So, we shall substantiate some of the things
that we have learnt ah in the previous class
and in order to do that let us start with
an experiment hm which is known as the Stern-Gerlach
experiment.
And this experiment is the first one ah to
actually establish the quantization of angular
momentum. And let us see how ah this experiment
ah actually ah talked about the quantization
of angular momentum and ah hence that all
these quantities or this physical observables
could take quantized values including energy
and ah other things.
So, ah they considered a simple setup which
is ah this is ah this is an oven and it contains
ah silver atoms. So, there silver atoms are
emitted in all directions and they are further
sent through collimators. So, that this thing
can pass through this and now, there are two
pole pieces of magnets. ah So, this is the
south pole, and it is a a specially designed
hm magnet where ah one of them is flat, ah
say the south is flat and the north pole has
sharp edges. And so basically these silver
atoms coming out of the collimator is ah they
undergo a deflection ah because of the ah
the inhomogeneous magnetic field that exists
in this region, and then they are collected
on the screen and they ah get deflected or
deviated like this, and they are ah so this
is a detector or a ah or a screen, and this
is a collimator and this part is the magnets
as told earlier.
So, this ah magnet let us see a schematic
ah representation of what this magnet does.
So, this a crosssection of the two magnets
here they look like this and there is a gradient
of magnetic field ah that ah is created and
this ah gradient of magnetic field. So, this
is say the Z direction. So, this dB dZ are
is actually negative ah which means that as
Z increases ah dB dZ decreases.
So, once again I just repeat that this is
an oven containing silver atoms ah we will
discuss this silver has 47 as the electronic
configuration they are made to pass through
they actually are coming out in all possible
directions. However, ah they collimated through
a series of collimators and then they are
made to pass through a magnet arrangement
ah which one of the pole pieces of the magnet
is say the south pole is flat while ah the
north pole is having pointed edges so that
one can have the inhomogeneous ah distribution
of the magnetic field as a function of Z.
ah And so basically this is ah the ah in this
figure. So, basically this is the ah you know;
ah and they are made to fall on the detector
hm and so they in principle as we discuss
that they should go and hit anywhere on the
on the screen here between two extreme values
ah pardon me. So, this is the detector. ah
However, they chose to actually impinge on
two ah distinct locations and how that corresponds
to quantization of angular momentum is what
we shall see.
So, then ah there is a ah the silver atoms
while passing through the magnetic ah region
where the magnets exist, ah there is the experience
ah force a force F and that F is given by
equal to minus gradient of U, where U is the
potential energy. Now, here U is equal to
ah minus mu dot B where mu is the magnetic
moment of the silver atoms.
So, if you put this u here. So, there will
be a minus a gradient of minus mu dot B which
ah since mu ah the magnetic moment can be
taken to be a constant. So, this becomes equal
to mu dot ah dB ah dZ say for example, and
this can be written in a scalar form such
as mu Z and a dB, dZ. So, this is the force
that is exerted on the hm on the atoms. And
ah we have shown that this del B del Z is
negative. So, if mu Z is a negative then we
will have a positive force ah which is exerted
on the atoms and maybe they will be shifted
up from this mean position let us call this
as O.
And ah in the other case when mu Z is hm positive
then there will be a negative force that will
be exerted on the atoms and they will be ah
pushed down which is below O on the screen.
So, this is the idea, and if we follow classical
physics then hm then mu Z is equal to a mu
a cosine theta where theta is the angle that
mu ah makes with Z axis. And so ah mu Z that
way mu Z should take continuum of values 
of values between ah minus ah mu to plus mu
because the cos theta has limits of plus 1
and minus 1. So, there should have been all
the atoms should have been distributed hm
between these two values, but however, it
it does not happen, what happens is that there
are two spots on the screen, so all these
atoms they go get deflected and they impinge
on the screen at two discrete points.
Now, what is happening and how is it related
to the quantization of angular momentum let
us try to understand that. So, basically ah
put things in perspective 
and for that silver has 47 electrons and of
course, will say that there is a nucleus and
how that ah comes into the discussion we will
just see. Now, the atomic theory says that
46 of those electrons have total ah have total
angular momentum J which is equal to L plus
S is equal to 0, also also the ah orbital
angular momentum of the 47th electron 
is 0 as well.
So, the lone survivor is the spin angular
momentum 
of the 47th electron. Thus what should happen
is that, so we should get ah signature of
the spin angular momentum of the 47th electron
on the screen if we do this experiment. And
moreover ah the nucleus 
makes very negligible contribution to the
angular momentum 
ah because of because of the ah large mass
compared to the mass of the mass of an electron.
So, the ah the magnetic moment of the silver
atoms silver is written by Ag hm is effectively
due to the due to the spin angular momentum
of a single electron. So, this should correspond
to, so what we mean by this is that ah the
the experimental data that one gets which
is a too bright spots on the screen where
the silver atoms ah they impinge on. So, this
data should correspond to to the intrinsic
spin angular momentum 
momentum hm which can take two discrete values
namely plus h cross by 2 corresponding to
maybe the ah spot above the mean level ah
and a minus h cross by 2 corresponding to
the ah the spot that appears below the mean
level.
And these are ah conventionally written as
as up, which is also written as a two component
spinner and ah down which is 0, 1. ah So,
this 
conclusively demonstrates a quantization of
of angular momentum, ok. So, ah, but now,
it is shown for matter waves or other particles
such as electrons and so on, but it also happens
for light waves such quantization of angular
momentum and that is related to its polarization
properties that is the direction in which
the amplitude is pointing at. And so there
could be right circularly polarized ah light
waves which are photons or there could be
left circularly polarized hm photons, and
they are associated with integer ah number
of the Plank's constant that is a left circularly
say corresponds to a plus h cross and ah ah
the right circularly ah polarized photon corresponds
to an angular momentum of minus h cross.
Thus, this quantization is not a a priority
for the matter waves, but it can also happen
for light waves. So, also the quantization
holds for light waves, ok. So, ah this is
the birth of quantum mechanics in which ah
some of the observables are related to the
measurement of subatomic particles ah are
found to be quantized. And then we have also
talked about the wave particle duality ah
at length and it was conclusively told that
in some experiments ah particles show the
particle nature of flight. Whereas, in some
other experiments they show the wave nature
of light and there is an ambiguity and ah
and without this the description is not complete.
ah
There are examples of particle ah nature or
the corpuscular nature in which ah the ah
their ah Compton effect and their photoelectric
effect which shows this ah ah particular characters
whereas, the interference and diffraction
etcetera they show the wave character. And
we have seen elaborately the Young's double
slit experiment which is ah basically which
is because of the interference of light waves
or the photons ah the waves associated with
the photons. ah So, we want to understand
that what are the ah other ah features associated
with the wave function.
So, let us get into a discussion called as
the wave function and its interpretation.
So, ah we are going to talk about hm mainly
what it means, what does a wave function mean,
but also there are issues related to superposition
or hm the specific directional properties
of the amplitude which are known as polarization
etcetera. ah However, ah some of these issues
can be left out of discussion for now, and
we may come back to it later or to them later
and right now, let us talk about what is a
wave function and what are its interpretation.
So, let us say there is a wave function. So,
this wave function is associated with the
motion of a particle. So, let it is a function
and it is a function of space variables x,
y and z or you can write it in terms of the
ah this spherical ah polar variables r theta
and phi or the cylindrical variables rho phi
and z and so on, and also it is a function
of time. So, if ah this is a wave function.
So, the question that we ask is that ah what
precisely does it describe, second question
is ah is it a measurable quantity?
And so in part in particular which features
of the particle 
are related to this wave function. But certainly
this wave function is related to the existence
of the particle because in some region if
psi is equal to 0 then of course, we say that
the particle is not there. But if it has to
be ah directly the measure of finding a particle
at a given space time point ah x, y, z and
t then this quantity should always be ah positive
and real. Now, there is no problem with that
this quantity can always be positive and real,
but we know that this quantity is complex
because for it if it is not complex then we
cannot explain the interference properties
that we have seen in the Young's double slit
experiment.
So, this is certainly not directly the measure
of a particle or the existence of a particle,
but it is somehow related to it. So, we will
write down that these observations.
It is not ah directly ah related ah or it
is maybe it is a better way of saying it is
that ah ah it does not by itself ah denote
the presence of a particle at x, y, z and
t. For that to happen 
it has to be always I mean always and everywhere
when I say always it means that at all points
of x, y, ah z and t ah always are positive
and real, but the problem is that, but a real
function cannot describe interference properties
we will come back to this.
So, psi must be in general 
a complex quantity. ah Please do not get this
message that psi always a complex quantity
we will solve Schrodinger equation and more
often than not we find psi to be a real quantity
ah the at least a special part of it and and
so on hm. And this does not violate any of
the things that have being told here about
the interpretation of the wave function it
can still be a real quantity, but ah in general
it is a complex quantity. So, that is why
this in general it is written. ah So, if it
is in general a complex quantity it cannot
be associated it cannot be, ah as it is associated
with the with the existence of a particle,
ok.
So, then what should we do then how do we
get this psi into perspective or ah make it
useful for understanding the behavior and
the motion of the particle.
So, this ah clue is obtained from classical
ah electrodynamics. ah Just like E square
that is a electric field square and H square
which are the electric and the magnetic energy
densities hm denote while ah the vector E
and the vector H are just mathematical ah
forms of the electric and the magnetic fields.
ah The measurable quantities or the quantities
which have physical interpretation are these
densities or when they are integrated over
volume there called as a electric electrostatic
energy or the energy associated with the electric
field and the energy associated with the magnetic
field and so on. ah So, that way ah we can
write down psi mod square 
to denote probability of finding the particle
at x, y, z and t.
In fact, this was one hm bottlenecks with
the birth of quantum mechanics many people
including hm some of the renowned scientists
at that time felt very uncomfortable with
the ah non deterministic doctrine or the idea
associated with quantum mechanics that it
has to have a probabilistic interpretation.
But however, hm after a lot of deliberation
and looking at many experiments and looking
at and the development of the field during
that time they had to ah adopt these the probabilistic
interpretation. And so this is the ah meaning
or the ah physical interpretation associated
with the wave function that a mod square of
that is related to the probability ah of or
it is the probability of finding a a particle
at a given a point a a or at a given space
time point
And, so if we want to connect these to the
probabilistic interpretation ah of the Young's
double slit experiment then. So, we can understand
that the interference pattern occurs due to
the statistical effects are coming from a
large number of photons hm which are interfering
from the two slits ah on the ah and impinging
somewhere on the screen, ok. So, psi square
is the probability of finding a particle at
a given place on the screen.
So, psi square is the 
particle somewhere 
on the screen 
and we are superposing a large number of photons
coming and interfering from the two slides
and making that pattern ah occurring on the
screen which is kept at a distance from the
slits. So, the appearance of the interference
fringes ah that depends on the passage of
the wave through both the slits, but now,
if these waves are associated with the particle
it implies that the particle has ah gone through
either of the slits, ok.
And if there is a careful detection mechanism
or a careful monitoring mechanism which finds
out that which slit has it gone through, then
the interference pattern will be wiped off
because then it becomes deterministic and
determinism does not give rise to this interference
pattern as we know.
And the way we can understand this is this
ah that suppose a careful detection 
finds out that the photon has gone through,
say slit 1. Then the intensity would be proportional
to a psi 1 square and on the other hand it
can also go through slit 2 in which case the
intensity ah of the pattern would be proportional
to a psi 2 square. So, the total intensity
is ah proportional to psi 1 square plus psi
2 square and there is no interference.
So, the interference can only come from ah
by superposing. ah So, actually 
the intensities 
to ah psi 1 plus a psi 1 plus psi 2 square
which has in addition to these two terms that
are written above it has a term like psi 1
star psi 2 and a psi 1 psi 2 star these are
the interfering terms which gives rise to
interference pattern. So, this interference
is very crucial and for that interference
to happen the photon hm has to be hm in deterministically
passed through both these slits that is, you
cannot determine which state it has gone through
and the wave function by themselves will interfere
and will give rise to an interference pattern,
ok.
So, ah this is, so by this you will not have
interference, but with this only included
one will have interference pattern. So, this
is the way the probabilistic interpretation
has shaped up the or the development of quantum
mechanics and its very important to say that
psi or psi 1 or psi 2 they refer to a single
photon here or a single particle for that
matter psi mod square is a probability of
finding a particle. However, for probabilistic
interpretation to be valid or ah give rise
to a valid description one has to statistically
superpose many such events and this is what
we have ah seen even earlier that a large
number of photons actually come and they interfere
and they give rise to the pattern.
The other thing that is important is this
ah uncertainty principle which we have ah
mentioned and we have seen that how superposing
ah many waves ah gives rise to a completely
deterministic hm position of the particle.
So, the uncertainty in the position of the
particle goes away and giving very large uncertainty
to the momentum or the velocity of the particle.
So, ah if now, we have said that psi is related
to the existence of a particle at a given
space time point, and so that ah given by
the hm the probability of finding a particle
at a space time point is given by psi mod
square hm x say we are talking about if we
talk about in one dimension this is the. So,
this is the probability of finding a particle
However, ah hm so what does the experiment
say on this? So, the experiment say that even
if you want to find out the ah position of
the particle there has to be an uncertainty
in the position which is not related to the
position other or the uncertainty or the measuring
ah device or other the list count of the measuring
device ah it is completely a sort of fundamental
uncertainty that cannot be avoided. So, say
delta x is ah the uncertainty in the 
in the position of the particle, and there
has to be an uncertainty in the canonically
conjugate variable which in this case in the
momentum uncertainty; so is the uncertainty
in the momentum of the particle.
And uncertainty principle says that this delta
x multiplied by delta p x has to be greater
than ah the quantity called as h cross. So,
if you want to localize your particle ah to
a great degree of accuracy you will have to
give up ah a large ah or, so the momentum
uncertainty of the particle will be very large
which means that the particle actually can
be moving ah from minus infinity to plus infinity.
ah And if you want to actually talk very precisely
about the velocity of the particle ah at ah
given ah space time then you have to give
up the notion of specifying ah with accuracy
with any degree of accuracy the position of
the particle. And the uncertainty in the momentum
and the uncertainty in the position ah is
related by this where h crosses a ah the planks
constant.
And similarly there are other uncertainty
principles as well like the uncertainty in
energy and the uncertainty in time is also
related by this. And the uncertainty in the
angular momentum multiplied by the uncertainty
in the ah angular variable that is canonical
to the angular momentum ah ah is also ah obeys
this relation. And these are absolutely fundamental
and therefore, nothing to do with the limitations
of the ah or the accuracy of the apparatus
that are used to calculate these things. ah
Just a small demonstration or rather ah ah
explanation of how one can ah find this ah
uncertainties all one can prove for a given
case.
Suppose one has a wave function which is given
by psi of x hm it is it could be a function
of t also, but let us just put that time equal
to 0. So, a momentum and the position uncertainty
delta x is written as ah x square average
or expectation minus x expectation average
and similarly a delta p x is equal to ah p
x square average ah or expectation minus p
x expectation square and each one of them
can be found. So, a x square can be found
from the wave function which is x square.
ah Let us just write it clearly.
X square and the psi mod square from minus
infinity to plus infinity dx that should be
equal to ah something which is hm which should
give me the ah the expectation value of x
square whereas, the expectation value of x
is equal to the same thing with x ah psi square
dx. Now, if psi has a form which is like a
symmetric form about 0 then of course, the
this term will be equal to 0 and similarly
the p x square ah can be written as. ah So,
the p x operator is equal to minus i h cross
del del x. Now, this is a hm representation
of the momentum operator in position space
this we will talk about ah soon, and this
can be ah obtained as psi of this you cannot
put all these psi's together because this
is a psi star and there is a minus h cross
square d 2, d 2 dx two ah and ah psi ah dx
from again from minus infinity to plus infinity
whereas, p x is equal to ah minus infinity
to plus infinity as psi star ah minus i h
cross ah d psi dx. ah
So, this is your ah p x square operator which
is related to the kinetic energy of and this
is your p x. So, one can use this ah for a
given psi of x one can use this formula in
order to calculate all these quantities such
as x square expectation and x expectation
and so on. And then hence calculate ah delta
x from this formula 1 and delta p x from this
formula 2. And then one should be able to
show that this is the ah at least of the order
of h cross in order to have the uncertainty
principle to be value.
So, these are by and large the foundations
of quantum mechanics. We have elaborately
explained ah the probabilistic interpretation
and the wave notion or wave particle duality
of ah of not only light waves or photons,
but also matter waves ah through examples.
And one of the main examples that we have
taken is the Young's double slit experiment
in which photons do interfere and show an
interference ah pattern ah on the screen.
And these are some of the ah important way
the quantum mechanics actually developed in
these days.
And why we are doing this is that some of
these key ideas are ah will be actually applied
to modern topics such as quantum computation
and so on. And finally, this uncertainty principle
which we say that it is related to the ah
two of very fundamental fact that these are
coming because of the with ah all these the
wave interpretation and so on and ah and the
probabilistic interpretation. So to say, ah
because we can still talk about the wave function
ah or a particle with a well defined momentum
k, and we can also talk about ah ah position
of a particle given by psi of x.
However, we know that for a probabilistic
interpretation to take place there we have
to superpose many of this particle a very
large number of them, and in which case ah
these spread in the position of the particle
and spread in the momentum of the particle
are inevitable. And that is why this ah this
uncertainty principle due to Heisenberg is
an inevitable ah thing which is coming out
of these fundamental ah postulates and interpretation
of quantum mechanics.
