, Ψ(X, Y ) = ΨX(X)ΨY (Y ),the function ΨY (Y ) satisfies

(−∂2

Y + Y 2)

ΨY (Y ) = 1 ·ΨY (Y ) (4.49)

and thusΨY (Y ) ∝ e−Y

2/2.12 (4.50)As a consequence, we have

ΨX(X) ∝ e−X2/2 (4.51)

and for the internal wavefunction we obtain

Ψ(X, Y ) ∝ e−12 (X2+Y 2). (4.52)

This solution has the assymptotic form (for large a and small Φ, similarly as inprevious chapters)

Ψ(a,Φ) ∝ e−a22 (cosh2 Φ+sinh2 Φ) = e−

a22 (1+2 sinh2 Φ) ∼ e−a

2/2e−a2Φ2. (4.53)

Now we compute the wavefunction in the tunnelling region and match it with (4.53).

In the WKB region it is convenient to use Ψ =√aΨ, because we have

1a∂aa∂a

1√a

Ψ = 1√a

(∂2aΨ + 1

4a2 Ψ)≈ 1√

a∂2aΨ, (4.54)

10At this point we use the “original”/non-rescaled a variable.11This can be quickly derived from the well-known formula for the Laplace operator in polar

coordinates, ∂2x + ∂2

y = r−1∂rr∂r + r−2∂2ϕ, after setting X = x, Y = iy, a = r, Φ = −iϕ.

12This is the ground state of the harmonic oscillator in 1 dimension.

52

where the second term vanished due to the WKB approximation. The equation (4.18)then has the form

−∂2

a + a−2∂2Φ + a2

[1− λa2(1− ηΦ2)

]Ψ(a,Φ) = 0. (4.55)

We again search for the solution

Ψ(a,Φ) = B(a,Φ)q− 12 exp

(−∫qda− 1

4g(a)Φ2), (4.56)

whereq =

√a2(1− λa2). (4.57)

After inserting these expressions into (4.55), repeating the procedure from previouschapters, redefining

a→ λ−1/2a, Φ→ λ1/2Φ, q → λ−1/2q, g → λ−1g, (4.58)

and using ξ =√

1− a2, we arrive at

−2a2gξ = 4ηa4 + a−2g2, (4.59)

−2a3

ξqBξ +BΦΦ −

12gB − gΦBΦ = 0, (4.60)

q = a√

1− a2. (4.61)

These equations are very similar to their analogues in section 3, (3.8) – (3.10), whenthe signs for tunnelling outwards are chosen in them. In fact, the only differencebetween (3.9) and (4.60) is the opposite sign of the first term.

To linearise equation (4.59) we define

g = 2a4

vvξ (4.62)

and obtain the Gegenbauer differential equation

(1− ξ2)v′′ − 4ξv′ + ηv = 0. (4.63)

The general solution can be written in the form

v = 1a

(a1P

−1ν (ξ) + a2Q

1ν(ξ)

)(4.64)

with the coefficient ν given by

ν(ν + 1) = 2 + η. (4.65)

53

In the second equation, (4.60), we change the variables from ξ, Φ to ξ, ψ =Φvand arrive at

Bψψ = a4v2(

2Bξ + vξvB). (4.66)

This equation is almost identical to (3.14) as is the process of its solution. The result(for B(ξ, ψ) = Ξ(ξ)Ψ(ψ)) is

Ψ ∝

cos(√−Cψ) for C ≤ 0

cosh(√Cψ) for C > 0 (4.67)

Ξ(ξ) ∝ |v|−1/2 exp(C

2

∫ ξ

ξ0

dξ

a4v2

). (4.68)

Before continuing in our task, we examine the Vilenkin’s condition, i.e. thatthe dependence on Φ vanishes for small a. We are searching for a weight functionG(α) such that

∫ ∞

−∞dα|v|− 1

2 q−12a−

12 exp

(−1λ

∫qda− 1

4gΦ2)·

·G(α) exp(iαΦv

)exp

[−α

2

2

∫ ξ

ξ0

dξ

(1− ξ2)2v2

]∼ constant in Φ.

(4.69)

For v = 1aP−1ν , we quickly obtain (as in chapter 3)

F (α) ∝ δ(α), (4.70)

while for v = 1a(ηP−1

ν +Q1ν) we arrive at

F (α) ∝ eK2 α

2.13 (4.71)

The conclusion is that we have obtained a solution (considering the one with v =1aP−1ν ; the final form of wavefunction is the same as in (4.75)) of Vilenkin’s type

that seems to be more accurate than that of Vilenkin in [14].14 The reason, as wasmentioned before, is that using the methods of Banks, Bender and Wu [4], we haveneglected fewer terms in the equation than Vilenkin.

13In the latter case we must overcome a small obstacle during the computation. This arises dueto the fact that we have an opposite sign in the exponent of the right hand side of the equationanalogous to (3.27) (instead of r we now have a). However, we can define a = ib and after presumingb ∈ R perform the integration. After the analytic continuation we then obtain the answer for a ∈ R.Note that despite the fact that the weight function increases exponentially, this does not cause anydivergence in the integral, because the term is cancelled by a corresponding factor with oppositesign, present in the other exponential.

14In Vilenkin’s solution there are only elementary functions as opposed to the wavefunction weobtained.

54

Let us turn to our case, i.e. searching for a solution with the assymptotic(4.53). Hence, we need to examine

∫ ∞

−∞dα|v|− 1

2 q−12a−

12 exp

(−1λ

∫qda− 1

4gΦ2)·

·G(α) exp(iαΦv

)exp

[−α

2

2

∫ ξ

ξ0

dξ

(1− ξ2)2v2

]∼ e−a

2Φ2.

(4.72)

For v = 1aP−1ν we have

G(α) = constant in α (4.73)and for v = 1

a(ηP−1

ν +Q1ν)

G(α) ∝ δ(α). (4.74)We are interested in the latter case, because it contains a whole new class of solu-tions that differ in their very nature (i.e. in their assymptotic behaviour) from theVilenkin’s one. The wavefunction is

Ψ(a,Φ) ≈ B|v|−1/2q−1/2a−1/2 exp(−1λ

∫qda− a4

2vξv

Φ2). (4.75)

We now show that the results match correctly the internal solutions inboth directions (a and Φ). For the case v = 1

aP−1ν , G(α) ∝ δ(α), which satisfies the

Vilenkin’s condition of independence on Φ for small a, we have

|v|−1/2 ≈√

2, q−1/2 ≈ a−1/2, g ≈ −4a4v′′(a = 0), (4.76)∫ a

0qda ≈ a2

2 , (4.77)

Ψ ≈ constant · 1a

exp(− a

2

2λ

), (4.78)

which corresponds to (4.31).

In the case of the one-parametric class of solutions v = 1a

(ηP−1ν +Q1

ν),G(α) ∝ δ(α), the only difference is

|v|−1/2 ≈ a, g ≈ 4a2, (4.79)

which gives us

Ψ ≈ constant · exp(− a

2

2λ − a2Φ2

)(4.80)

in accordance with (4.53).

Although all values of η correspond to the same leading term in g(a) (andthus to the same leading behaviour in transversal direction), the difference between

55

various η can be seen clearly in the next-to-leading term. This follows from (A.19)and (A.29):

v = 1a

[ηP−1

ν

(√1− a2

)+Q1

ν

(√1− a2

)]

= − 1a2 + η

2 + ν(ν + 1)2

[log a2 + Ψ(ν + 1) + γ − 1

2

]+ o(1), (4.81)

va = 2a3 + ν(ν + 1)

2a + o(1a

), (4.82)

g = 4a4[1 + ω(a)] + o(a4), (4.83)

whereω(a) = ν(ν + 1)

2 log a2 + η

2 + ν(ν + 1)4 + Ψ(ν + 1) + γ − 1. (4.84)

The result is in agreement with [3].

56

Conclusion

In the thesis we focused on the topic of quantum cosmology. More specifically, westudied the solutions of the Wheeler–DeWitt equation for the case of a closed FLRWuniverse with spatially homogeneous scalar field with quadratic potential. Firstly,we described the procedure, developed by Banks, Bender and Wu [4], for finding atunnelling solution in two-dimensional quantum mechanics. We than showed how toalter the technique (according to [3]) so that it covers the case of an incoming beamof particles, which tunnel through the barier to the inside of the well.

We contributed to the calculations with our own arguments, for examplewhen dealing with the separation constant (analysis around formula (1.62) and similarones in the following chapters). Using the technique of analytic continuation [9,p. 164], we also constructed the form of the incoming beam of particles correspondingto the tunnelling solutions. We demonstrated that some solutions are not physicallyacceptable and we found a neccessary condition for a solution to be well-behaved.

In the last chapter we studied the case of a “universe created from nothing”.We first described the approach of Vilenkin [14], in which he used the condition thatthe dependence of the wavefunction on the values of the scalar field vanishes as theradius of the universe approaches zero. Motivated by [8], we then chose a specialvalue of the ordering parameter, p = 1. In this case, we proposed an alternativeboundary condition. Finally, employing the techniques from the previous chapterswe constructed a new one-parametric class of solutions.

Using the method of analytic continuation it should be again possible tofind the forms of the respective solutions in the Lorentzian domain (i.e. region of largeradii). As we are working in the WKB approximation, it is plausible that a classicaltrajectory can emerge by interference of waves [6] and finding it we would obtain theinformation about the very early evolution of the universe with scalar field. This maybe useful when investigating possible inflationary scenarios.

57

Appendix A

Associated Legendre functions

In this appendix we use formulae from [1], [2] and [5, Appendix].

Consider the differential equation

(1− w2

)u′′(w)− 2wu′(w) +

[ν(ν + 1)− µ2

1− w2

]u(w) = 0, (A.1)

which is called the associated Legendre equation. The solutions can be written in theform

u(w) = a1Pµν (w) + a2Q

µν (w), (A.2)

where a1 and a2 are some coefficients and P µν , Qµ

ν are the associated Legendre functionsof the first and second kind, respectively.

The following identities hold for the functions P µν and Qµ

ν :

P µν (w), Qµ

ν (w) ∈ R for µ, ν ∈ R, w ∈ (0, 1), (A.3)

P µν (w) = 1

Γ(1− µ)

(1 + w

1− w)µ/2

2F1

(−ν, ν + 1; 1− µ; 1− w

2

), (A.4)

Qµν (w) = π

2 sin(πµ)

(cos(πµ)P µ

ν (w)− Γ(µ+ ν + 1)Γ(−µ+ ν + 1)P

−µν (w)

), (A.5)

where 2F1 is the hypergeometric function defined by the series

2F1(α, β; γ; z) = Γ(γ)Γ(α)Γ(β)

∞∑

n=0

Γ(n+ α)Γ(n+ β)n!Γ(n+ γ) zn (A.6)

and in (A.4), (A.5) we have to take the respective limits in µ when we are interestedin the case µ ∈ −1, 1.

58

Observe also the following useful identities:

P 1ν = −ν(ν + 1)P−1

ν (w), (A.7)Q1ν = −ν(ν + 1)Q−1

ν (w), (A.8)P µ−ν−1(w) = P µ

ν (w), (A.9)Qµ−ν−1(w) = Qµ

ν (w) + cos(πν)Γ(µ− ν)Γ(µ+ ν + 1)P−µν (w). (A.10)

These imply that it is possible to choose and fix some ν and some µ (for either of thefunctions P−1

ν and Q1ν we have two options for each index).

We will now prove the equalities used in section 2.2. We presume thatP µν (w) and Qµ

ν (w) are analytic (in the region we are interested in)1. Let us considera solution u(w) of (A.1) with ν, µ ∈ R, which is real on (0, 1). It is therefore of theform u(w) = αP µ

ν + βQµν , α, β ∈ R. The key point is to realise that the function

v(w) = u(w) is then equal to u(w) for all w ∈ (0, 1) and also in some region aboveand under the real axis. The reason is that one can expand u(w) into Taylor seriesaround any point x0 ∈ (0, 1) and in this expansion all coefficients are real (becauseall the derivatives of u(w) are real on (0, 1)). The claim then follows from

∞∑

n=0anz

n =∞∑

n=0anzn, when an ∈ R ∀n. (A.11)

Finally, due to the uniqueness of analytic continuation, we have that there holds

u(w) = u(w) (A.12)

in the entire region.

The other equality we used in (2.2),

u′(w) = u′(w), (A.13)

follows readily from

u′(w) = limε→0

u(w + ε)− u(w)ε

= limε→0

u(w + ε)− u(w)ε

=

= limε→0

u(w + ε)− u(w)ε

= u′(w).(A.14)

1This is of course true, but we will not prove it in this text.

59

In section 2.3 the following formulae were needed:

P µν (w = 0) = 2µ

√π

Γ(

1−µ−ν2

)Γ(1− µ−ν

2

) (A.15)

Qµν (w = 0) = − 2µ−1π3/2

Γ(

1−µ−ν2

)Γ(1− µ−ν

2

) tan(π(µ+ ν)

2

)(A.16)

∂P µν (w)∂w

= 1w2 − 1 [wνP µ

ν (w)− (µ+ ν)P µν−1(w)] (A.17)

∂Qµν (w)∂w

= 1w2 − 1 [wνQµ

ν (w)− (µ+ ν)Qµν−1(w)] . (A.18)

The assymptotic formula for P−1ν (√

1− x2), x → 0 can be derived quicklyfrom (A.4) and (A.6):

P−1ν (√

1− x2) = x

2 + 116[2− ν(ν + 1)]x3 + o(x3). (A.19)

To obtain the assymptotic formula for Q1ν , x → 0 we have to perform the

limit µ→ 1. For the sake of completeness, we first construct the whole series expan-sion. In other words, we compute

limε→0

π

2 sin [π(1 + ε)]

(cos [π(1 + ε)]P 1+ε

ν (w)− Γ(ν + 2 + ε)Γ(ν − ε) P−1−ε

ν (w))

= (A.20)

= limε→0

π

2 sin(πε)

(cos(πε)P 1+ε

ν (w) + Γ(ν + 2 + ε)Γ(ν − ε) P−1−ε

ν (w)).

The factor in large brackets equals zero for ε = 0 because of (A.7). As a result

Q1ν(w) = 1

2∂

∂ε

∣∣∣∣∣ε=0

(cos(πε)P 1+ε

ν (w) + Γ(ν + 2 + ε)Γ(ν − ε) P−1−ε

ν (w))

=

= 12∂

∂ε

∣∣∣∣∣ε=0

(P 1+εν (w) + Γ(ν + 2 + ε)

Γ(ν − ε) P−1−εν (w)

).

(A.21)

We introduce the digamma function

Ψ(z) = Γ′(z)Γ(z) (A.22)

and using

zΓ(z) = Γ(z + 1), (A.23)

Γ(ε) = 1εΓ(1 + ε) = 1

ε+ o

(1ε

)(A.24)

60

we write

∂

∂ε

∣∣∣∣∣ε=0

P 1+εν (w) = ∂

∂ε

∣∣∣∣∣ε=0

(1 + w

1− w) 1+ε

2 1Γ(−ν)Γ(ν + 1) ·

·∞∑

n=0

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n− ε)

(1− w2

)n]=(1 + w

1− w)1/2 1

Γ(−ν)Γ(ν + 1) ·

·[

12 log 1 + w

1− w∞∑

n=1

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n)

(1− w2

)n− Γ(−ν)Γ(ν + 1)+

+∞∑

n=1

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n) Ψ(n)

(1− w2

)n], (A.25)

∂

∂ε

∣∣∣∣∣ε=0

Γ(ν + 2 + ε)Γ(ν − ε) = Γ(ν + 2)

Γ(ν) [Ψ(ν + 2) + Ψ(ν)] , (A.26)

∂

∂ε

∣∣∣∣∣ε=0

P−1−εν (w) = ∂

∂ε

∣∣∣∣∣ε=0

(1 + w

1− w)− 1+ε

2 1Γ(−ν)Γ(ν + 1) ·

·∞∑

n=0

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n+ 2 + ε)

(1− w2

)n]=(1 + w

1− w)−1/2 1

Γ(−ν)Γ(ν + 1) ·

·[−1

2 log 1 + w

1− w∞∑

n=0

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n+ 2)

(1− w2

)n+

+∞∑

n=0

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n+ 2) Ψ(n+ 2)

(1− w2

)n]. (A.27)

The result is

Q1ν(w) = 1

2Γ(−ν)Γ(ν + 1)

√1 + w

1− w[12 log 1 + w

1− w ·

·∞∑

n=1

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n)

(1− w2

)n−

− Γ(−ν)Γ(ν + 1) +∞∑

n=1

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n) Ψ(n)

(1− w2

)n]+

+ ν(ν + 1)√

1− w1 + w

[(Ψ(ν + 2) + Ψ(ν)− 1

2 log 1 + w

1− w)·

·∞∑

n=0

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n+ 2)

(1− w2

)n−

−∞∑

n=0

Γ(n− ν)Γ(n+ ν + 1)n!Γ(n+ 2) Ψ(n+ 2)

(1− w2

)n]. (A.28)

61

Now we can deduce

Q1ν(√

1− x2) = −1x

+ 12ν(ν + 1)

[log x2 + Ψ(ν + 1) + γ − 1

2

]x+ o(x), (A.29)

where we have used Γ′(1) = −γ (γ being the Euler–Mascheroni constant) and theformula

Ψ(z + 1) = 1Γ(z + 1) lim

ε→0

Γ(z + 1 + ε)− Γ(z + 1)ε

=

= 1zΓ(z) lim

ε→0

(z + ε)Γ(z + ε)− zΓ(z)ε

= (A.30)

= 1zΓ(z) lim

ε→0

[z

Γ(z + ε)− Γ(z)ε

+ Γ(z + ε)]

= Ψ(z) + 1z.

Another equation, which is closely related to associated Legendre equation,is the Gegenbauer differential equation,

(1− w2)u′′(w)− 2(µ+ 1)wu′(w) + (ν − µ)(ν + µ+ 1)u(w) = 0. (A.31)

This equation arises for example when one deals with the Sturm-Liouville problemfor the Laplace operator on a (n-dimensional) sphere considering functions dependenton only one spherical angle [5]. Making a substitution

u(w) = (1− w2)−µ/2v(w), (A.32)

computing the respective derivatives (for simplicity we deal only with cases µ 6= 0,−2)

u′ = (1− w2)−µ/2[µw(1− w2)−1v + v′

],

u′′ = (1− w2)−µ/2[µ(1− w2)−1v + µ(µ+ 2)w2(1− w2)−2v + (A.33)

+ 2µw(1− w2)−1v′ + v′′],

and inserting the expressions into (A.31), we arrive at the associated Legendre equa-tion (A.1). The solution to (A.31) is therefore

u(w) = (1− w2)−µ/2 (a1Pµν (w) + a2Q

µν (w)) . (A.34)

62

Bibliography

[1] http://functions.wolfram.com/HypergeometricFunctions/.

[2] http://mathworld.wolfram.com/GegenbauerDifferentialEquation.html.

[3] V. Balek. From ’nothing’ to inflation and back again. arXiv preprintarXiv:1401.4870 [gr-qc], 2014.

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