In the last lecture, I mentioned that we will
study quantum mechanics by studying 5 postulates,
we have looked at two postulates so far. So,
let’snow look at the third postulate. This
postulate is about the results of a measurement
in quantum mechanics. Recall that in classical
mechanics, if one measures an observable quantity,
Let’ssay we measure energy of a particle,
we can get any possible value of that energy.
There is no restriction on the values that
one obtains when one mix a measurement on
a classical particle.
However, that is different in quantum mechanics.
And that is what postulate 3 is about. So
the statement of postulate 3 is that in any
measurement of an observable which has the
associated operator A the only values that
one will ever obtain are the eigenvalues of
A. So, let’slook at what eigenvalues mean
and what one can get when you make a measurement
on A.
So, the eigenvalues of A are these constants
here. Let’s examine that. So, suppose you
have an operator A in general, when that acts
on function f you get another function g.
However, there are certain functions, which
when you operate with the operator A for example,
let’ssay the function b, then you get a
constant times b then this function b is called
an eigenfunction of A .
And this constant here is called the eigenvalue
corresponding to the eigenfunction b. So,
all functions are certainly not eigenfunctions
of an operator, let say A, but the ones that
are that satisfy the equation written here
are called eigenfunctions. So, coming back
to the measurement of the observable corresponding
to the operator A, you see here that we have
solved the eigenvalue equation of the operator
A.
These phi are the eigenfunctions and this
constant A is the eigenvalues and you see
of course, that there are multiple eigenvalues
and eigenfunctions denoted by n = 1, 2 and
so on. Now, the postulate states that when
we make a measurement, the outcome of the
measurement will be one of these values a
1, a 2, a 3 and so on. So, you cannot get
any value like in classical mechanics, there
is a restriction on the values that you can
get.
And in particular, the values that you can
get are eigenvalues of the operator A. The
equation that you see here is called the eigenvalue
equation of A. So, you solve the eigenvalue
equation of A and that tells you what possible
values you can get when you measure the observable
corresponding to A. Now, note here that it
does not matter what the wave function of
the system is.
It just depends on the operator. So, if you
are measuring the energy, the eigenvalue corresponding
to energy or what you can measure, when you
make a measurement of energy on the system,
it does not depend on what the wave function
of the system is. But the wave function as
we have seen contains all information about
the system. In particular, it will tell us
which eigenvalue is obtained when you make
the measurement.
So, let’slook into that question. So, here
we have rewritten the eigenvalue equation
again and the eigenvalues of the operator
are a 1 a 2 a 3 and so on. The question is
which eigenvalue is actually obtained when
you measure a. In general there is uncertainty
about which eigenvalue is obtained if we make
a single measurement on a single classical
system. Now, this of course, is not contrast
with classical mechanics.
Where if you have multiple replicas of an
identical system, then when you go and measure
a property, you will get the same value of
that property when you make a measurement
on two identical systems. This is different
in quantum mechanics. And therefore, quantum
mechanics is very non intuitive, that even
when you have identical systems, or if you
imagine identical replicas of a system, and
you make measurements on those systems.
You can actually get different eigenvalues
when you make a measurement. So, there is
uncertainty about which eigenvalue is obtained.
However, even though these eigenvalues would
be different in these multiple replicas, there
is something which is certain and that is
that these eigenvalues have very definitive
probability for occurring. So, for example,
if you call the probability of obtaining eigenvalue
a as p of a 1.
Probability of obtaining eigenvalue as p of
a 2 these probabilities have a distribution.
So if you plot this, and you write eigenvalue
a 1, a 2, a 3, a 4 and so on this axis, and
on this axis, you plot the probability of
obtaining a certain eigenvalue, let’ssay
a i, then it could be that a 1 has this probability
a 2 has that probability a 3 has a certain
different probability a 4 might have 0 probability
and so on.
This information is contained in the wave
function. And this distribution of eigenvalues
is absolutely definite. So, although the result
of each individual measurement is uncertain,
and is not definite, when we make a large
number of measurements on multiple replicas,
then the distribution of probabilities of
each eigenvalue is completely definite. That
is what the next postulate is about. It tells
us what this distribution of probability will
be.
So, let’slook at postulate 4 let me mention
that we will consider a slightly less general
statement of this postulate in this course,
because that will be sufficient for our purpose
of understanding spectroscopy and it will
save us from some more complicated mathematics.
So, the statement that we will consider as
postulate 4 is that, when we make a measurement
of the observable corresponding to the operator
A.
And the wave function of the quantum system
is represented by the normalized wave function
psi then the average value of the observable
obtained is given by the integral here, which
is psi star A psi d tau integrated from - infinity
to infinity, which in shorthand notation is
written like this, the Dirac notation which
we have seen before in the previous lecture,
or you know the notation which denotes average,
which is here on the left side.
The point is that the wave function contains
information about the probability of every
individual eigenvalue. So, you could have
a distribution of eigenvalues and this could
be how the distribution looks like of the
different eigenvalues. Now, the wave function
contains the information about what these
different probabilities are but we will restrict
ourselves to only now what the average value
of this distribution is. And that is what
this form of postulate 4 is telling us.
To summarize postulate 3 and 4, when a measurement
of a property is made on a quantum system,
the only values obtained are eigenvalues of
the operator corresponding to the property
we are measuring. The wave function contains
information about the properties of the system.
In particular, it tells us what the distributions
of eigenvalues are and therefore, what the
average value of the property is.
The postulate tells us what this average value
of a particular property is. So, again if
you measure let say energy of a quantum particles
it will tell you what the average value of
that energy is. The energy itself might have
different values and it would have a distribution.
postulate 4 tells us what that average value
is. If psi happens to be an eigenfunction
of the operator A.
Then the measurement will yield only one value
corresponding to the eigenvalue in that case,
there is no uncertainty let’s understand
this. So, if you have the operator A and let’s
say phi i is an eigenfunction with eigenvalue
a i. Now, suppose the wave function of the
system is happens to be some phi 3 here the
eigenfunctions of the operator or let’s
say phi i = 1 2 3 and so on.
Now, suppose the wave function of the system
happens to be phi 3, then if you make a measurement
of the property A on the system with a function
phi 3, the eigenvalue that you will get or
the result of the measurement that you will
get is a 3 and only a 3. So the average value
which is returned as phi 3 a phi 3 will be
phi 3 a 3, which is a constant time phi 3,
this constant comes out of the integral.
So you have a 3 phi 3 phi 3 and because phi
3 is normalized this is simply equal to a
3. So, the average value is simply a constant
in this case, and every time you make a measurement
of the property a on this system, you are
going to get the value a 3 interestingly this
average value that we have been talking about
is sometimes referred to as expectation value.
Now, these are just synonyms.
And average value is the more accurate term,
because when you make a measurement, you expect
to get a distribution of values not this average
value, but because of convention, the average
value is also referred to as expectation value
and it is a commonly accepted terminology.
Let’s now look at postulate 5. This postulate
tells us about the time evolution of a quantum
system. Recall that in classical mechanics,
the time evolution of the position and the
momentum of a particle was given by Newton’s
laws in quantum mechanics, the time evolution
of the state of the system or the wave function
which is a function of position and time is
given by the Schrodinger equation.
Which is the following here i h bar del psi
/ del t = H psi. H as we have talked about
before is the Hamiltonian operator which corresponds
to the total energy, the kinetic energy and
the potential energy. So, this is the total
energy operator. So, we can see that the Hamiltonian
operator is particularly important for quantum
mechanics, it tells us how this wave function
evolves in time.
The left hand side of this equation is the
first derivative with respect to time and
the right hand side is the Hamiltonian operating
on the wave function.
Let’s consider a special case which will
be particularly important in the study of
chemistry, this is the case when the Hamiltonian
does not depend on time. These systems are
called conservative systems, because you can
show that in this case the average energy
of the system does not change with time. So,
for systems where the Hamiltonian is, time
independent, let’s see how the wave function
evolves in time.
Let’s try the following special form of
the wave function here and see whether it
is a solution. So, the special form is that
this wave function which is a function of
position and time is written as a function
of only position and multiplied by a function
of only time. Let’s substitute this into
the Schrodinger equation and see whether this
can be possible solution. So, substituting
this in the Schrodinger equation, we get my
i h bar del psi / del t = H psi.
That is the Schrodinger equation and if I
substitute I get i h bar del phi r chi t / del
t = H of phi r chi of t. Now, you notice that
the function phi of r depends only on the
position and therefore, this derivative of
time does not operate on it. So, you can take
the phi r out of this derivative and write
this as i h bar phi r del chi t / del t on
the right hand side the Hamiltonian does not
depend on time.
So, this function which is a function of only
time chi of t can be taken out and move to
the left like this and you get chi t = H of
phi of r. Now, we collect parts with the same
variable on each side. So, on the left hand
side let’s collect terms which are functions
of time. So, chi t / del t, 1 over chi t on
the left side, and on the right hand side,
we have 1 over phi of r H of phi of r.
Now you notice that the left hand side of
this equation depends only on time and the
right hand side depends only on position now,
the position and time are two independent
variables, they will vary independent of each
other and not be connected. So, for this equation
to be true where one part depends only on
time and the other part depends only on position.
The only way that this can be satisfied at
all times is if these individual parts.
The left hand side and the right hand side
are both constants. So, let’s say they equal
to a constant which we will call E. Now, let’s
see what we get by equating the left hand
side and the right hand side separately to
this constant E. if we consider the left hand
side equal to a constant. We get i h bar del
chi t / del t, 1 over K of t = E and this
is del chi t / del t = - i / h bar E of chi
t.
This is now a first order ordinary differential
equation, where you have a function chi, which
when you take a first derivative gives you
a constant times the function chi. Now, you
know that a particular solution of this can
be chi of t = e to the power of - i E / h
bar t. So, you have this constant here multiplied
by t, this will be a solution of this equation
and you can very easily check by substituting
it back into this equation.
If you substitute the solution here on the
left hand side, you will get the answer. So,
here is the solution which works for this
equation. Now, if you take the right hand
side to be constant we get 1 over phi r H
of phi r = E which is saying that any phi
r which is an eigenfunction of the Hamiltonian,
you see this year that will satisfy the original
Schrodinger equation.
So, a solution of the Schrodinger equation
psi of phi r, t = phi of r multiplied by e
to the power of - i Et / h bar where phi of
r is an eigenfunction of H that is this statement
and you have the time part which is just here,
which is sometimes called a phase factor.
We have seen that the following form of the
wave function is a solution of the Schrodinger
equation i h bar del psi / del t = H psi when
the Hamiltonian does not depend on time. So,
here these phi are just eigenfunctions of
the time independent Hamiltonian and the E
is the corresponding eigenvalue so, here is
the eigenfunction and this E is the eigenvalue.
Now, you can very easily show that a wave
function of the form phi r , t.
Which is a linear combination of the following
form j C j phi j of r, e to the power of - i
E j t / h bar is all also a solution of the
Schrodinger equation. I have used the variable
j instead of i, because there is a i here
which corresponds to the imaginary number
i. So just to avoid confusion I have used
E j. Now, this result that wave function of
this form is a solution is a fairly powerful
result.
Because it gives you a way, to write the time
evolution of any general wave function. For
example, let’s say the r wave function at
a time t = 0 for a system is f of r. Now,
you can write this wave function as a linear
combination of the eigenfunctions of the Hamiltonian,
because we have seen in postulate 2 that the
eigenfunctions form a complete basis 
then the time evolution of this wave function
is really simple.
Because you have f of r time t, you have to
take this linear combination and just add
the appropriate phase factors like this. We
have seen that this will be a solution of
the Schrodinger equation.
So, if H does not depend on time, it is very
easy to solve the dynamical equation for a
quantum system. The solution effectively reduces
to solving an eigenvalue equation for the
time independent Hamiltonian. This eigenvalue
equation is called the time independent Schrodinger
equation and sometimes in the context of quantum
chemistry, this equation is just called the
Schrodinger equation.
So, you have to know whether Schrodinger equation
refers to this equation or whether it refers
to the dynamical equation just based on context.
Note that the wave function always evolves
in time even if the Hamiltonian is time independent,
the wave function evolves in time.
We will now look at a further special case
of conservative systems. This is the situation
When the wave function is an eigenfunction
of the Hamiltonian and it is a very relevant
situation for chemical systems now if the
wave function at some time say time t = 0
has this eigenfunction phi of phi n of r where
this phi n is an Eigen satisfies the eigenvalue
equation of the Hamiltonian.
Then we have seen that at time t, the wave
function has the following form. So, it is
the wave function at time t = 0 multiplied
by the phase factor which tells you what this
wave function will be at a future time t.
Now the probability density in this case very
interestingly psi star psi does not depend
on time and that is because when you take
complex conjugate psi star, the time part
becomes e to the power of + I E n t / h bar.
And that cancels with the time part in psi
and this wave function becomes independent
of time. Because this probability density
does not depend on time, such states are called
stationary states, we will use the idea of
stationary states throughout this class. The
other property of stationary states is that,
if we make a measurement of a property, which
has an observable, which is time independent.
Then we get the same value at all times, the
property measurement does not depend on time.
We can contrast this with a situation where
the wave function is not an eigenfunction
of the Hamiltonian, but a linear combination
of eigenfunctions of the Hamiltonian. Let’s
say we take a simple situation where it is
a linear combination of two eigenfunctions
phi 1 and phi 2 then the probability density
looks like psi star psi = c 1 phi 1 r square
+ c 2 phi 2 r square + the cross terms.
C 1 star c 2 phi 1 star r phi 2 r e to the
power of i E 1 - E 2 t / h bar + c 1 c 2 star
phi 1 r phi 2 star r e to the power of – i
E 1 – E 2 t / h bar. We can see here that
the probability density does depend on time.
Unlike the situation where, in the case of
stationary states, the probability density
did not depend on time.
