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BURLINGTON

Four Lectures on Wave Mechanics

BLACKIE & SON LIMITED
50 Old Bailey, London 17 Stanhope Street, Glasgow
BLACKIE & SON (INDIA) LIMITED
Warwick House, Fort Street, Bombay
BLACKIE & SON (CANADA) LIMITED
1 1 18 Bay Street, Toronto

Four Lectures on
Wave Mechanics
Delivered at the Royal Institution, London, on 5th, 7th, i2th, and 14th March, 1928

BY

Dr.

ERWIN SCHRODINGER
Professor of Theoretical Physics in the University of Berlin

BLACKIE & SON LIMITED
LONDON AND GLASGOW
1928

C33
Printed in Great Britain by Blackie Gf Son, Ltd., Glasgow

T)edicated
to the memory of Fritz Hasenohrl
m 13^

T

.

Contents

FIRST LECTURE

Page
1 Derivation of the fundamental idea of wave mechanics

from Hamilton's analogy between ordinary me-

chanics and geometrical optics -

-

-

-

i

2. Ordinary mechanics only an approximation, which no

longer holds for very small systems -

-

-6

3. Bohr's stationary energy-levels derived as the fre-

quencies of proper vibrations of the waves -

-9

SECOND LECTURE

4. Rough description of the wave-systems in the hydrogen

atom. Degeneracy. Perturbation -

-

- 14

5. The physical meaning of the wave function. Explana-
tion of the selection rules and of the rules for the

polarization of spectral lines

-

-

-

- 16

6. Derivation of the wave equation (properly speaking)

which contains the time -

-

-

-

- 22

7. An atom as perturbed by an alternating electric field - 23

THIRD LECTURE

8. Theory of secondary radiation and dispersion -

- 27

9. Theory of resonance radiation, and of changes of the state of the atom produced by incident radiation

whose frequency coincides, or nearly coincides,

with a natural emission frequency -

-

- 31

viii

CONTENTS

Page
10. Extension of wave mechanics to systems other than a

single mass-point -

-

-

-

-

- 35

11. Examples: the oscillator, the rotator -

-

-

- 38

FOURTH LECTURE

12. Correction for motion of the nucleus in the hydrogen

atom

43

13. Perturbation of an arbitrary system -

-

-

- 45

14. Interaction between two arbitrary systems

-

- 49

15. The physical meaning of the generalized c];-function - 51

Four Lectures on
Wave Mechanics

FIRST LECTURE

1. Derivation of the fundamental idea of wave mechanics from Hamilton's analogy between ordinary mechanics and geometrical optics.

When a mass-point in moves in a conservative field of

force, described by the potential energy V(x^y, z)^ then,

if you let it start from a given
A point with a given velocity,

i.e. with a given energy E^

you will be able to get it

into another arbitrarily chosen
B point by suitably " aiming ",

i.e. by letting it start in a

quite definitely chosen direc-

tion. There is in general one definite dynamical orbit

A B which leads from

to

zvith a given energy. This

orbit possesses the property that

hC2Tdt=^0,
JA

... (1)

and is defined by this property (Hamilton's principle in
the form given to it by Maupertuis). Here T means the

2

WAVE MECHANICS

kinetic energy of the mass-point, and the equation means:
A B consider the manifold of all orbits leading from to

and subject to the law of conservation of energy

= V {T -{-

E); among them the actual dynamical orbit

is distinguished by the fact that, for it and for all in-

finitely adjacent orbits of the manifold, the / has the

same value up to small quantities of the second order

(the words *' infinitely adjacent" being taken to define

w = the first order of smallness). Calling

ds -j the velocity

of the mass-point, we have

2T=mw'' = m (j\ ^ = 2{E-V) = ~ ^2m{E-V),

by means of which equation (1) can be transformed into

s}^^2m{E- V)ds = 0.

. . (2)

This form has the advantage that the variational principle is applied to a purely geometrical integral, which does not contain the time-variable, and further, that the condition of constant energy is automatically taken care of.
Hamilton found it useful to compare equation (2) with Fermafs principle, which tells us that in an optically
non-homogeneous medium the actual light rays, i.e. the
tracks along which energy is propagated, are determined
by the " law of minimum time " (as it is usually called). Let fig. 1 now refer to an optical medium of arbitrary
non-homogeneity, e.g. the earth's atmosphere; then, if you have a searchlight at A, furnishing a well-defined beam, it will in general be possible to illuminate an
B arbitrarily chosen point by suitably aiming at it with
the searchlight. There is one definite light-path leading

FIRST LECTURE

3

A from to B, which obeys, and is uniquely defined by,

the law

C^ ds

\ii='

(^)

Here ds, as before, means the element of the path, and

u is the velocity of light, a function of the co-ordinates

X, y, z.
The two laws contained in equations (2) and (3) respectively become identical, if we postulate that

=, u= ^ ,

... (4)

where C must be independent of x, y, z but may depend
on E. Thus we have made a mental picture of an optical

medium, in which the manifold of possible light-rays

coincides with the manifold of dynamical orbits of a
m £ mass-point moving with given energy in a field of
force V{x,y, z). The fact that 11, the velocity of light, de-
pends not only on the co-ordinates but also on £", the total

energy of the mass-point, is of the utmost importance.

This fact enables us to push the analogy a step farther
E by picturing the dependence on as dispersion, i.e. as a
dependence on frequency. For this purpose we must

attribute to our light-rays a definite frequency v, de-

We pending on E.

will (arbitrarily) put

E=hv

(5)

{h being Planck's constant), without dwelling much on this assumption, which is very suggestive to modern physicists. Then this non-homogeneous and dispersive medium provides in its rays a picture of all the dynamical
orbits of our particle. Now we can proceed a stage
farther, putting the question: can we make a small

4

WAVE MECHANICS

" point-like " light-signal move exactly like our masspoint? (Hitherto we have only secured the geometrical
identity of orbits, quite neglecting the question of time-
rate.) At first sight this seems impossible, since the
velocity of the mass-point,

a' = ^V2m(£-F), ... (6)

is (along the path, i.e. with constant E) inversely pro-
C portional to the light-velocity u (see equation (4); depends on E only). But we must remember that ii is of
course the ordinary phase-YtXocity , whereas a small light-signal moves with the so-called group-velocity, say g, which is given by

g dv \u/' or, in our case, following equation (5), by

'
g

dME i)

<'>

^ We will try to make g w. The only means we have

at our disposal for this purpose is a suitable choice of
E C, the arbitrary function of that appeared in equation
= (4). From (4), (6), and (7), the postulate g w becomes

_^ /EV2m{E- V) \

dE\

C

/

hence

^2m{E-V) djE^ {V2m{E-V));

(^-l) ^2m(E-V)

V is constant with respect to E. Since

contains the

E C co-ordinates and must be a function of onlv, this

—

FIRST LECTURE

5

relation can obviously be secured in a general way only by making the first factor vanish. Hence

^-1 = 0, or C=E,

which gives equation (4) the special form

^_"

E
^2m{E-Vy

(8)

This assumption about phase-velocity is the only one which will secure absolute coincidence between the dynamical laws of motion of the mass-point and the optical laws of motion of light-signals in our imagined light-propagation. It is worth while mentioning that,
according to (8),
^^_energy_ ^g,^ momentum

There is still one arbitrariness in the definition of ?/,
E viz.: may obviously be changed by an arbitrary additive
constant, if the same constant is added to V{x, jy, z). This arbitrariness cannot be overcome in the non-
relativistic treatment and we are not going to deal with
the relativistic one in the present lectures.
Now the fundamental idea of wave-mechanics is the
following. The phenomenon, of which we believed we
had given an adequate description in the old mechanics by describing the motion of a mass-point, i.e. by giving its co-ordinates x, y, z as functions of the time variable ty
— is to be described correctly according to the new ideas
by describing a definite wave-motion, which takes place among waves of the type considered, i.e. of the definite frequency and velocity (and hence of the definite wavelength) which we ascribed to what we called *' light " in

6

WAVE MECHANICS

the preceding. The mathematical description of a wave-

motion will be furnished not by a limited number of

functions of the one variable t, but by a continuous

manifold, so to speak, of such functions, viz. by a func-

tion (or possibly by several functions) of Xy y, z, and t.

These functions will be subject to a partial differential

equation, viz. to some sort of wave equation.

The statement that what really happens is correctly

described by describing a wave-motion does not neces-

sarily mean exactly the same thing as: what really exists

We is the wave-motion.

shall see later on that in general-

izing to an arbitrary mechanical system we are led to

describe what really happens in such a system by a

wave-motion in the generalized space of its co-ordinates

(^-space). Though the latter has quite a definite physical
meaning, it cannot very well be said to " exist "; hence "
a wave-motion in this space cannot be said to *' exist

in the ordinary sense of the word either. It is merely

an adequate mathematical description of what happens.

It may be that also in the case of one single mass-point, with which we are now dealing, the wave-motion must
not be taken to " exist " m.too literal a sense, although

the configuration space happens to coincide with ordinary

space in this particularly simple case.

2. Ordinary mechanics only an approximation, which no longer holds for very small systems.

In replacing the ordinary mechanical description by a wave-mechanical description our object is to obtain a theory which comprises both ordinary mechanical phenomena, in which quantum conditions play no appreciable part, and, on the other hand, typical quantum phenomena.
The hope of reaching this object resides in the following

FIRST LECTURE

7

m analogy. Hamilton's wave-picture, worked out the
way discussed above, contains something that corresponds
to ordinary mechanics, viz. the rays correspond to the
mechanical paths, and signals move like mass-points. But the description of a wave-motion in terms of rays is
merely an approximation (called " geometrical optics " in the case of light-waves). It only holds if the structure of
the wave phenomenon that we happen to be dealing
with is coarse compared with the wave-length, and as
long as we are only interested in its " coarse structure ". The detailed fine structure of a wave phenomenon can
never be revealed by a treatment in terms of rays (" geometrical optics "), and there always exist wave-phenomena which are altogether so minute that the ray-method is. of no use and furnishes no information whatever. Hence
in replacing ordinary mechanics by wave mechanics we may hope on the one hand to retain ordinary mechanics
as an approximation which is valid for the coarse " macro-mechanical " phenomena, and on the other hand to get an explanation of those minute " micromechanical " phenomena (motion of the electrons in the atom), about which ordinary mechanics was quite unable to give any information. At least it was unable to do so without making very artificial accessory assump-
tions, which really formed a much more important part
of the theory than the mechanical treatment itself.*

* To give an example: the actual application of the rules for
quantization to the several-electron problem was, strange to say,
not hindered by the fact that nobody in the world ever knew how
We to enunciate them for a non-conditionally periodic system!
simply took the problem of several bodies to be conditionally periodic, though it was perfectly well known that it was not. This shows, I think, that ordinary mechanics was not made use of in a very serious manner, otherwise the said application would have been as impossible as the application of penal law to the motion of the planets.

)

8

WAVE MECHANICS

The step which leads from ordinary mechanics to
wave mechanics is an advance similar in kind to Huygens'
We theory of light, which replaced Newton's theory.
might form the symbolic proportion:
Ordinary mechanics : Wave mechanics
= Geometrical optics : Undulatory optics.

Typical quantum phenomena are analogous to typical

wave phenomena like diffraction and interference.

For the conception of this analogy it is of considerable

importance that the failure of ordinary mechanics does

We occur in dealing with very tiny systems.

can im-

mediately control the order of magnitude at which a

complete failure is to be expected, and we shall find

that it is exactly the right one. The wave-length, say A,

of our waves is (see equations (5) and (8)

A = = = A

^^

V

^

^2m{E— V)

mw' (9)^

'

'

^

i.e. Planck's constant divided by the momentum of the

Now mass-point.

take, for the sake of simplicity, a cir-

cular orbit of the hydrogen-model, of radius «, but not
necessarily a " quantized " one. Then we have by

ordinary mechanics (without applying quantum rules):

— mwa = n h ,

where n is any real positive number (which for Bohr's

quantized circles w^ould be 1, 2, 3 . . . ; the occurrence
of h in the latter equation is for the moment only a con-

venient way of expressing the order of magnitude).

Combining the last two equations, we get

A_ 277

an

( D 929 )

FIRST LECTURE

9

Now in order that we may be justified in the appli-

cation of ordinary mechanics it is necessary that the

dimensions of the path calculated in this way should

turn out to be large compared with the wave-length.

This is seen to be the case as long as the '' quantum number " w is large compared with unity. As n becomes

smaller and smaller, the ratio of A to « becomes less and

A less favourable.

complete failure of ordinary me-

chanics is to be expected precisely in the region w^here

we actually meet with it, viz. where n is of the order of

unity, as it would be for orbits of the normal size

of an atom (10~^ cm.).

3. Bohr's stationary energy-levels derived as the frequencies of proper vibrations of the waves.

Let us now consider the wave-mechanical treatment of

a case which is inaccessible to ordinary mechanics; say, to

fix our ideas, the wave-mechanical treatment of what in

ordinary mechanics is called the motion of the electron

in the hydrogen atom.

In what way are we to attack this problem?

Well, in very much the same way as we would attack

the problem of finding the possible movements (vibra-

tions) of an elastic body. Only, in the latter case

the problem is complicated by the existence of two types
of waves, longitudinal and transverse. To avoid this

complication, let us consider an elastic fluid contained

in a given enclosure. For the pressure, p, say, we should have a wave equation

V^/'-^J =

(10)

u being the constant velocity of propagation of longi-

tudinal waves, the only waves possible in the case of a

(D929)

2

—,

10

WAVE MECHANICS

We fluid.

should have to try to find the most general

solution of this partial differential equation that satisfies

certain boundary conditions at the surface of the vessel.

The standard way of solving is to try

which gives for ip the equation

^V v^^ +

= o,

. . . (10')

i/j being subject to the same boundary conditions as p.
We then meet with the well-known fact that a regular

solution ip satisfying the equation and the boundary

conditions cannot be obtained for all values of the co-

efficient of ip, i.e. for all frequencies v, but only for an

infinite

set

of

discrete frequencies

v^,

Vo,

Vg,

. . . , j^/^,

.. .

which are called the characteristic or proper frequencies

(Eigenfrequenzen) of the problem or of the body. Call

— ^k the solution (ordinarily unique apart from a multi-
plying constant) that belongs to Vk, then since the equation and the boundary conditions are homogeneous

k
will, with arbitrary constants Ck, Ok, be a more general solution and will indeed be the general solution, if the
set of quantities {ipky i^k) is complete. (For physical appli-
cations we shall of course have to use the real part of the
expression (H).)
In the case of the waves which are to replace in our thought the motion of the electron, there must also be some quantity p, subject to a wave equation like equation
(10), though we cannot yet tell the physical meaning of p. Let us put this question aside for the moment.

FIRST LECTURE

ii

In equation (10) we shall have to put (see above)

u= ^ ,

(8)

This is not a constant; it depends (1) on E, that is, essen-

tially on the frequency v {= E/h); (2) on the co-ordinates

X, y^ z, which are contained in the potential energy V.

These are the two complications as compared with the

simple case of a vibrating fluid body considered above.

Neither of them is serious. By the first, the dependence

on £", we are restricted in that we can apply the wave

equation only to a function p whose dependence on the

time is given by

\rnEt.
p -- e h

whence

= p

p— 4:7T^E^
p

(12)

We need not mind that, since it is precisely the same
assumption (Ansatz) as would be made in any case in the standard method of solution. Substituting from (12) and (8) in (10) and replacing the p letter by i/j (to remind us that now, just as before, we are investigating a function of the co-ordinates only), we obtain
VV-f^^2'"(^-n'A = 0. . . (13)

We now see that the second complication (the depen-
dence of u on Vy i.e. on the co-ordinates) merely results in a somewhat more interesting form of equation (13) as compared with (10'), the quantity multiplying ip being no longer a constant, but depending on the co-ordinates. This was really to be expected, since an equation that is to embody the mechanical problem cannot very well help

12

WAVE MECHANICS

A containing the potential energy of the problem.

sim-

m plification

the problem of the " mechanical " waves

(as compared with the fluid problem) consists in the

absence of boundary conditions.
1 thought the latter simplification fatal when I first

attacked these questions. Being insufficiently versed in
mathematics, 1 could not imagine how proper vibration

frequencies could appear without boundary conditions.

Later on I recognized that the more complicated form

of the coefficients (i.e. the appearance of F(.t, y^ z) ) takes charge, so to speak, of what is ordinarily brought about

by boundary conditions, namely, the selection of definite

values of E.

I cannot enter into this rather lengthy mathematical

discussion here, nor into the detailed process of finding the solutions, though the method is practically the same

as in ordinary vibration problems, namely: introducing

an appropriate set of co-ordinates (e.g. spherical or

elliptical, according to the form of the function V) and

putting j/f equal to a product of functions, each of which

contains one co-ordinate only. I will state the result

straightforwardly for the case of the hydrogen atom.

Here we have to put

F=-- + const.,

. . . (14)

r being the distance from the nucleus. Then it is found
that not for all, but only for the following values of E, is it possible to find regular, one-valued, and finite solu-
tions ip:

= - -^^^ = (A) En const.

w
;

1, 2, 3, 4 . . . ]

> E (B)

const.

J

^^^,^

FIRST LECTURE

ij

The constant is the same as in (14) and is (in non-relativistic wave mechanics) meaningless, except that we
cannot very well give it the value which is usually adopted for the sake of simplicity, viz. zero. For then all the
values (A) would become negative. And a negative
frequency, if it means anything at all, means the same as the positive frequency of the same absolute value. Then
it would be mysterious why all positive frequencies should
be allowed, but only a discrete set of negative ones. But the question of this constant is of no importance here.
You see that our differential equation automatically
selects as the allowed £'-values (A) the energy-levels of the elliptic orbits quantized according to Bohr's theory; (B) all energy-levels belonging to hyperbolic orbits. This is very remarkable. It shows that, whatever the waves
may mean physically, the theory furnishes a method ot
quantization which is absolutely free from arbitrary postulates that this or that quantity must be an integer. Just to give an idea how the integers occur here: if e.g. (/) is an azimuthal angle and the wave amplitude turns
m out to contain a factor cos m</), being an arbitrary conm stant, then must necessarily be chosen integral, since
otherwise the wave function would not be single-valued.
You will be interested in the form of the wave
functions ijs which belong to the £'-values mentioned above, and will inquire whether any observable facts can be explained by them. This is the case, but the matter
is rather intricate.

SECOND LECTURE
4. Rough description of the wave -systems in
the hydrogen atom. Degeneracy. Perturbation.
The chief property of the amphtude functions is that
those which belong to the discrete set of values En (" elliptic orbits ") fall off very rapidly with the distance from the nucleus, viz. like an exponential g- const, r^ which practically restricts them to a region of precisely the same order of magnitude as the corresponding Bohr
orbit. The others, which belong to hyperbolic levels, fall off much less rapidly, viz. only like r~^.
The detailed behaviour of the " elliptic " functions
within the said region cannot very well be described in
a unique way, for the following reason. To one value
En there belongs in general not only one, but precisely
n^ independent solutions of the wave equation. From
the mathematical point of view this is an exception due to the particular form of the potential energy F, especially to its spherical symmetry. This multiplicity of solutions belonging to one proper value corresponds to the well-
known multiplicity of orbits belonging to the same
energy-level in Bohr's theory. It is there called '' de-
generacy ", and we will keep this expression in wave mechanics also. Now, since the equation is linear and
homogeneous, any linear aggregate with quite arbitrary coefficients will also be a solution belonging to the same

SECOND LECTURE

15

proper value. It is well known that In such a case no set of solutions is in any way distinguished from any other set, derived from the f^rst by forming a set of independent linear aggregates, equal in number to the first set. By this process of forming linear aggregates we can reach
solutions which exhibit a very different behaviour. To
give an example: from a set of solutions whose node-
surfaces are (1) concentric spheres, (2) co-axial cones,
(3) planes passing through the cone-axis, you can form other solutions, in which the concentric spheres and the co-axial cones are replaced by two sets of confocal paraboloids. This is only one of the simplest cases. In general, taking arbitrary coefficients, the system of node-surfaces will be much more complicated.
This multiplicity of solutions, or, as is often said, of
the proper values (which, by the way, is well known from ordinary vibration problems), is of the utmost importance in the case of the atom. If there is no multi-
= plicity (e.g. for the lowest frequency, n i), then a
slight alteration of the potential energy F, corresponding
e.g. to the application of a weak external electric field, will cause nothing but a slight displacement of the proper
— value and a slight alteration of the proper solution just
as a small piece of metal attached to a tuning fork would slightly alter its pitch and its form of vibration. But a multiple (say a-fold) proper value shows its actual multiplicity in this case in that it splits up into a slightly different proper values; every one of them has now a quite definite proper function, which differs very little from a quite definite linear aggregate of the proper functions that belonged to the multiple value. This splitting up may, theoretically, be caused by the very slightest disturbance, yet may differ widely for two

i6

WAVE MECHANICS

disturbances that are different in character. For instance,

a homogeneous electric field produces the parabolic

node-surfaces mentioned before, whereas a magnetic

field produces the spheres and cones.

It need hardly be said that this splitting up corresponds

in the two cases just mentioned to the splitting up of the

hydrogen lines in the Zeeman and Stark effects. The

displacement of the lines is quantitatively described by the new theory just as it was by the older one. But some-

thing more is described, which was inaccessible to the

older theory, namely, the state of polarization of the

lines, their intensities, and, in particular, the absence of
a lot of lines which we should expect to appear if we

took into account all the possible differences of the split

We energy levels.

shall see this presently.

5. The physical meaning of the wave function. Explanation of the selection rules and of the rules
for the polarization of spectral lines.
The high importance of the perturbation effects con-
sists in the fact that as soon as the degeneracy is removed
we have to deal with uniquely defined proper functions ijjk and can now more easily test any hypothesis about the
physical meaning of the quantity called ijj. Let us call
= Ek hvk and iPkix.y.z)

the proper values, proper frequencies, and proper func-
V tions of a problem, whose potential energy we suppose

sufficiently unsymmetrical to do away with all degeneracy.

Then

= ilj Zckilfke^'''^"^'+^^K
k

. . . (15)

with arbitrary constants Ck, 9k, will describe the most

SECOND LECTURE

17

general '* vibration " of the system/" In order to avoid ambiguity, since every ifj^ in itself is only defined apart
from an arbitrary multiplying constant, we shall subject
the j/f/e's to the normalizing condition

/// xPk^dxdydz=l,

. . (16)

Perhaps this is the place to mention a very important property which the ipk^ possess automatically, viz. they are '* orthogonal " to each other:

/// xjjkipidxdydz^.^ for k^l, . (17)
and they form a complete orthogonal set; a function which is orthogonal to them all must necessarily vanish. (These properties are important for the development of an arbi-
trary function in a series in terms of the ?/fy^'s, but we shall not enter upon that here, as we do not need it for
the moment.)
Now return to the general vibration function (15). We put the question: is it possible to ascribe a definite
physical meaning to the quantity in such a way that the
emission of light with frequencies

— becomes intelligible? Yes, it is, but strange to say-
only if we make use of the complex j/f-function as it stands,
instead of its real part, as we are accustomed to do in
ordinary vibration problems.
* Here we have not taken into account the " continuous spec-
trum ", corresponding to the hyperbohc orbits. We may either
suppose these modes of vibration to be absent or we may take the Z to include, as a hmiting case, the integral, which would have to
«
be added in order to take proper account of the continuous region of proper values. At all events I wished to avoid encumbering the formulae more than necessary.

i8

WAVE MECHANICS

The hypothesis which we have to admit is very simple,
namely that the square of the absolute value of i/j is proportional to an electric density, which causes emission of light according to the laws of ordinary electrodynamics. Since the square of the absolute value of ip is formed by multiplying ip by the conjugate complex quantity (which
we w^ill call ip), a glance at the expression (15) shows that
the terms which compose ip^p contain the time in the
— form of cosine factors of the desired frequencies Vk Vk'.
More precisely, let us put, for the charge- density />,
p= -eip'^= -eI.ZckCk'ipk^k'e-'''^^'^-'^'^'+'^-'^^'\ (18)
k k'
where e means the absolute electronic charge. Integrating this over the whole space and making use of equations (16) and (17), we find for the total charge

k
which shows that we shall have to postulate

k
in order to make the total charge equal to the electronic charge (which we feel inclined to do).
It was said before that and ?/f, hence p, is practically confined to a very small region of a few Angstrom units.
— Since the wave-lengths of the light-radiations v^ Vk'
are very large compared with this region, it is well known
that the radiation of the fluctuating density p will be very nearly the same as that of an electric dipole whose
electric moment has (e.g.) the ^-component
= M^ \ \ \ zpdxdy dz
(and similarly formed x- and jy- components). Calcu-

SECOND LECTURE

19

M^ lating

from (18), we find after an easy reduction

k
- -2i:ck Ck' auk' cos [2 77 {v^ v^) t-\-dk- 0,] .
{k,k')

(19)

Here akk' is an abbreviation for the following constant:
= akk' e^\jz^kh'dxdydz, . . (20)

and S means a sum over all the pairs {k, k'). Hence the
{k,k')
squares of these integrals (and the corresponding in-

tegrals relating to the x- and ^-directions) determine the

— intensity of emitted light of frequency v;^ The Vy^' .

|

|

intensity is not determined by them alone; the amplitude-

constants Ck also come into play, of course. But this is

quite satisfactory. For the integrals akk' are determined

by the nature of the system, i.e. by its proper functions,

regardless of its state, akk' is the amplitude of the cor-

responding dipole, which would be produced by the

proper vibrations ipk^ ^k', if only these two were excited,

= and with equal strength lck= Ck'

•
"v^)

The first sum in (19) is of no interest in our investi-
gation of the emitted radiation, since it means a component of electric moment that is constant in time.

The correctness of our 0i/f-hypothesis has been
checked by calculating the akk''^ in those cases where the i/f/e's are sufficiently well defined, namely in the case of
the Zeeman and Stark eff"ects. The so-called rules of
selection and polarization and the intensity-distribution in these patterns are described by the ^aa's in the following obvious way, and the description is in complete agreement with experiment:

20

WAVE MECHANICS

The absence of a line which might be expected to
occur (*' selection-rule ") is described by the vanishing of the corresponding akk'y cind of the two other constants relating to the x- and j^-directions.
The linear polarization of a line in a definite direction
is described by the fact that only the constant akk' relating to this direction differs from zero, whereas the two other constants vanish. In a similar way the circular polarization, say in the x^-plane, is indicated by (1) vanishing of the ^-constant, (2) equality of the x- and ^/-constants, and (3) a phase-difference of 77/2 between the corresponding cosine-functions in equation (19).
Finally, the intensity relations between the non-vanish-
ing components in the Stark and Zeeman patterns of hydrogen are correctly indicated by the relations between the squares of the ^^/^-'s in question; which is satisfactory,
since the assumption that the CkS will be equal for the fine-structure components of one level is very suggestive, notwithstanding our lack of knowledge of the c^s in other respects.
Of course it is impossible to set forth in this lecture
any of the calculations that led to the results just given; they would fill pages and pages, and are not at all difficult, but very tedious. In spite of their tediousness, it is rather fascinating to see all the well-known but not understood '' rules " come out one after the other as the result of very familiar elementary and absolutely cogent analysis,

like e.g. the fact that / cos mcj) cosn^dcf) vanishes unless
—_
n m. Once the hypothesis about i/jip has been made, no accessory hypothesis is needed or is possible; none could help us if the " rules " did not come out correctly.
But fortunately they do.

SECOND LECTURE

21

I think I ought to draw attention to another fact which was only briefly mentioned at the beginning, namely, that the very fundamental " frequency-rule " of Bohr,

= — vkk' ""^k— n'

{Ek Ek')y

1^

may also be said to be explained by the «/fi/f-hypothesis.
Something exists in the atom which actually vibrates with the observed frequency, viz. a certain part of the electric density-distribution or, if you prefer,
of l/jl/j. This might lead us to suspect that only the square
of its absolute value, and not the ?/f-function itself, has a
real meaning. And this suspicion again might arouse the
desire to replace the w^ave equation by an equation
which describes the behaviour of j/f«/f directly. To remove
this desire, I will remind you of a case in which a similar desire might occur for exactly similar reasons; yet all of you will confess that it would be fatal to pursue
it.
Maxwell's equations describe the behaviour of the electromagnetic vectors. But these are not really accessible
to observation. The only things that are observable are
the ponderomotive forces, or, if you please, the energy, since the forces are caused by virtual energy- differences. But all these quantities (energy, Maxwellian-stresses) are
quadratic functions of the field-vectors. Therefore we
might desire to replace Maxwell's equations by others, that determine the observable quadratic functions of the field-vectors directly. But everyone w'ill agree that this would at all events mean an immense complication and that it would not really be possible to do without Max-
well's equations.

22

WAVE MECHANICS

6. Derivation of the wave equation (properly speaking) which contains the time.
The equation
V'^ + ~{E-V)i. = 0, . (13)

which we have used for the investigation of the hydrogen

atom, only furnishes the distribution in space of the

ampHtude of the vibration, the dependence on time

always being given bv

2ntEt

- i/j

e~^

(21)

The value of the frequency, E, is present in the equation, so that we are really dealing with a family of equations, each of the members being valid for one particular frequency only. The state of things is exactly the same
as in ordinary vibration problems; our equation corresponds to the so-called " amplitude equation " (see section 3, equation (10')),

+ ~4'-o, ^''l'

. . . (10')

and not to

.... V^-p-^p = 0,

(10)

from which the former is derived in the manner described

above (namely by supposing /) to be a sine-funciion of

the time). In our case the problem is to make the analo-

gous step in the reverse direction, i.e. to remiove the
parameter E from the amplitude equation and introduce

time- derivatives instead. This is easily done. Take one

of the family (13) (with a particular value of E), then by

(21) we have

277/^'
;

h)

SECOND LECTURE
Using this, we get from (13)
V20 ^_ ,/,___^ = 0.

23
. (22)

The same equation is reached whatever the value of E may have been (for E has been ehminated). Hence
equation (22) will be valid for an arbitrary linear aggregate of proper vibrations, i.e. for the most general wavemotion that is a solution of the problem.
We may tentatively go a step farther and try to use it
also in the case where the potential energy V contains the
time-variable explicitly. It is by no means obvious that

this is a correct generalization, for terms with F, &c.,
— might be missing they could not possibly enter into

equation (22), in view of the way we have reached this

equation. But success will justify our procedure. Of

course it would have been nonsense to introduce the

V assumption that

contained the time explicitly in

equation (13), since the condition (21), by which this

equation is restricted, would make it impossible to

satisfy (13) in the case of an arbitrarily varying F- function.

7. An atom as perturbed by an alternating
electric field.

This generalization enables us to solve the important
problem: how does an atom behave under the influence
of an external alternating electric field, i.e. under the influence of an incident wave of light? This is a very important question: for it contains not only the mechanism of secondary radiation and, in particular, of resonanceradiation, but also the theory of the changes of state of the atom under the influence of an incident radiation of appropriate frequency, and in addition the theory of

24

WAVE MFXHANICS

refraction and dispersion; for it is well known that
— dispersion I mean the phenomenon of a refractive — index is caused by the superposition on the primary

radiation of all the secondary wavelets, which every

single atom of the body emits under the action of, and

in phase with, the primary radiation. If an incident
E electric vector causes every atom to emit a secondary

wavelet such as would be emitted by a dipole of the electric

moment

M = aE

(23)

Z (a being a constant), and if atoms are present in unit

volume, then they produce an increase in the refractive

index of

27rZa

(24)

Hence, studying the value of a (which usually depends

on the frequency) means studying the phenomena of

refraction and dispersion.

To investigate the behaviour of an atom in an external

V alternating electric field, let us take

in equation (22)

to be composed of two parts, one describing the internal

electrostatic field of the atom, Fq, and one describing the light-field, Aez cos ZTrvt; A, v mean the amplitude
and the frequency of the light-field, which we suppose

polarized in the direction of z. (The negative sign of the

electronic charge has been taken account of; our ^ is a

positive number.) Hence equation (22) becomes

_ - + = V2^ 1^-''^ ^^'"^ (Fo Aez cos27Tvt)ilj 0. (25)

We A shall take to be very small compared with the
internal field (described by Vq) and solve the equation
A by approximation. If were zero, by assuming (21) we

SECOND LECTURE

25

should get back to equation (13) (only with the notation

We Vq instead of V).

shall assume the problem of the

unperturbed atom to be completely solved, its normal-

ized proper functions and proper values being

ijjk and Ek{=hvk).

= Hence the most general solution of (25), when ^4 0, is

= .... ilj i:c,rPke'^^^^\

(26)

k

the CkS being arbitrary complex constants.

We A shall try to satisfy equation (25), with

also

present, by (26), but with the Ck's varying slightly with

time (method of variation of constants). Taking this into

account, and also the facts that ipk, hvk are proper func-
tions and proper values of the unperturbed equation, we

easily obtain, by substituting (26) in (25):

= ^-^^^ S ^'Aa^'""^'

^^^ 27rvtZckiljke-''''^\ (27)

k

'*'

k

This equation will be satisfied if (identically with respect to time) all the coefficients of the expansion of its left-hand side with respect to the complete system of orthogonal functions, ijjk^ are identical with the respective coefficients of the expansion of its right-hand side. Hence multiply by ijji and integrate over the whole space. Put for abbreviation (see section 5):

= aki e i \ i ifjk^izdx dy dz.

, (20)

Then, owing to the normalization and orthogonality of

the 0;i's, we get

^ ^ 2 e2-'V,f
ci

277Z
n

^^^ 2 ^^t aki Ck ^-"'•''t'
k

(D929)

(/= 1,2^3,4,...). . . . (28)
3

26

WAVE MECHANICS

This infinite set of ordinary differential equations is
equivalent to (27). Isolating q and splitting up the cosine into exponentials, we write it as follows:

= + !^ S r;

akiCk [^2-^-(.,-.',+ .)^

_|_

^2 .f (.,-.,-.).]
^

(28')

Hitherto we have not made use of any approximation

We process.

will now do this in two different ways, one

leading to the theory of secondary radiation (excluding

the case of resonance) and of dispersion, the other furnish-

ing the case of resonance and the changes of state of the

atom.

THIRD LECTURE
8. Theory of secondary radiation and dis-
persion.
In equation (28') we shall at first assume that all the
aggregates

which appear in the exponents are large as compared with the order of magnitude of

h'

This means that the difference between the incident

frequency and any one of the frequencies of spontaneous

emission is large compared with the frequency that

would correspond to the potential energy which the atom

acquires in the external field. (Exclusion of exact or

near resonance.) With this assumption the equations (28')

show that all the time-derivatives of the q's are small

compared to the time-derivatives of the exponentials.

After having stated this, let us take any one of the ex-

ponentials on the right-hand side of any one of the

We equations (28').

may assume its coefficient Ck to be

constant during a period of the exponential. Hence this

term will only cause a small periodic oscillation of the

Ci (on the left), which is restored (or nearly so) after
27

28

WAVE MECHANICS

the exponential has gone through a period. But the

same holds for all the exponentials. Hence all the c's

execute a vast number of small oscillations around their

mean values, oscillations which would of course vanish

We with vanishing A.

may therefore replace the t's on

the no'/?^-hand side of equation (28') by constants, viz.

by their mean values, since by neglecting the small

We oscillations here only terms in A^ are dropped.

shall

write CfP for the said constants. The equations are now

We easily integrated.

get

Hence the /th term in our solution (26) will be:

— ^-^i:a,,cA^

+ -"

. (29)

Though we have not yet reached a point that can be compared with experiment, we will give in words the
description of what happens, according to equation (29), under the influence of an incident light-wave. Every proper vibration j/f/, whether it is itself excited from the
beginning or not, is compelled to execute a multitude of small additional forced oscillations, namely two '' in honour " of every proper vibration ipk that is excited
appreciably {ck^ 4= 0). The frequencies of the two forced oscillations that 0/ executes *' in honour " oi ipk, as we
i said, are Vk v, i.e. the sum and diff"erence of the incident
frequency and the frequency of the " honoured " proper
vibration. Their amplitudes are proportional to the amplitudes both of the external field and of the '* honoured "
vibration; they also contain as a factor a^h the constant

THIRD LECTURE

29

which governs the intensity of the spontaneous emission

— of frequency v^ \

yi\. Further, in the two forced

ampHtudes two " resonance-denominators " appear, caus-

ing one of the two ampHtudes to increase rapidly when

the incident frequency approaches the frequency of

spontaneous radiation \ Vk— '^^i]-
Before forming the complete solution from (26) and
(29) we will restrict ourselves to the most important case,
viz. that in which only one free vibration is excited, say ipk-

Ck^=l cP=0 for l=^k.

We may think of i/jk as corresponding to the normal state.

Then on the right-hand side of equation (29) the first
= term (except for I k) and the summation sign are

dropped, and we get for the complete solution (equation

(26), in which k is to be replaced by /):

= + ^ ^,.^-".'

E «,;0, -^

— + -^

. (30)

(Note that now the exponentials are independent of the index of summation, /; only two frequencies of forced
vibration are present.)
To get information about the secondary radiation we form the component * M^ of the resultant electric moment
from (30). Neglecting small terms of the second order
(proportional to ^^), we find after reduction:

= — — M^

e I j iljiljzdxdydz= dkk

h

(31)
I {vi—Vkf—v^

* In general, for an anisotropic atom, there will be an My and

M^ an

(orthogonal to the polarization of the incident radiation)

We as well.

will not deal with them here.

30

WAVE MECHANICS

The first term (—akk) is independent of the time; it is the constant electric moment due to the excitation of the

free vibration i/j^. It is of no interest here. The second

term determines the secondary wavelet. It is seen to

coincide in frequency with the incident electric force

(A cos27Tvt). Its phase is the same or opposite, depend-
— ing on whether v '^ vi Vky just as in the classical

theory. (This holds if ijjk corresponds to the normal state,
— so that vi Vk is always positive; if it is negative, the

reverse is true; Kramers' terms of the dispersion

formula.) The quantity a of equation (23), which by the

expression (24) determines the contribution to the re-

fractive index, is found from the second term on the
A right-hand side of (31) by dropping cos27Tvt. The
— — denominators (v/ VkY ^^ furnish the phenomenon of

anomalous dispersion in the neighbourhood of all those

emission (or absorption) frequencies that involve the
— index k of ipk remember that we supposed only this

one free vibration to be excited. The quantity aki^ in the

numerator is the same as that which determines the

— intensity of spontaneous emission i^^ i^z . In all these

|

|

respects the formula is a complete copy of the old Helmholtz formula (supplemented by Kramers' " negative "

terms) and is thought to be in complete agreement with

experiment.
Two additional points are worth mentioning. You know that Thomas and Kuhn formed a hypothesis con-

cerning the sum of all the coefficients in the dispersion

formula, in our case

2

According to them it is to be equal to the value of the

THIRD LECTURE

31

coefficient for one elastically bound electron, i.e. it must be equal to

4:7T^m
(multiplied by one, in our case, for we are dealing with
the o;z£'-electron atom; in general, multiplied by an
integer). The equality of the two above-mentioned
— quantities can be proved for our dispersion formula but
the proof is a little lengthy, and I will therefore omit it.
The second remark is the following. Perhaps you remem.ber the statement, first made by Smekal, that
there should also exist secondary radiations, whose frequencies differ from the frequency v of the incident radiation (therefore without phase relation, therefore
without influence on the refraction phenomenon). The
frequencies expected are

Secondary radiations of precisely these frequencies are
furnished by the present theory, if we give up our sim-
plifying assumption that only one free vibration is excited, and suppose at least two of them, say ipk and to i/r/j-, be present.
9. Theory of resonance radiation, and of changes of the state of the atom produced by incident radiation whose frequency coincides, or nearly coincides, with a natural emission frequency.
At the beginning of the last section we had to make
the assumption that all the aggregates like

are of appreciable size, which means that the frequency

32

WAVE MECHANICS

of the incident light, v, is excluded from the immediate

neighbourhood of any natural frequency of the atom

We under consideration.

will now consider an incident

frequency which is very close to one of the natural

frequencies. To fix our ideas, let

— + > ^yfe

J^z

i^ be very small and vi

Vk

(" very small " means: of the order of magnitude of Aukijh or smaller, possibly vanishing). Returning to
equation (28'), you will now find on the right-hand side
of this system of equations altogether two exponentials which vary slowly, viz.

the former appearing in the /th equation, the latter in the

^th equation. These terms (as we shall see presently) now
cause very appreciable " secular " changes in the two
A quantities c^ and q, however small the amplitude of
the incident wave may be. All the other exponentials will

only cause small periodic disturbances, as before. It is

therefore reasonable to drop them altogether, since we

are now dealing with a much coarser phenomenon (viz.

We appreciable secular variations of c^ and q).

might

even suppose all the other c's to be zero; this would have

no effect, since they are certainly constant within the

degree of accuracy we are aiming at. For determining

Ck and Ci we get from (28') the two simple equations

Ck=i(JCie ''\
with the abbreviations
T^Aaki
,

(32)
/.JON

THIRD LECTURE

33

To solve them, we introduce new variables x, y by
putting

ci=xe-,

= Ck ye '^,

. . (34)

The result can be written

i€\ lay.

(d _u-\\y = i(jx.
\dt 2/
These equations have constant coefficients and are readily
solved by familiar methods. The solution can be written
in the following form:

(35)

with the abbreviations

Y+

= vJ + ''^ f^=^'

(^6)

whereas p, p\ (/>, ^' are arbitrary real constants, non-

We negative if you like.

can put (35) in the form:

x=e 2
= y e 2

[{p-irp.p')co^e-\-i{p-iip')s>me]A
— + + [(/xp p') cDS^ z(/xp p') sm^],J

with the abbreviation
e=y,+i+i:.

. . . (38)

From (37) we can easily form the squares of the absolute
values of x and y, that is (by equation (34) ) of ci and Ck,
and we can thus get information about the varying dis-

34

WAVE MECHANICS

tribution of intensity between the two vibrations in

— We question which is the point of main interest.

obtain

= h = - + U/ 1'

I' (P H^Py ^f^Pp' cos2 9,

Ck\'=\y\'={^p-py-^4fjipp'sm^e. ' ^^^^

The sum of the intensities is constant^ as might have been
anticipated. It may be taken to consist of three parts,
two " portions " fixed invariably to the two vibration-
levels, the third (viz. 4/x/)/)') oscillating slowly between
them. To ^-k. our ideas, let us take the case where at
a certain time all the intensity was stored up in one
vibration, say the lower one, Ck. Choosing the corre-
= sponding value of t so as to make cos ^ 0, this requires

= ^.
P'

We then find for the ratio between the oscillating portion
of the intensity and its total amount

(by using the fact, obvious from (36), that

We = see that when €

the total intensity is oscillating.

= By (33), e

means the case of sharp resonance. If

the resonance is not complete, then (40) shows that only

a certain fraction of the intensity oscillates, and that this

fraction becomes inappreciable when the lack of re-

sonance, e, becomes large compared with the quantity

THIRD LECTURE

35

a defined by (33). (The order of magnitude of o- is the

potential energy (divided by h) which the atom acquires

in the electric field of the light- wave, owing to the electric

moment which is due to the co-operation of the Ath and

/th modes of vibration.) The quantity o- would, in a certain

sense, give a measure of the natural sharpness of the

resonance-line, if it were possible to form a universal

A We idea of the amplitude of the incident light.

shall

not enter upon this question here.

The theory put forward here in its rough features de-

scribes both the change of state of the atom produced by

radiation of appropriate frequency and the appearance

of resonance-radiation. For of course the presence of the

two vibrations ijjk and ipi will give rise to their natural

emission. It is worth while mentioning that on account

of the exponentials appearing in equation (34) this
— emission should not have exactly the frequency vi Vky

but a frequency exactly equal to v, the frequency of the

incident light- wave.

10. Extension of wave mechanics to systems other than a single mass -point.

Hitherto we have applied the method of wave me-

chanics only to a very simple system, viz. a single mass-

point moving in a field of force which was either constant

We or varying with the time.

will now proceed to a quite

We arbitrary mechanical system.

might have done this

before; all that has been said about the influence of an

alternating field would apply with very slight modifi-

cation to an arbitrary system, e.g. to the many-electron

atom. But I thought it better to have a clear and simple

case before our mental eye.

The derivation of the fundamental wave equation

36

WAVE MECHANICS

put forward in the first lecture is very easily generalized

to a quite arbitrary system, the only difference being that the *' space " in which the wave-propagation takes place

is no longer ordinary three-dimensional space but the
** configuration space ".

Let us recall the Hamilton-Maupertuis principle

from which we started, namely,

SJ'2Tdt =

.... (1)

and which we transformed into
Sr^2m{E-V)ds = 0, ... (2)
JA
by putting
2r- mw' = m (jY = 2{E -V) = j^ V2mXE^T).

We then compared it with Fermat's principle for a wave-

propagation:

which led us to

aff^o, ..... (3) J^ u

^ V2m{E-V)

... (4)

Now, in general T is not of the simple form ^ (^) but

2T=I.^b„q,q„
Ik

. . . (41)

where the hi^s, are functions of the generalized co-

We ordinates qi.

now define a line-element ds in the

generalized ^-space by

or

= ds'- i:i.bikdqidqk.

. . . (42)

THIRD LECTURE

37

The generalized non-Euclidean geometry, which is de-

fined by the latter formula, is exactly the one which

Heinrich Hertz used in his famous mechanics and which

allowed him to treat the motion of an arbitrary system

formally as the motion of a single mass-point (in a non-

Euclidean, many-dimensional space). Introducing this

geometry here, we easily see that all the considerations of

the first lecture which led us to the fundamental wave

equation may be transferred, even with a slight formal

w = simplification, viz. that we have to put

1. In exactly

the same way as before we obtain

E

V2{E- V)

and finally for the wave (or rather amplitude) equation:

V^ + ^{E-V)^=0. . . (43)

For the wave equation properly speaking we get, just as

before (section 6),

„„ ,

4:771 ;

Stt^V ,

^

,i .X

But, of course, V^ is now to be understood not as the
simple Laplacian in three dimensions nor as the simple Laplacian in a many-dimensional Euclidean space (i.e.
the sum of the second derivatives with respect to the
single co-ordinates), but it is to be understood as the well-known generalization of the Laplacian in the case of a general Hne-element like (42). In the treatment of
general problems we can usually avoid writing down the explicit expression for this operation; we need only know
that it is a self-adjoint differential operator of the second
order. (Never mind whether you know what '' self-

38

WAVE MECHANICS

adjoint " means, it is of no importance for the moment.)
Yet for the sake of completeness I will put down the
general expression for V^. Let aik be the minor corres-
sponding to hiky divided by the determinant S zb ^/y^- Let « be the determinant of the aikS. Then

^''^
^'-'^'^kk'^'i^'^)- •
In the case of a single mass-point of mass 7;z, treated
—m in Cartesian co-ordinates, this reduces to times the
+ + elementary V^-operator (viz. d^/dx^ d^/dy^ d^/dz^).
Or, if you chose to describe the motion of a single masspoint by any other co-ordinates, e.g. polar or elliptic, you
— would get times the expression for the elementary V^
transformed to those co-ordinates. If the system consists
of n free mass-points, you get the sum of their elementary
V^-operators each divided by the appropriate mass.
The theory in its present form is applicable to systems of any number of degrees of freedom more than, equal
to, or less than, three. I shall give a rapid account of a few examples without going through the details of calculation unless they present some physical interest.
11. Examples: the oscillator, the rotator.
Take the ow^- dimensional harmonic oscillator. The expression for the energy in ordinary mechanics may be
taken to be

(we have expressed the coefficient of the potential energy in terms of the classical proper frequency Vq which it

THIRD LECTURE

39

produces). This easily leads to the amplitude equation:

It can be shown that this equation has solutions which

are finite along the real ^-axis, for the following values of

E only:

= + = E„ {n i)hvo; 72 0,1,2,3. . (46)

The proper functions are the so-called Hermite ortho-

gonal functions

= i/r„ (2"«!)"2^"2/f„(A;)

. . r47^

with

•^W'?

A H„{x) is the so-called wth Hermite polynomial.

graph

of the first five functions (47) is given in the figure.

>+CG
The first five proper vibrations of the Planck oscillator according to undu-
— + latory mechanics. Outside of the region 3 < x_< 3 represented here,
all five functions approach the x-axis in monotonia fashion.
Though theoretically they extend to infinity, they are practically restricted by the exponential to a domain of the
order of magnitude of the amplitude of the corresponding

40

WAVE MECHANICS

We classical mass-point. (This is very easy to prove.)
have not discussed the physical meaning of our generalized i/f-function. Yet the following statement is of interest. If the i/r„'s were the proper functions of a one-electron
problem and q one of the rectangular co-ordinates, we
would (following our j/fj/f-hypothesis) estimate the in-
— tensity of emission of frequency -j\En Ek\y polarized

in the direction of ^, by the square of the integral

j qi/jki/j^^dq.
If we try to do the same here^ we get a most satisfactory
result, viz. the integral vanishes^ unless
\k-n\ = l.

This means that all the emission frequencies except

We 1 . vq are excluded.

shall return later to the question

of the physical meaning of in the general case.

Take as a second example another one-dimensional

problem: the simple rotator with its axis fixed in space.

Here all the energy is kinetic, viz.:

2 \dt

A = = where

moment of inertia, ^ angle of rotation.

The amplitude equation becomes

which has the solutions:
sin
^ COS
Obviously ifj must be restricted to be periodic in ^ with

)

THIRD LECTURE

41

period 27r. Hence the coefficient of (/> must be an integer;
this condition furnishes the proper values

^" = £fi' « = 0.1.2,3

,

(48)

in complete agreement with the older form of quantum theory. Let us try to get an estimate for the intensity of radiation in the same formal way as before. If, in ordinary mechanics, an electrified particle were fixed to the rotator at a distance a from the centre of gravity, its rectangular co-ordinates would be

x\

/cos\

,

Now form

Since

the

p^roduct

of

the

first

two

^^ \
cosj

functions

can

.

^

always be expressed by the sum or difference of

\

+ {n k)(f>, it is easily recognized that none of the eight

quantities comprised in the above formula diflfers from

+ — zero, unless either ;z ^ or \n k\ is unity; or,

|

|

what amounts essentially to the same, unless

\n-k\ = l.

This is the well-known selection-rule for the rotator. It is interesting to treat the rotator again without the
We assumption that its axis is rigidly fixed in direction.
find for the amplitude equation

Here V\,/, means that part of the elementary V--operator

D ( 9:d9

4

42

WAVE MECHANICS

(when expressed in polar co-ordinates) that contains the differentiations with respect to the angles 9, </> only. It is known that the above equation only has finite singlevalued solutions w^hen the constant is the product of two successive integers:

= —^2" =^{^i+ 1); w 0, 1,2, . . .,

and that the solution is a spherical harmonic of order n,
+ (The proper value £„ is {2n l)-fold degenerate, since + there are 2w 1 independent spherical harmonics of
order n.) This furnishes the proper values

this means essentially that *' half-integers " are to be inserted in place of n in the " classical " formula (48).
+ = + — (For «(w 1) (w J)^ i, and a common constant
in all the E„'s cancels out in forming their differences.)
It is known that the representation of band-spectra very often compelled the use of " half-integers ", and it seems that all of them are compatible with the new formula. (Of
course formula (49) is the correct one to use, and not (48), because the axis of a molecule is never rigidly fixed.)
The selection rule comes out in exactly the same way as
in the former case, only by a more troublesome calcu-
lation.

FOURTH LECTURE

12. Correction for motion of the nucleus in the hydrogen atom.
In the first lecture we treated the hydrogen atom as a
one-body problem, as if the nucleus were fixed in space.
Ill ordinary mechanics it is well known that if we start
with the problem of two bodies (of masses m and M), we
can split it in two, viz.: (1) Uniform rectilinear motion of the centre of gravity
(inertial motion).
(2) Keplerian motion around a fixed centre of a body, with the " combined mass " /^, such that

l-i + m

(50)

According to Bohr's theory, this refined treatment of the hydrogen atom is quantitatively supported by the slight difference in frequency between the Helium + -lines and those hydrogen-lines which would exactly coincide with them if the nucleus had infinite mass. (In other words, the
slight difference between the Rydberg constant for He +
H and for is quantitatively accounted for by taking into
account the slight movement of the nucleus; Sommerfeld.)
We meet with exactly the same state of affairs in
wave mechanics. The six-dimensional amplitude equa-
tion for the two-body problem is:

^V,2^ + l^^i^-V ^-fiE-V)^=0. (51)
43.

44

WAVE MECHANICS

By Vi^ and Vg^ we mean the elementary Laplacians with
respect to the co-ordinates of the electron {x^, j^, z^) and
V of the nucleus {x2, y2, ^2)- About we need only make
the assumption that it depends on

only. Now, instead of x^, . . ., z^, introduce the co-

ordinates of the centre of gravity (f , 7^, f) and the relative

m M We co-ordinates of with respect to (say Xy j, z).

can

easily prove that

The meaning of the V-'s is obvious; /x is given by (50). By inserting this in (51) we get an equation which can be split up by supposing ijj to be the product of a function
of f , 7?, ^ only (say ^) and one of x, y, z only (say x). In the splitting up an arbitrary constant is introduced, which is represented by Et in the following equations.
For cf) we get
;;^V^.„.^ + '#'^ = 0, . (62)
and for x
j^V\,,^X + ^(E-E,-V)x = 0. (53)

The former describes the motion of the centre of gravity

under no forces, according to wave mechanics; the

constant Et corresponds to its translational energy and

E — can have any non-negative value.

Et corresponds to

the internal energy. The second equation is exactly that

of the one-body problem for a mass-point with mass /x

moving in a fixed field V. Hence for the proper values

corresponding to the internal energy there will be no

FOURTH LECTURE

45

m difference other than that is replaced by /x (see (14')) in
the formula for the Rydberg constant. Thus Sommerfeld's important result, mentioned above, is re-stated in wave mechanics. Owing to the analytical simplicity of this deduction, there has not been much ado about it in the literature. But it really is one of the most immediate
proofs that there must be something true in the many-
— dimensional wave-treatment however irritating the latter
may be at first.

13. Perturbation of an arbitrary system.
The theory of the perturbation of an arbitrary system really presents no new features as compared with the
perturbation theory of the one-electron atom, a special
case of which has been discussed in sections 7-9; but we
shall widen our outlook by stating it afresh in a concise
form. The general wave equation (44) of section 10 can
be written:

We H will write for the operator

OTT

V) (Fas an operator means: " to multiply by

Then by

(43), section 10, the proper functions ifjk are precisely
those which are reproduced by the operator H, apart from

a multiplying constant, which is the proper value:

ffM = £,^,

(55)

Equation (54) takes the simple form

4>~Hm

(56)

:

46

WAVE MECHANICS

Now, adding to F a small perturbing field, which may

or may not contain the time explicitly, means altering the
H H operator slightly. (Of course an alteration of might

also be produced in another way, e.g. by altering one of

the masses, &c. It will do no harm if this more general

We case is included in our treatment.)

shall call the

H H\ altered operator -\-

bearing in mind that H' is to

We be a '' small " operator.

have to solve

i.= ^-p{H[>l.]+H'W). . . (57)

Tentatively substituting
= S<:,^,e'^; ^.= 5,

• • (58)

with slowly varying time-functions Ck^ we obtain in the
first instance

k

nk

This equation will be satisfied if it is orthogonal to all the ipis *. Multiply by j/j/ and integrate over the whole
configuration-space

ci=-j^Y.Ckaike

/= ,

1, 2, 3, 4, . . .,

(59)

where

= aik j (lqH'[iPk]iph

• • • (60)

and / dq always means a multiple integral over the whole
configuration-space. The af,i's are s?nall quantities.
We will suppose the perturbation to be conservative.

We *

take it for granted that with respect to the completeness

and orthogonaUty of the proper functions the general case behaves

We like the simple hydrogen case. That is quite safe.

also, as

there, avoid encumbering our formulae by exphcitly taking account

of a continuous spectrum of proper values.

FOURTH LECTURE

47

Then the aki's are constants; just as in the special cases
treated before, only the exponentials with vanishing exponent will cause appreciable variations of the q's.
First take the system to be non - degenerate. Then, dropping the other terms, which only furnish slight oscillations, you get, for every q,

c,= ^cr, c,= c?e~\ . . (61)

which, if you substitute it in (58), merely means that the frequency is slightly altered by the amount

h'

Now take a case of degeneracy. Let the amplitudes
Ci, Ci+i, . . . , Q+a-i belong to a different proper functions, all belonging to the same proper value Ei, or proper
frequency v/. Then in each of the equations relating to them you will have not only one, but a, vanishing exponents, which give rise to secular changes. Hence these
a amplitudes wdll be determined by the following set of
equations:

= = — L ci+p

-r- S Ci+^ai+pj+^; /)

a 0, 1, 2, . . .,

(62)

These equations show that under the influence of a slight perturbation there will in general be an exchange of amplitudes between degenerate modes of vibration which belong to the same proper value. It is correct to talk of an exchange, since it is easily proved from equation

(62) that

= '2 |r/+p|'

const.

p=

Yet when thinking of this exchange we must remember

48

WAVE MECHANICS

= — that the set of proper functions ipupip a 0, 1, . . . , 1)

is arbitrary up to an orthogonal hnear substitution of

determinant 1. This induces a similar substitution of the

amplitudes q. Given a definite perturbation, i.e. definite

values of the quantities fl/+A,/+p, it is always possible to

find at least one orthogonal substitution of the j/f/+p's

which brings the equations (62) into the simple form (61)
of the non- degenerate case. Then these particular proper

functions, selected in a way that suits this particular form

of perturbation, will under its influence have constant

amplitude-squares, but will in general belong to slightly

different proper frequencies. The a-fold proper value

has been split up into a slightly differing proper values;

the degeneracy is removed by the disturbing field, and

the particularly chosen proper functions of the degenerate problem are the non- degenerate proper functions " in zero approximation " to the single proper values of the

perturbed problem. The a slight alterations in proper

value

can

be

shown

to

be

the

a

roots

of

the

" '' secular

equation

— Ull

X,

(llji-l,

' ' ', «/, /+aJ

=

Of course it may happen that these roots are not all
We different; a certain degeneracy is then retained.
may either say that the members of an arbitrarily chosen
set of the degenerate functions all vibrate with the un-
— perturbed frequency, but exchange their amplitudes or
that the members of the appropriately chosen set have
constant amplitudes, but that each function has a slightly
different frequency: these two assertions are of course

FOURTH LECTURE

49

— — identical. For as we may put it either: a vibration

of varying amplitude has not really got the frequency

which we ascribe to it; or: two or more slightly different frequencies, when superimposed, lead to a " beat

phenomenon ", i.e. to a varying amplitude.

14. Interaction betw^een two arbitrary systems.

Take now two arbitrary systems, at first without interaction, one of which is described according to wave
mechanics (see equation (56) ) by

and the other by

Multiply the first by ^, the second by ip, and add the resulting equations; you get

H since the operator

L does not affect cf) and does not

affect ip. The latter equation is the wave equation of the

" combined system ", i.e. of the system formed by

mentally uniting the two systems to form one. (The

process is exactly the reverse of what is so often done in " splitting up " an equation by supposing the solution to

be the product of two functions, dependent on different

individual variables.) The proper functions of the com-

bined system are the products of any one of the proper

functions of the first system and any one of the second

system. The proper value that belongs to such a product

is easily seen to be the sum of the respective proper values.

(This corresponds to the additivity of energy in ordinary

50

WAVE MECHANICS

mechanics.) By the addition of proper values a 7iezo degeneracy may be caused in the combined system, even

though the single systems were non-degenerate. (Let

us suppose the latter case, for the sake of simplicity.)

Let £", E' be two proper values of the first system,

F, F' two of the second system, and suppose that
£' + F = F + i^=G

or

E-E' = F-r.

Hence: if a common difference of proper values exists

between the two systems, it will give rise to a two-fold

G degenerate proper value

of the combined system.

For simplicity's sake, suppose that other relations of the

same kind are absent, and now suppose that a slight

interaction of the two systems takes place, changing the

H H ^ L operator -\- into

L -\- T, where T will of course

F'-
contain the variables both of the first and of the second
system. Then the amplitudes belonging to E -^ F' and F to E' -}- will show a slow secular interchange, all the
others remaining essentially constant. The sum of the
squares of the two amplitudes in question is also constant. Interpreted in the single systems, this cannot very well have any other meaning but that e.g. the amplitude of
F increases at the expense of that of F' and, so to speak,
to compensate for the amplitude of E' increasing at the expense of that of E. This seems to be the appropriate wave-mechanical description of what in the older

FOURTH LECTURE

51

form of the quantum theory was called the transfer of a
— — quantum of energy E E' {= F F') from one system
to the other.

15. The physical meaning of the generalized
i/f-function.

Perhaps the latter conclusions are obscured by the fact

that we have hitherto avoided putting forward any definite

assumption as to the physical interpretation of the func-

tion ip{q^, q^y qn, . . . , t) relating to a system whose configuration in terms of ordinary mechanics is described by

the generalized co-ordinates ^1, ^2? • • • > ?«• This interpretation is a very delicate question. As an obvious generaliza-

tion of the procedure of spreading out the electronic

charge according to a relative density function «/f j/f (which

furnished satisfactory results in the one-electron problem;

see section 5), the following view would present itself

in the case of a general mechanical system: the real

natural system does not behave like the picture which

ordinary mechanics forms of it (e.g. a system of point-

charges in a definite configuration), but rather behaves

like what would be the result of spreading out the

system,

described

by

^1,

.

.

. ,

9„,

throughout its con-

figuration-space in accordance with a relative density

function 0j/f. This would mean that, if the ordinary

mechanical picture is to be made use of at all, the actual

system behaves like the ordinary mechanical picture,

present in all its possible configurations at the same time, though '* stronger " in some of them than in others.

I maintained this view for some time. The fact that

it proves very useful can be seen from the one-electron
problem (see section 5). No other interpretation of the

i/f-function is capable of making us understand the large

—

52

WAVE MECHANICS

amount of information which the constants aki furnish about the intensity and polarization of the radiation. Yet this way of putting the matter is surely not quite satisfactory. For what does the expression '* to behave like " mean in the preceding sentences.^ The '' behaviour " of the j/f-function, i.e. its development in time, is governed by nothing like the laws of classical mechanics; it is governed by the wave-equation.
An obvious statistical interpretation of the ^-function
has been put forward, viz. that it does not relate to a single system at all but to an assemblage of systems,
ijj determining the fraction of the systems which happen to be in a definite configuration. This view is a little unsatisfactory, since it oflFers no explanation whatever
why the quantities aki yield all the information which they
do yield. In connexion with the statistical interpretation it has been said that to any physical quantity which would have a definite physical meaning and be in principle {principiell) measurable according to the classical picture of the atom, there belong definite proper values (just as e.g. the proper values Ek belong to the energy); and it has been said that the result of measuring such a quantity will always be one or the other of these proper values,
but never anything intermediate. It seems to me that
this statement contains a rather vague conception, namely
that of measuring a quantity (e.g. energy or moment of momentum), which relates to the classical picture of the atom, i.e. to an obviously wrong one. Is it not rather bold
to interpret measurements according to a picture which
we know to be wrong? May they not have quite another
meaning according to the picture which will finally be forced upon our mind? For example: let a beam of electronic rays pass through a layer of mercury vapour, and

FOURTH LECTURE

53

measure the deflection of the beam in an electric and in a magnetic field before and after the beam has traversed
the vapour. According to the older conceptions this is interpreted as a measurement of diflferences of energy-
levels in the mercury atom. The wave-picture furnishes
another interpretation, namely, that the frequency of part of the electronic waves has been diminished by an amount equal to the difference of two proper frequencies of the mercury. Is it quite certain that these two interpretations do not interfere with one another, and that the
old one can be maintained together with the new one?
Is it quite certain that the conception of energy, indispensable as it is in macroscopic phenomena, has any
other meaning in micro-mechanical phenomena than the number of vibrations in h seconds?

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