www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com

www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


www.pdfgrip.com


XIX

order contributions are difficult to obtain. Nonetheless, we also consider at
various points of the text comparisons with WKB approximations, also for
the verification of results.
In writing this text the author considered it of interest to demonstrate

the parallel application of both the Schrodinger equation and the path integral to a selection of basic problems; an additional motivation was that a
sufficient understanding of the more complicated of these problems had been
achieved only in recent years. Since this comparison was the guide-line in
writing the text, other topics have been left out which are usually found in
books on quantum mechanics (and can be looked up there), not the least for
permitting a more detailed and hopefully comprehensible presentation here.
Throughout the text some calculations which require special attention, as
well as applications and illustrations, are relegated to separate subsections
which — lacking a better name — we refer to as Examples.
The line of thinking underlying this text grew out of the author's association with Professor R. B. Dingle (then University of Western Australia,
thereafter University of St. Andrews), whose research into asymptotic expansions laid the ground for detailed explorations into perturbation theory
and large order behaviour. The author is deeply indebted to his one-time
supervisor Professor R. B. Dingle for paving him the way into this field
which — though not always at the forefront of current research (including
the author's) — repeatedly triggered recurring interest to return to it. Thus
when instantons became a familiar topic it was natural to venture into this
with the intent to compare the results with those of perturbation theory.
This endeavour developed into an unforeseen task leading to periodic instantons and the exploration of quantum-classical transitions. The author has
to thank several of his colleagues for their highly devoted collaboration in
this latter part of the work over many years, in particular Professors J.-Q.
Liang (Taiyuan), D. K. Park (Masan), D. H. Tchrakian (Dublin) and Jianzu Zhang (Shanghai). Their deep involvement in the attempt described here
is evident from the cited bibliography.*
H. J. W. Miiller-Kirsten
*In the running text references are cited like e.g. Whittaker and Watson [283]. For ease of
reading, the references referred to are never cited by mere numbers which have to be identified e.g.
at the end of a chapter (after troublesome turning of pages). Instead a glance at a nearby footnote
provides the reader immediately the names of authors, e.g. like E. T. Whittaker and G. N. Watson
[283], with the source given in the bibliography at the end. As a rule, formulas taken from Tables
or elsewhere are referred to by number and/or page number in the source, which is particularly
important in the case of elliptic integrals which require a relative ordering of integration limits and

parameter domains, so that the reader is spared difficult and considerably time-consuming searches
in a source (and besides, shows him that each such formula here has been properly looked up).

www.pdfgrip.com


Chapter 1

Introduction
1.1

Origin and Discovery of Quantum Mechanics

The observation made by Planck towards the end of 1900, that the formula
he had established for the energy distribution of electromagnetic black body
radiation was in agreement with the experimentally confirmed Wien- and
Rayleigh-Jeans laws for the limiting cases of small and large values of the
wave-length A (or AT) respectively is generally considered as the discovery
of quantum mechanics. Planck had arrived at his formula with the assumption of a distribution of a countable number of infinitely many oscillators.
We do not enter here into detailed considerations of Planck, which involved
also thermodynamics and statistical mechanics (in the sense of Boltzmann's
statistical interpretation of entropy). Instead, we want to single out the vital
aspect which can be considered as the discovery of quantum mechanics. Although practically every book on quantum mechanics refers at the beginning
to Planck's discovery, very few explain in this context what he really did in
view of involvement with statistical mechanics.
A "perfectly black body" is defined to be one that absorbs all (thermal)
radiation incident on it. The best approximation to such a body is a cavity
with a tiny opening (of solid angle d£l) and whose inside walls provide a diffuse distribution of the radiation entering through the hole with the intensity
of the incoming ray decreasing rapidly after a few reflections from the walls.
Thermal radiation (with wave-lengths A ~ 10~ 5 to 10 - 2 cm at moderate

temperatures T) is the radiation emitted by a body (consisting of a large
number of atoms) as a result of the temperature (as we know today as a
result of transitions between a large number of very closely lying energy levels). Kirchhoff's law in thermodynamics says that in the case of equilibrium,
the amount of radiation absorbed by a body is equal to the amount the body
1

www.pdfgrip.com


2

CHAPTER 1. Introduction

emits. Black bodies as good absorbers are therefore also good emitters, i.e.
radiators. The (equilibrium) radiation of the black body can be determined
experimentally by sending radiation into a cavity surrounded by a heat bath
at temperature T, and then measuring the increase in temperature of the
heat bath.

Fig. 1.1 Absorption in a cavity.
Let us look at the final result of Planck, i.e. the formula (to be explained)
u(u,T) = 2*?£(-?-)kT,
y
J
c 3 \ex - l )

where x = ^ = ^ - .
kT kXT

(1.1)

'

y

Here u(v, T)du is the mean energy density (i.e. energy per unit volume) of
the radiation (i.e. of the photons or photon gas) in the cavity with both
possible directions of polarization (hence the factor "2") in the frequency
domain v, v + dv in equilibrium with the black body at temperature T. In
Eq. (1.1) c is the velocity of light with c = u\, A being the wave-length of
the radiation. The parameters k and h are the constants of Boltzmann and
Planck:
k = 1.38 x 1(T 23 J K'1, h = 6.626 x 10 - 3 4 J s.
How did Planck arrive at the expression (1.1) containing the constant h
by treating the radiation in the cavity as something like a gas? By 1900 two
theoretically-motivated (but from today's point of view incorrectly derived)
expressions for u(u, T) were known and tested experimentally. It was found
that one expression agreed well with observations in the region of small A (or
AT), and the other in the region of large A (or AT). These expressions are:
(1) Wien's law.
u(u,T) = dv3e-C2U/T,
(1.2)
and the

www.pdfgrip.com


1.1 Origin and Discovery of Quantum Mechanics

3


(2) Rayleigh-Jeans law:
u(i>,T) =

2^C3T,

(1.3)

Ci, C2, C3 being constants.
Considering Eq. (1.1) in regions of a; "small" (i.e. exp(x) ~ 1+x) and "large"
(exp(—x) < 1), we obtain:

u(i/, T)
u(i/,T)

2^^kT,
.47TZ/ 2

{x small),

e xhv,

(x large).

We see, that the formulas (1.2) and (1.3) are contained in Eq. (1.1) as approximations. Indeed, in the first place Planck had tried to find an expression
linking both, and he had succeeded in finding such an expression of the form

u(v,T) =

av
e6"/T-i'


where a and b are constants. When Planck had found this expression, he
searched for a derivation. To this end he considered Boltzmann's formula
S — klnW for the entropy S. Here W is a number which determines the
distribution of the energy among a discrete number of objects, and thus over
a discrete number of admissible states. This is the point, where the

Fig. 1.2 Distributing quanta (dots) among oscillators (boxes).
discretization begins to enter. Planck now imagined a number TV of oscillators
or iV oscillating degrees of freedom, every oscillator corresponding to an
eigenmode or eigenvibration or standing wave in the cavity and with mean
energy U. Moreover Planck assumed that these oscillators do not absorb or
emit energy continuously, but — here the discreteness appears properly —
only in elements (quanta) e, so that W represents the number of possible
ways of distributing the number P := NU/e of energy-quanta ("photons",
which are indistinguishable) among the N indistinguishable oscillators at

www.pdfgrip.com


4

CHAPTER 1. Introduction

temperature T, U{T) being the average energy emitted by one oscillator.
We visualize the iV oscillators as boxes separated by N — 1 walls, with the
quanta represented schematically by dots as indicated in Fig. 1.2. Then W
is given by

{N + p-iy.


w = (N

(1.4)

- 1)!P!

With the help of Stirling's formula*
IniV! ~ JVlniV-iV + O(0),

N -* oo,

and the second law of thermodynamics ((dS/dU)v
Example 1.1)

u

= 1/T), one obtains (cf.

= vmrri

(L5)

as the mean energy emitted or absorbed by an oscillator (corresponding to
the classical expression of 2 x kT/2, as for small values of e). Agreement
with Eq. (1.2) requires that e ex is, i.e.
e = his,

h = const.


(1.6)

Fig. 1.3 Comparing the polarization modes with those
of a 2-dimensional oscillator.
We now obtain the energy density of the radiation, u(i>,T)dv, by multiplying U with the number nvdv of modes or oscillators per unit volume with
frequency v in the interval v, v + dv, i.e. with
riydu — 2 x —w—dv,

(1.7)

*See e.g. I. S. Gradshteyn and I. M. Ryzhik [122], formula 8.343(2), p. 940, there not called
Stirling's formula, as in most other Tables, e.g. W. Magnus and F. Oberhettinger [181], p.3. The
Stirling formula or approximation will appear frequently in later chapters.

www.pdfgrip.com


5

1.1 Origin and Discovery of Quantum Mechanics

where the factor 2 takes the two possible mutually orthogonal linear directions of polarization of the electromagnetic radiation into account, as
indicated in Fig.1.3. We obtain the expression (1.7) for instance, as in electrodynamics, where we have for the electric field
E oc elwt \ J eK sin KI^I sin K2X2 sin K3X3
K

with the boundary condition that at the walls E = 0 at Xi = 0, L for i = 1,2,3
(as for ideal conductors). Then L^j = nrii, rii = 1,2,3,...,
2
2 2

r2
L K — 7T n ,

where^

(lvL\A
0
I
I = rr.

2 [2-KUY
,
KT = I
J , so that

The number of possible modes (states) is equal to the volume of the spherical octant (where n^ > 0) in the space of n^,i = 1,2,3. The number with
frequency v in the interval v, v + dv, i.e. nvdv per unit volume, is given by
,.,
dj\l

3

dM ,
.
d
™4*±\IL>
— -—dv
=
n
dv

v
dv
dv |_8 3 \ c /
14 8 2
4TTV2
=

83

^

^

=

dv

^ ^ '

as claimed in Eq. (1.7). We obtain therefore
u^T)

= Unv = 2^-fJ^—i.

(1.8)

This is Planck's formula (1.1). We observe that u(v,T) has a maximum
which follows from du/dX = 0 (with c = vX). In terms of A we have
u(X,T)dX =


^ehc/*kT_idX,

so that the derivative of u implies (x as in Eq. (1.1))

The solutions of this equation are
^max = 4.965 and xmin = 0.
'''From the equation

I -\ JW - V 2 ) E = 0, so that

www.pdfgrip.com

- UJ2/C2 + K? = 0,UJ = 2-KV.