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Quantum Mechanics: An Introduction
Arun Kumar Bhattacharya
Lecture 1
The Breakdown of Classical Physics
In the early part of the twentieth century it was realized that the ideas of
classical physics could not offer a cogent explanation of the results from a number
of experiments. The following observations signaled the breakdown of classical
physics.
(1) The anomalous stability of atoms. According to classical mechanics and
electromagnetic theory, electrons orbiting the nucleus of an atom should spiral into
the nucleus by continuous emission of radiation. Classical physics could not
explain why such a thing is not found to happen. The characteristic line spectrum
of an atom also could not be explained on the basis of classical theories.
(2) The ultraviolet catastrophe connected with blackbody radiation. According to
the classical theory, put forward by Rayleigh and Jeans, the energy density of
radiation emitted by a blackbody diverges as  2 . Experiments showed that the
energy density is finite for    .
(3) Wave-particle duality. Interference phenomenon of light, photoelectric
phenomenon and the experiments by Davison, Germer, and Thomson on the
diffraction of electrons led to the conclusion that radiation and matter exhibit both
wave and particle-like characteristics. This wave-particle duality is an unfamiliar
concept in classical physics.
(4) Specific heat of solids at low temperature. Classical physics could not explain
the fact that the specific heat of solids C v goes to zero as T  0 .
A proper understanding of the behaviour of radiation and matter at the atomic
scale was possible only after a revolutionary change in the theoretical structure of
physics. The efforts of Bohr, Schroedinger, Heisenberg, Born, Dirac, and many
others led to the development of quantum theory which
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provided a consistent description of the submicroscopic world. Since this
theory introduced concepts which are very different from the ones derived
from our everyday experience of macroscopic phenomena, quantum theory
appears strange and mysterious.
Two-slit experiment
We take up a discussion of the two-slit experiment because it encapsulates
many of the important features of quantum systems.
(A) Experiment with bullets
Let us first perform the experiment using familiar macroscopic objects
like bullets and water waves. We then perform the same experiment using
electrons and compare the results of the various experiments. In the
experiment with bullets the set-up is as shown in Figure 1.
Figure 1
A machine gun G sprays bullets with a fairly large angular spread towards a
wall containing two holes just big enough to let the bullets through. Beyond
this wall is a backstop which absorbs any bullet hitting its thick wall. There
is a detector of bullets which can slide along the surface of the backstop. The
detector has an arrangement for keeping a count of bullets entering into it.
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We can thus specify how many bullets accumulate in the detector in a given
time interval. Using this data we can find out the probability that the bullets
passing through the holes arrive at the backstop at a distance x from its
centre. This probability is just the ratio of the number of bullets collected at
location x to the total number of bullets reaching the backstop in the same
time interval, assuming that the gun fires at the same rate. The experimental
observations are as follows:
(i) The bullets always arrive in identical lumps. The size of the lump is
independent of the rate of firing. If the rate of firing of the gun is very small
at any instant either nothing arrives or exactly one bullet reaches the
backstop.
(ii) The probability P12 that a bullet arrives at the detector located at a
distance x from the centre varies as shown in the Figure. P12 has its
maximum value at x=0 but is small when x is large.
(iii) If the experiment is repeated by blocking holes 2 and 1 in turn, the
corresponding probability distributions P1 and P2 are as shown in the Figure.
(iv) P12 is the sum of the probabilities P1 and P2 . The effect with both holes
open is the sum of the effects produced with each hole open in turn. This
result signifies that in the experiment with bullets there is no interference.
(B) Experiment with water waves
We now perform the two-slit experiment with water waves. We have an
arrangement (Please see Figure 2) for producing concentric spherical waves
in a large body of water. The waves advancing towards right encounter a
wall with two small holes. Further right we have another wall which fully
absorbs the waves reaching there. A wave detector can move along the
surface of the absorbing wall. The detector records the square of the height
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of the wave and thus directly measures the energy carried by the wave or the
intensity of the wave.
Figure 2
Our observations are as follows:
(i) The intensity can have any value depending on the amplitude of the wave
generated at the source. There is no “lumpiness” in the intensity of the wave
detected. The variation of the intensity I12 as a function of the location x of
the detector is shown in the accompanying Figure.
(ii) The curves I 1 and I 2 respectively represent the intensities obtained by
blocking hole 2 and hole 1, respectively. We note that
I12  I1  I 2 .
There is interference of the waves coming out of holes 1 and 2. In regions
where the waves arrive in phase there is constructive interference ( I12 is
large); at places where the two waves arrive with a phase difference of 180  ,
there is destructive interference.
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(iii) Let the instantaneous height of the water wave with hole 2 and hole 1
closed in turn be respectively h1 e it and h2 e it ; the corresponding intensities
2
2
I 1  h1 ; I 2  h2 .
When both holes are open the wave height is
h1 e iwt  h2 e i (t  ) and the corresponding intensity is
I12  I1  I 2  2 I1 I 2 cos  .
Here  is the phase difference between h1 and h2 . The last term on the righthand side of the expression for I12 is called the interference term.
(C) Experiment with electrons
We now perform the two-slit experiment with electrons. It may be
mentioned that since 1990 experiments of this nature are being performed in
a routine manner.
Nearly monoenergetic electrons coming out of an electron gun are
accelerated towards a wall with two holes (Please see Figure 3). The
electrons passing through the holes are detected by a detector which may be
an electron multiplier connected to a loudspeaker.
Figure 3
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The following observations are noteworthy:
(i) We always hear sharp clicks of the same size from the loudspeaker. No
half-clicks are ever heard.
(ii) The clicks are random. If we repeatedly count the number of clicks in a
sufficiently long period of time we find more or less the same number. We
can, therefore, meaningfully talk of an average count rate. When the detector
is moved along the x-direction the average count rate changes but not the
click sounds. Whatever arrives at the detector comes in lumps of the same
size.
(ii) The curve P12 represents the relative probability that an electron arrives at
x.
From the above observations we conclude: The electrons reaching the
detector can be divided into two classes, namely, (a) the ones that come
through hole 1 and (b) the ones that come through hole2. We repeat the
experiment once with hole 2 closed and the next time with hole 1 closed.
The resulting probability distributions are P1 and P2 , respectively. We find
that P12  P1  P2 . There is thus interference of electrons.
To explain the interference effect we may argue that it is not true that
whatever reaches the detector goes through either hole 1 or hole 2. We may
speculate that the electrons reach the detector by following very complicated
paths around hole 1 and hole 2. We, however note that there are some points
where very few electrons arrive when both holes are open but which receive
many electrons if we close one hole. So, closing one hole seems to increase
the number of electrons going through the other hole. Also notice that at x =
0, P12 is more than twice as large as P1  P2 . It is as though closing one hole
has decreased the number coming through the other hole. We cannot explain
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both the two possibilities by assuming that electrons follow complicated
paths.
The mathematical relation between P12 and P1 , P2 is of the same nature
as that between
I12 and I 1 , I 2 that we obtained with water waves. We,
therefore conclude that electrons arrive as lumps (that is, as particles) but
the possibility of arrival has the same variation as the intensity of a wave.
So electrons have both particle and wave chacteristics. The proposition that
electrons go through either hole 1 and hole 2 seems to be questionable.
Watching the electrons
In the experiment with electrons we make the modification shown in
Figure 4 below.
Figure 4
Immediately to the right of the wall with holes we place a strong light
source so that we see flashes in the vicinity of hole 1 or hole 2 whenever an
electron comes out of hole 1 or hole 2. We observe that every time we hear a
click, we see a flash either near hole 1 or hole 2, but never flashes near both,
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indicating that electrons do indeed reach the detector passing through either
hole 1 or hole 2. Let P1 and P2  be the respective probability distributions
when the flashes are seen near hole 1 and hole 2. The distributions are
similar to the ones we found in our experiment with bullets. So, we have to
rule out the possibility that electrons reach the detector by going through
both holes. Electrons that come through hole 1 are not affected by whether
hole 2 is open or closed. The total possibility of an electron reaching the
detector by passing through either hole 1 or hole 2 is given by



P12  P1  P2
We conclude that when we watch through which hole an electron emerges
there is no interference pattern. The distribution of electrons at the backstop
when the electrons are under watch is different from the distribution
obtained when there is no provision for seeing the electrons. It seems that
the interaction between light and electron disturbs the state of the electron.
The electron receives a jolt due to light causing the interference pattern to
disappear.
Let us try to eliminate the disturbance in electron motion by making the
light source dimmer. When a very dim source is used the flashes do not
become weaker. The only noticeable change is that sometimes we hear a
click at the detector but see no flash. The electrons sometime go by without
being seen. Light, which we earlier considered as being spread out like
waves, now appears lumpy, just like the electrons. The lumps of light are
called photons. Reducing the intensity of the light source merely decreases
the number of photons but not their “size.” So, when the source is very dim
some electrons manage to get by without encountering a photon.
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When using a very dim source we record our observations in three
separate columns. In columns 1 and 2 we keep a count of electrons reaching
the detector attended by a flash near hole 1 and hole 2, respectively. In
column 3 we record those electrons which reached the detector without
producing a flash. The data in columns 1 and 2 give rise to the probability
distributions P1 and P2  . But the data in column 3 which correspond to unseen
electrons give rise to a distribution characteristic of waves. So, if the
electrons are not jolted by photons we have an interference pattern. The
disturbance caused by photons washes out the interference pattern.
We may ask: Is there a way to see the electrons without disturbing them?
The disturbance due to the photons will obviously depend on the momentum
of a photon which is given by p  h /  . So, to reduce the disturbance we
should use light of longer wavelength rather than light of low intensity. Let
us see what happens when we go on increasing the wavelength of the light
used for seeing the electrons. Because of its wave nature, light can
distinguish two spots as being separate only when the distance between the
spots exceeds the wavelength of light. So, when we use light of wavelength
larger than the separation between the holes we see a big fuzzy flash spread
over both holes. We can no longer say through which hole the electron goes.
For light of such long wavelength the jolts received by the electrons become
negligibly small so that P12 shows interference.
From the discussion above, we conclude that it is impossible to keep the
probability distribution undisturbed when light is so arranged as to provide
us information regarding the hole traversed by the electron. Heisenberg was
the first to realize that this is a fundamental limitation which cannot be
overcome by experimental ingenuity. Heisenberg proposed a general
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principle which in the context of our present experiment may be stated thus:
“It is impossible to design an apparatus to determine which hole the electron
goes through, that will not at the same time disturb the electrons enough to
destroy the interference pattern.” This is a basic aspect of nature.
If one still persists with the question: Is or is not true that the electrons go
through either hole 1 or hole 2? The answer to the question depends on
whether or not in the experimental arrangement there is a provision for
determining which hole an electron goes through. In making predictions
regarding the outcome of experiments performed on submicroscopic systems
we must always keep in view the actual arrangement in the experimental setup.
There is one more question that we have to answer. If all matter has
wavelike behaviour, why did we not see any interference pattern in the
experiment with bullets? It turns out that for a macroscopic object like a
bullet the wavelength is so small that the interference pattern becomes too
finely spaced (Please see Figure 5). We cannot resolve the separate maxima
and minima. We see only a kind of average distribution characteristic of
classical particles.
Figure 5
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Summary
The following is a summary of the main conclusions of the various
experiments discussed so far:
(i) The probability P of an event is given by  2 , where  is a complex
quantity, called the probability amplitude.
(ii) When an event can take place in several alternative ways, then the
probability amplitude of the event is the sum of the probability amplitudes
for each possible way considered separately. There is interference:
  1   2
2
; P  1   2 ,
assuming that there are only two alternative ways.
(iii) If the experiment has an arrangement for ascertaining which of the
alternatives is actually realized, then the probability of the event is the sum
of the probabilities corresponding to each alternative. There is no
interference:
P  P1  P2 .
At the quantum level we are forced to talk in terms of probability. It is, in
general, impossible to predict exactly what would happen in a given
situation.
The uncertainty principle
Heisenberg stated the principle of uncertainty in the following form: If we
measure the x-component of the momentum of a submicroscopic particle
with an uncertainty p , then we cannot know the x-coordinate of the particle
with a precision larger than x  h / p, where h is Planck’s constant. It has
been already pointed out that this uncertainty is not due to the limitation of
the apparatus or the ingenuity of the experimenter. We now know that there
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are similar uncertainty relations involving other pairs of dynamical
quantities.
Let us explore the implications of Heisenberg’s uncertainty relation for
our experiment with electrons. For this purpose we make some modification
in the experimental set-up (Please see Figure 6). With the help of rollers we
allow the wall with holes to move freely along the x-direction. Watching
carefully the motion of the wall we can ascertain through which hole an
electron goes.
Figure 6
Suppose the detector is at x = 0. The x-component of the momentum of an
electron which reaches the detector via hole 1 changes by p x in the
downward direction. So the wall gets an upward kick and its momentum
change is p x , in the upward direction. If the electron goes through hole 2 its
momentum changes by the same amount in the upward direction. In this case
the momentum of the wall changes by px in the downward direction.
Without any further disturbance to the electron we can ascertain the path
chosen by it simply by focusing our attention on the movement of the wall.
Suppose that the momentum of the wall changes by px . Then according
to the uncertainty principle the position of the centre of the wall or the two
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holes cannot be known with arbitrary accuracy. For every electron passing
through the holes there will be slight shifts in the location of the holes. As a
consequence the centre of the fringe pattern will keep on changing due to the
passage of electrons. So, the interference pattern will be smeared out. This is
in accordance with our earlier conclusion: ‘which path’ information washes
out the interference pattern.
The Stern-Gerlach Experiment.
We now consider an experiment of a different nature – the Stern-Gerlach
(SG) experiment. This experiment clearly brings out the necessity for going
beyond the ideas of classical physics. A careful study of an essentially twostate system considered in the SG experiment will be helpful in analyzing
problems involving many more states.
Figure 7
Silver atoms are heated in an oven with a small hole. The atoms coming
out of the oven are collimated and are then subjected to a strongly
inhomogeneous magnetic field. The field gradient is in a direction at right
angles to the direction of propagation of the atoms. A silver atom has a
magnetic moment equal to the spin magnetic moment of an electron. The
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

magnetic moment of the atom  is proportional to the electron spin S .
Because of the magnetic field gradient along the z-direction, the force
Fz acting on the atom is given by
Fz   z
B
.
z
Since the atom is quite heavy we assume that its motion can be described
classically. If the electron were a classical spinning particle,  z would vary


between   to +  continuously. We would then expect a fuzzy spot on
the detector screen. But we find that the SG apparatus splits the atomic beam
into two components which give rise to two sharp spots on the screen. The
corresponding values of S z are   / 2 . If the field gradient is in the ydirection, the beam splits into two components: a S y  component and an
S y  component.
We now let the atomic beam pass through two or more SG apparata
sequentially. In arrangement (a) the beam coming out of the oven goes first
through an SG ẑ apparatus in which the magnetic field gradient is along the
ẑ -direction. We then block the emerging S z  component. The surviving
S z  component then passes through another SG ẑ apparatus. As expected, we
find only one beam –the S z  beam – coming out of the second
SG ẑ apparatus.
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Figure 8
In the arrangement shown in Figure (b) the first SG apparatus is the same
as in case (a) but the second one is an SG x̂ apparatus. The S z  beam
coming out of the first SG ẑ splits into two components of equal intensity
corresponding to S x  and S x  . It is tempting to conclude that half of the
atoms in the S x  beam coming out of the first SG ẑ is characterized by S z 
and S x  while the remaining half has S z  and S x  . That this is an
erroneous conclusion will become clear from the following consideration.
We consider the set-up (c) which has a third SG ẑ apparatus. We observe
that in the final stage there are two components. The emerging beams have
both S z  and S z  components. This is very surprising because we totally
filtered out the S z  component coming out of the first SG ẑ apparatus. How
then does the S z  component reappear in the final emerging beam?
Obviously we made a mistake in assuming that the beam entering the third
SG ẑ apparatus can be characterized as having S z  and S x  . This example
illustrates that in a quantum description we cannot assign definite values to
S z and S x simultaneously. Further the process of selection of the S x 
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component by the SG x̂ apparatus destroys the previous information about
S z . We should emphasize that the limitation on the simultaneous
specification of S z and S x is a fundamental aspect of the submicroscopic
world.
Summing up our analysis, we see the need to develop a new theory for a
coherent description of submicroscopic phenomena. The new theory, known
as quantum theory ( or quantum mechanics), has been presented in three
different forms: (i) The wave mechanics of Schroedinger in which the

probability amplitude  (r , t ) satisfies a differential equation known as the
Schroedinger equation. This approach makes extensive use of differential
equations which are used frequently in many areas of classical physics. (ii)
The approach of Heisenberg, on the other hand, exploited the algebra of
linear vector spaces, operators, and matrices. Dirac then developed a hybrid
approach which synthesized the approaches of Schroedinger and
Heisenberg. (iii) The third formalism of quantum theory, known as
Feynman’s path integral method, came much later. This method is based on
a geometrical picture of the happenings in the microworld.
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Lecture 2
The hypothesis of de Broglie associates a wave with a material particle.
De Broglie gives us the wavelength of the matter waves but tells us nothing
about the propagation of the wave. We expect that there should be a wave
equation governing the behaviour of matter waves. Such an equation was
first proposed by Scroedinger in 1925. We shall arrive at this equation by
using simple-minded plausibility arguments.
Let us consider a free particle constrained to move along the x-direction.
The momentum p and the de Broglie wavelength   h / p of the particle are
constant. We use a sinusoidal traveling wave
x
 ( x, t )  A sin 2 (  t )  A sin( kx  t )

(1)
or a wave packet, which is a combination of such sinusoidal waves, to
represent the wave function associated with the particle. We have put
k  2 /  ;   2 in equation (1). The frequency  obeys Einstein’s relation
  E / h , where E is the total energy of the associated particle. If the particle
is subjected to forces, its momentum p will not be of constant magnitude.
The wave function of such a particle will have a form more complicated than
the sinusoidal wave represented by (1).
We expect the wave equation to be a differential equation involving space
and time derivatives of  ( x, t ) . Assuming the form (1), we see that
 ( x, t )
 2 ( x, t )
 kA cos( kx  t ) ;
  k 2 ( x, t )
2
x
x
 ( x, t )
 2 ( x, t )
 A cos( kx  t ) ;
  2 ( x, t ).
2
t
t
(2)
We now make the following reasonable assumptions:
(i) The wave equation must be consistent with de Broglie-Einstein relations
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  h / p ;  E / h .
(3)
and the expression
E
p2
V
2m
(4)
for the total energy of a non-relativistic particle of mass m in a potential
V(x).
(ii) The wave equation must be linear in  ( x, t ) so that we can superpose
wave functions to produce interference effects observed in the experiments
of Davisson and Germer, and of Thomson.
In equation (4) the potential energy V is a function of x and t. Using (3)
we obtain from equation (4)
2k 2
 V ( x, t )   ,
2m
(5)
where   h / 2 . The supposed linearity of the wave equation implies that
the differential equation cannot contain any term independent of  ( x, t ) nor
can it have any term proportional to [ ( x, t ) ]n with n  1.
The sinusoidal form of the wave function implies that taking the second
space derivative of  ( x, t ) amounts to multiplying it by –k2 , and taking the
first time derivative introduces the multiplicative factor -  . Since the
differential equation we are seeking must be consistent with (5) we try the
following form

 2 ( x, t )
 ( x, t )
,
 V ( x, t ) ( x, t )  
2
t
x
(6)
where  and  are constants. For a free particle V(x,t)=V0. We thus get
  sin( kx  t )k 2  sin( kx  t )V0    cos( kx  t ).
(7)
Constants  and  can satisfy this equation only for special combinations of
x and t such that sin( kx  t )  cos( kx  t ) . This restriction arises because the
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differentiation of the chosen form of  ( x, t ) changes sines to cosines and vice
versa. So, instead of (1) we try the following combination of sine and cosine
functions
 ( x, t )  cos( kx  t )   sin( kx  t ),
(8)
where  is a constant. From equations (6) and (8), we obtain
[k 2  V0  ] cos( kx  t )  [k 2   V0   ] sin( kx  t )  0 . (9)
which implies
 k 2  V0    ,
(10)
and
 k 2  V0   /  .
(11)
Equations (10) and (11) yield
  1 /     i .
(12)
Substituting the value of  in (10), we get
 k 2  V0  i .
(13)
Comparison of (13) and (5) gives
   2 / 2m ,
(14)
and
   i .
(15)
There are two possible choices of sign in (12) which are of no significant
consequences. We choose the plus sign so that   i . Then   i and the
wave equation reads
 2  2 ( x, t )
 ( x, t )

 V ( x, t ) ( x, t )  i
.
2
2m x
t
(16)
Equation (16) has been determined assuming that V(x,t)=V0. We postulate
that this equation remains valid in the general case where the potential is a
function of x and t. Equation (16) is the one-dimensional form of the
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Schroedinger equation. Schroedinger was led to this equation by using
arguments more subtle than the ones employed by us.
A few comments are now in order.
C1. The generalization of (16) to three dimensions leads to the following
form of the time-dependent Schroedinger equation



2 2 
 (r , t )
.

  (r , t )  V (r , t ) (r , t )  i
2m
t

 ( r , t ) is, in general, complex.
(17)
C2. If  1 ( x, t ) and  2 ( x, t ) are two solutions of (16) for a particular V(x,t), then
 ( x, t )  c1 1 ( x, t )  c2 2 ( x, t ) ,
(18)
where c1 and c2 are arbitrary constants, is also a solution of (16). So, the
Schroedinger equation is linear, as required.
C3. The Schroedinger equation is of first order in t. Hence its solution

requires the knowledge of  (r , t ) at time t = 0, but not its time derivative. The
equation does not resemble the classical wave equations for elastic or
electromagnetic waves. In fact, for V = 0, (except for the factor i on the
right-hand side) it is identical with the diffusion equation for heat flow, for
example. The presence of i ensures that the quantum mechanical equation
has wave-like solutions.

C4. The wave function  (r , t ) must in some sense be a measure of the
presence of a particle. We do not expect to find the particle in regions where

  0 . Since  ( r , t ) is not positive everywhere, it cannot be a direct measure

of the likelihood of finding the particle at r at time t. In electromagnetic

theory of light, interference patterns result from the superposition of E field.

The intensity of the fringes is, however, proportional to E 2 . Pursuing this

analogy, it is reasonable to assume that the quantity  (r , t ) is a measure of
2
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
the probability of finding the particle at r at time t. This probabilistic

interpretation of  (r , t ) was introduced into quantum mechanics by Max
Born.
C5. We can obtain an interesting alternative form of the wave equation by
making the substitution


 (r , t )  exp[ iS (r , t ) / ] .
(19)
Equation (17) then gives




S (r , t ) (S (r , t )) 2 i 2 
[


 S (r , t ))  V (r , t )] exp( iS (r , t )) / )  0 (20)
t
2m
2m

The function S (r , t ) is, in general, complex. From (20) we conclude that

S ( r , t )) satisfies the nonlinear differential equation



S (r , t ) (S (r , t )) 2 i 2 


 S (r , t )  V (r , t ) = 0
t
2m
2m
(21)
The third term on the left-hand side of (21) contains i and  which carry the
signature of quantum mechanics. If this term becomes negligible, equation
(21) reduces to the Hamilton-Jacobi equation of classical mechanics.
C6. The ultimate justification of the Schroedinger equation (17) comes from
the fact that predictions based on this equation are in agreement with
experiment.
The time-independent Schroedinger equation
We apply the method of separation of variables to reduce the onedimensional time-dependent Schroedinger equation (TDSE)
 2  2 ( x, t )
 ( x, t )

 V ( x, t ) ( x, t )  i
2
2m x
t
(22)
21
to two ordinary differential equations. We look for solutions of TDSE in the
form
 ( x, t )   ( x)  (t ) .
(23)
Solutions of this form will exist if V does not depend explicitly on time,
which we assume in what follows. Combining equations (22) and (23) we
get

2
d 2 ( x)
d (t )
.
 (t )
 V ( x) ( x)  (t )  i ( x)
2
2m
dt
dx
Dividing both sides by  ( x)  (t ) , we obtain
1
 2 d 2 ( x)
1 d (t )
.
[
 V ( x) ( x)]  i
2
 ( x) 2m dx
 (t ) dt
(24)
We note that the left-hand side of (24) depends only on x, while the righthand side depends on t only. Consequently, the common value of the two
sides cannot depend either on x or t. It must be a constant which we call K.
We thus have two equations

 2 d 2 ( x)
 V ( x) ( x)  K ( x)
2m dx 2
(25)
1 d (t )
K.
 (t ) dt
(26)
and
i
The two ordinary differential equations are coupled because both of them
involve the same separation constant K.
Equation (26) has the solution
 (t )  A exp( iKt / ) .
(27)
 (t ) is an oscillatory function of time with frequency   K / h . But according
to the Einstein-de Broglie postulate  must be equated to E/h, where E is the
total energy of the particle associated with the wave function whose time-
22
dependent part is  (t ) . We, therefore, take the separation constant K to be
equal to E. Using this value of K, we have

 2 d 2 ( x)
 V ( x) ( x)  E ( x) .
2m dx 2
(28)
The product form of the wave function becomes
 ( x, t )  A ( x) exp( iEt / ) .
(29)
Equation (28) is called the time-independent Schroedinger equation (TIDE).
In three dimensions, the TIDE has the form
 

2 2 
(30)
  (r )  V (r ) (r )  E (r ) .
2m

The functions  (x) or  (r ) are called eigenfunctions. The form (29) implies

that the corresponding probability density  ( x, t )   * ( x, t ) ( x, t ) is time
independent. In fact, all the dynamical quantities are constant in time.
Properties of eigenfunctions
With a well-behaved form of V(x) and proper boundary conditions, the
TIDE will have acceptable solutions only for certain values of the total
energy E. The quantization of the energy in quantum theory thus follows
quite naturally from the Schroedinger equation. To be an acceptable solution
 (x)
and its derivative must satisfy the following properties:
(i)  (x) must be finite, single-valued and continuous everywhere.
(ii) d dx must be finite, single-valued and continuous (assuming V(x) has
no infinite discontinuity).
23
The above conditions must be fulfilled so as to ensure that measurable
quantities like position, momentums etc. are well-behaved. The continuity of
 (x) implies that d dx is finite. Since
d 2 2m
 2 [V ( x)  E ] ( x) ,
dx 2

(31)
d 2 ( x) dx 2 must be finite for finite V(x), E, and  (x) . This in turn demands
that d dx must be continuous. At a point where V(x)  ,  ( x) must go to

zero. Depending on the boundary conditions imposed on  (r ) , we can have

two types of solutions: (i) bound states for which  (r )  0 as r   (ii)
scattering states.
Importance of the separable solutions of the TDSE
A general solution of the one-dimensional TDSE will not have the form
 ( x, t )   ( x)  (t ) .
(32)
We mention below some important features of the separable solutions of the
TDSE.
(i) Such solutions correspond to stationary states. Although the wave
function
 ( x, t )   ( x) exp( iEt / )
depends on t, the probability density
 ( x, t )   ( x )
2
2
is independent of time. The same is true of the expectation value of any
dynamical quantity f(x, p). In a stationary state, the mean value of the
position x is constant and the mean value of linear momentum p is zero. This
24
renders the concept of motion of a quantum mechanical system in a
stationary state rather tenuous.
(ii) Separable solutions of TDSE of the type (32) represent states of definite
energy. When V is independent of time the Hamiltonian or the total energy is
given by
H
p2
 V ( x)
2m
The TIDE can then be written as an operator equation
2 d 2
Hˆ   E  with Hˆ  
 V ( x) .
2m dx 2
The mean value of the total energy is given by
2
 H    * ( x) Hˆ  ( x) dx  E   ( x) dx  E ,
L
L
assuming that  is normalized, that is,

*
L
( x) ( x)dx  1,
where L specifies the region of integration.
Since
Hˆ 2  Hˆ ( Hˆ  )  Hˆ ( E )  E 2 ,
we get
 H 2    * Hˆ 2dx  E 2 .
L
So, the variance of H is
H   H 2    H  2  E 2  E 2  0 .
This means that every measurement of the total energy is certain to yield the
value E.
(iii) Let us assume that the TIDE has a set of solutions 1 ( x), 2 ( x),  ,  N ( x)
each with its associated total energy E1 , E2, …, EN , respectively. Note that
25
N can be denumerably infinite. Because of the linearity of the TDSE, we can
write a general solution of the separable form as
N
 ( x, t )   c n n ( x) exp( iE n t / ) .
(33)
n 1
Note that the solution (33) which is a superposition of stationary states does
not correspond to a time-independent probability density. For the sake of
illustration suppose that the state of a particle is just a superposition of two
stationary states
 ( x, t )  c11 ( x) exp( iE1t / )  c22 ( x) exp( iE2t / ) ,
where c1 , c2 ,1 ,2 are assumed to be real. The probability density
 ( x, t )  c1212  c 22 22  2c1c 21 2 cos[( E 2  E1 )t / ] .
2
In this case the mean values of x and p will be oscillatory functions of time.
So, a linear superposition of states of different energy seems to be necessary
for ‘motion’ of a quantum mechanical system.
26
Lecture 3
Born’s Interpretation of Wave Function
We have already touched upon Born’s interpretation of the wave function.
We pick up the thread of our earlier discussion and discuss its implications.

The wave function  (r , t ) does not have a physical existence in the sense that

water waves have. Being a complex quantity  (r , t ) cannot be measured by

any physical instrument.  (r , t ) , however, encodes all possible information
about the associated particle. It is to be looked upon as a computational
device needed for the development of Schroedinger’s theory.

Born’s postulate regarding the physical significance of  (r , t ) can be
stated thus: If a position measurement is made at time t on a particle



associated with the wave function  (r , t ) , the probability P(r , t )dr of finding


the particle in a volume element d r around the point r at time t is



proportional to  (r , t ) dr . The quantity P (r , t ) has the significance of
2
probability density



P(r , t )   * (r , t ) (r , t ) .
We may mention that in an electromagnetic field the density of photons is
connected with the square of the electric field vector. In quantum mechanics

it is necessary to take the amplitude  (r , t ) times its complex conjugate to
make the probability real. We could construct other functions out of

 ( r , t ) which would be positive everywhere but they will lead to unphysical
behaviour of the probability density.
27
Since the motion of the particle is tied up with the propagation of an
associated wave function, the particle must be located in a region where the
wave function has an appreciable value.
One often encounters situations where a particle is confined to a limited
region. For example, a particle may be confined in a box with impenetrable
walls or an atomic electron may be held close to a nucleus by electrostatic
forces of attraction. In such cases, the particle is certainly to be found if the
entire space is investigated. We thus stipulate that
 
 P(r )dr  1,

(1)
where the integral extends over all space and we have assumed that P is
time-independent. This relation is satisfied if we set

P(r ) 
 *
.
*


d
r

(2)

If  is multiplied by any constant factor there is no change in the value of

P (r ) . This signifies that two wave functions differing by a constant
multiplicative factor describe identical states of a physical system. It is
convenient to introduce a factor so as to make the denominator on the righthand side of (2) equal to unity. To this end we define
1

N
,
(3)
where

N    * dr .

(4)
The function  satisfies the equation

   dr  1
*

(5)
and is said to be normalized.
Note that the wave function
28


 (r , t )  A exp i(k .r  t )
which represents a free particle is not normalized. The quantity  * in this
case has to be interpreted as a relative probability density. The ratio of the
magnitudes of

  dr
*
in two different regions of space gives the relative probability that these
regions are occupied by the particle. Alternatively, we can adopt a procedure
known as box normalization to get around the above difficulty.
Time independence of normalization
We now prove an important theorem concerning the normalization of
wave functions. This theorem asserts that for every wave function satisfying

the Schroedinger equation with time-independent and real potential V (r )




2 2 
 (r , t )
,
  (r , t )  V (r ) (r , t )  i
2m
t
(6)
the integral

N    *dr

is constant in time.
Taking the complex conjugates of both sides of (6), we have

 * 
2 2 * 
 * (r , t )

  (r , t )  V (r ) (r , t )  i
2m
t
(7)
Now

dN
  *
  ( *

 )dr

dt
t
t
Using equations (6) and (7), we can write



dN 1
2 2
2 2 *
  { * [
   V (r ) ]   [
   V (r ) * ]}dr
dt i 
2m
2m
29

i * 2
(      2 * )dr .
 2m

By using Green’s theorem we can write
dN
i
 *
* 

(

)dS
dt
2m S
n
n
(8)
in which  n is the normal derivative, and the surface S encloses the
volume  of integration. Now, the integral (4) for N exists only if the
function  vanishes at large distances rapidly. So, the surface integral in (8)
approaches zero as the surface of integration recedes to infinity.
Hence dN dt  0 ; the normalization of  does not change with time. This is a
statement of conservation of probability.
Probability current
Consider the Scroedinger equation



2 2 
 (r , t )

  (r , t )  V (r ) (r , t )  i
2m
t
(9)
Assuming V to be real, the complex conjugate of this equation is

 * 
2 2 * 
 * (r , t )
(10)

  (r , t )  V (r ) (r , t )  i
2m
t


Multiplying both sides of (9) by  * (r , t ) and those of (10) by  (r , t ) and
subtracting we get
2


 *
( * 2   2 *)  ( *

).
2m
i
t
t
We can rewrite this equation in the form






i

.[ * (r , t ) (r , t )   (r , t ) * (r , t )]  [ * (r , t ) (r , t )]
2m
t
(11)
Equation (11) has the form of the equation of continuity


P(r , t )
 div J  0 ,
t
(12)
30


where P (r , t ) is the probability and J is the probability current given by









J 
[ * (r , t ) (r , t )   (r , t ) * (r , t )]  Im[ * (r , t )  (r , t )] (13)
2im
m
Analogue of equation (13) arises in any theory in which an extensive
quantity (like mass, charge, or heat flow) is subject to a conservation law.
Considering the flow of a compressible fluid we take P to represent the


density of the fluid and J  Pv as the current of fluid crossing unit area

normal to the direction of the fluid velocity v . Then equation (12) signifies
that the change in the total amount of fluid in any fixed volume element is
accounted for by the flow of fluid through the surface enclosing the volume
element. In the context of quantum theory, equation (13) implies that the
decrease of the probability of finding a particle in some volume is accounted
for by the flow of probability current through the surface enclosing the

volume. For a stationary state, div J  0 and the probability density is
constant in time.
Expectation value of a dynamical quantity
The identification of  2 as probability density enables us to calculate the
average or expected result of a position measurement, performed on a
particle associated with wave function  . The expectation of a function

f (r ) of the position coordinates of the particle is given by
 f  


 
P(r ) f (r )dr 
 f




dr
.
2 
dr
2

The quantity <f > is a weighted average of the possible values of f (r ) , the

weight being given by the probability that the particle is located at r .
To find the expectation values of other dynamical quantities like linear
momentum, angular momentum, and energy we need to have some idea
31
regarding the state space and the action of various operators acting on
functions belonging to the state space. Quantum theory postulates that
dynamical observables are represented by Hermitian operators and the
possible results of measurements are the eigenvalues of relevant operators.
Operators and operator equations
We define an operator  as being a mathematical object which acting on
a function of x , say  (x) , turns into another function. The simplest example
is a multiplicative operator, where  is itself a function of x. Thus we might
have
 (x) = x.
The operator x acting on any function  (x) produces the new function x (x) .
A less elementary example is the differentiation operation. In this case  is a
function of  x , Â (  x ) . We may consider more general functions of the
type  (x,  x ). Consider, for example the operator
 (x,  x ). =

x.
x
For any function (x) ,
(

 ( x)

x) ( x)   ( x)  x
 (1  x ) ( x) .
x
x
x
So, we have the operator equation


x  1 x .
x
x
An operator equation of the form



Aˆ ( x, )  Bˆ ( x, )  Cˆ ( x, )
x
x
x
implies that for any  (x) , we have
[ Aˆ ( x,



)] ( x)  [ Bˆ ( x, ) ] ( x)  [Cˆ ( x, )] ( x) .
x
x
x
32
Eigevalues and eigenfunctions
Associated with any operator Â(x,  x ) is a set of numbers  n , and a set
of functions n (x) , defined by the equation
Â(x,  x ) n (x) = n n (x) ,
(14)
where  n is an eigenvalue and n (x) is the corresponding eigenfunction. An
operator acting on its eigenfunction gives the same function multiplied by a
numerical factor. Equation (14) is the eigenvalue equation of the operator
Â(x,  x ). The eigenvalues depend on the boundary conditions imposed on
the solutions of equation (14).
The commutator of two operators  and B̂ is defined by
[ Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ .

In general, [ Aˆ , Bˆ ]  0 . For example, suppose Aˆ  x, Bˆ  . Then, for any
x
function  (x)
[ x,





] ( x)  ( x  x) ( x)  ( x  1  x ) ( x)  (1) ( x) .
x
x x
x
x
We thus conclude that
[ x,

]  1.
x
Postulates of quantum mechanics in an elementary form
Quantum mechanics is developed on the basis of a number of postulates
which are given below in a somewhat loose manner:
(1) Each observable (or measurable dynamical quantity) is associated with a
linear operator. The operator acts on functions which represent states of the
quantum system. The form of the operator corresponding to an observable
can be obtained by appealing to correspondence principle. Since the
measurement of a dynamical quantity is certain to yield a real value, the
33
operator representing an observable must have only real eigenvalues.
Hermitian operators satisfy this requirement.
(2) The possible results of an observable A are the corresponding
eigenvalues  n of the operator Â. If the system is in eigenstate n (x) of the
operator Â. , the measurement will yield a certain result, namely,  n . The
average value of repeated observation of an observable A in an arbitrary
state  (x) is
 a 




 * ( x) Aˆ ( x,  x) ( x)dx



 * ( x) ( x)dx
Operators for linear momentum and total energy
Let us see how we can find the operator for the x-component of linear
momentum by exploiting the correspondence principle. This principle
asserts that the average motion of a well-defined wave packet must coincide
with the classical motion of the associated particle. So, the definition of the
momentum operator must be such that the equation
 pˆ  m
d
 xˆ 
dt
(15)
holds. We have seen that

 xˆ   x *dx ,

assuming  to be normalized.
(16)
Combining (15) and (16), we get
 pˆ  m



d 
 *


*
x


dx

m
(
x
dx   x *
dx)  2m Re[  x *
dx]





dt 
t
t
t
Considering the case of a free particle, we have
34

 2  2

.
 i
2
2m x
t
(17)
We thus get for a free particle

 pˆ  Re [i  x *

d 2
dx] .
dx 2
(18)
Integrating by parts the integral in (18), and using the fact that  goes to zero
as x   , we can write

 pˆ  Re [i  ( *  x

d * d
)
dx] .
dx dx
The quantity
d * d
dx
dx dx

 i  x

is purely imaginary and drops out of the expression for  p̂  . So, we can
write

 pˆ  Re [   * (i

d
)dx] .
dx
(19)
The integral on the right-hand side of (19) is purely real. This becomes clear
when we note that

2i Im[   * (i



d
d
d *
)dx]    * (i
)dx    (i
)d x


dx
dx
dx

d *
(  )dx  (i)[ * ]   0
  dx
= (i) 
So, we can do away with the symbol Re in (19) and write

 pˆ    * (i

d
)dx .
dx
(20)
If  were not normalized, we would have obtained
 pˆ 


d
)dx
dx
.

*


dx

 * (i

(21)

35
We assume the validity of (20) and (21) for a particle subject to a potential
V(x).
Equation (20) can be justified by adopting a different strategy. We start
with the free particle wave function in the form
 ( x, t )  A exp i(kx  t ) .
(22)
Since k  p / , we have
 ( x, t )
p

 i  ( x, t )  p[ ( x, t )]  i [ ( x, t )] .
x

x
Now




 pˆ    * ( x, t ) p ( x, t )dx    * ( x, t )( i

) ( x, t )dx .
x
(23)
Equation (20) or equation (23) indicates that there is an association between
the dynamical quantity p and the differential operator  i  x . The effect of
multiplying  ( x, t ) by p is the same as applying the differential operator
 i  x on  ( x, t ).
A similar association can be established between the total energy E and
the differential operator i  t . Since   E / , the differentiation of the free
particle wave function given by (22) with respect to t yields
E ( x, t )  i

[ ( x, t )]
t
so that E corresponds to the operator i  t . For a particle in a potential V(x)
p2
 V ( x)  E .
2m
Replacing p and E by their associated operators, we have

2 2

 V ( x)  i .
2
2m x
t
The operator equation signifies that when applied to some wave function
 ( x, t ) , the following differential equation will result
36

 2  2 ( x, t )
 ( x, t )
 V ( x) ( x, t )  i
2
2m x
t
which is just the Schroedinger equation. We thus conclude that postulating
the associations
p  i


and E  i
x
t
straightway leads to the Schroedinger equation. The expectation value of any
dynamical quantity f ( x, p, t ) is given by

 f ( x, p, t )    * ( x, t ) f op ( x,i  x , t ) ( x, t )dx ,

where the operator
f op ( x,i  x , t )
is obtained from the function
f ( x, p, t ) replacing p everywhere by  i  x .
References
1. R. Eisberg and R. Resnick, Quantum Physics.
2. D. J. Griffiths, Introduction to Quantum Mechanics.
3. J. J. Sakurai, Modern Quantum Mechanics.
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