Letís our formalism of quantum mechanics
but today, let me start by comparing classical
physics with quantum physics and write down
what the differences are between the two and
how you make a transition from one to the
other. So letís go back and look at Hamiltonian
mechanics because the whole of quantum mechanics
also presumes that you have a Hamiltonian
for the system and quantum mechanics is really
concerned with evolution in the presence of
a Hamiltonian.
So classically you start by defining dynamical
variables. And these dynamical variables are
of two kinds. There are q 1 q 2 etc and then
there are p 1 p 2 etc. so they go in canonically
conjugate pairs. And then these dynamical
variables satisfy Poisson bracket relations.
The Poisson bracket relations are {qi, pj}=
delta ij and {qi, qj}= {pi, pj} = 0. Now of
course no dynamics has been put in. so far,
it just says you have a set of variables and
they obey these relationships among each other.
The dynamics comes in in prescribing the time
evolution equations. So you introduce a Hamiltonian
which is a function of q and p. There exists
a Hamiltonian which is a function of the dynamical
variables such that qi dot is delta H over
delta pi and pi dot is - delta H over delta
qi.
This is the Hamiltonian formalism in classical
mechanics. the statement is that in phase
space, in the space of the qís and pís,
specifying initial data corresponds to specifying
a point in this phase space after which you
are supposed to solve these equations and
if this is autonomous Hamiltonian, then the
trajectory of the point that you started with
in phase space will tell you what the future
values of the dynamical variables are.
And therefore use this information together
with initial conditions to find out what happens
to dynamical variables as time evolves or
elapses. Now the important thing to note here
is that there are two pieces of information.
One of them is this dynamical piece of information
which tells you what the equations of motion
and how things evolve in time.
The other piece of algebraic information is
the input. It tells you a basic algebraic
relation in this phase space between the qís
and the pís. And it is not hard to see that
these relations are valid at all times. in
other words at any instant of time, qi at
time t with pj at time t is delta ij if you
start with that, it continues so. So the Poisson
bracket relations themselves remain preserved
as time goes along. The qís and pís adjust
themselves such that they always obey the
Poisson bracket relationship which is why
you could use any of the points as the initial
state. So this is very important to understand
there are two sets of relations. One is the
structure, the Poisson bracket, the canonical
variables and the other is the evolution equations.
Now when you come to quantum mechanics, our
assumption is going to be that the algebraic
information is replaced by self-adjoint operators
acting on the vectors of some Hilbert space.
So, just as you had phase space in classical
physics, in the quantum version itís no longer
a phase space. Itís meaningless to talk about
points in phase space due to the uncertainty
principle. So this is replaced by the Hilbert
space. So the appropriate space in which things
happen in quantum mechanics is the Hilbert
space of states. We still got to say where
a state fits into this. But the dynamical
variables are replaced by self ñadjoint operators.
There an analog of Poisson bracket relations
here and that is a relationship between the
operators and they are replaced by commutator
brackets. Then a commutator of two operators
is the difference of operating with the two
operators, one on the other in different orders.
So the commutator here would be [qi pj] which
is a shorthand for (qi pj- pj qi). if these
are matrices, then it says you have two matrices
A and B and the commutator AB - BA is also
a matrix. So what happens in quantum mechanics
is the Poisson brackets are simply replaced
by the commutators divided by ih cross. We
need that factor there for a reason i will
come to it.
And this is equal to delta ij but then these
are operators. So there must be an operator
on the right hand side and the appropriate
operator here is the unit operator. It doesnít
do anything just acts on a state and leaves
it as it is. But the i and j are the labels
which specify the dynamical variables. They
could be Cartesian components, labels of the
dynamical variables, the phase space variables
in the classical case or the canonical variables.
So of course the other two commutators are
0. So [qi, qj] = zero and [pi, pj] = 0. These
two are also part of the algebra.so algebraic
relationships between the operators replace
the algebraic relationships for a canonical
variable. The Poisson brackets are replaced
by commutators. The answer why i need that
ih cross is technical. These quantities are
physical quantities which are real variables.
The corresponding thing here would be self-adjoint
operators. But you see the commutator of two
self-adjoint or Hermitian operators is not
Hermitian itself. Suppose you have a matrix
A and B which are Hermitian, they both would
represent some physical operator physical
quantities.
Then [A , B] =C, which is some other operator.
Then C is AB ñ BA. But C dagger = B dagger
A dagger - A dagger B dagger. But thatís
equal to BA - AB because B and A are themselves
Hermitian. But this is equal to- C. so a commutator
becomes anti-Hermitian. On the other hand,
if itís a physical quantity, the combination
AB - BA is also an observable. Then you must
make it Hermitian. Thatís the reason why
you need this i here because the moment you
put the i, that also changes sign and it makes
this quantity a Hermitian quantity. The h
cross is there for dimensional reasons. Because
you can see that the product of any generalized
coordinate and the corresponding conjugate
momentum has always got the dimensions of
energy multiplied by time. And there is a
fundamental quantum constant called Planck's
constant which has the same dimensions. So
this makes it dimensionless here. You could
ask why not minus ih cross. + ih cross turns
out to be the right prescription and we will
see as we go along what would happen if you
had a minus here.
So the algebraic structure in classical goes
to the algebraic structure in quantum. The
next question is what about these equations.
You again have a Hamiltonian. The presumption
is you again have a Hamiltonian which is a
function of qís and pís, except these are
operators and therefore the Hamiltonian is
also an operator. And you can immediately
know that if q and p are complicated operators
like derivatives and so on and then you take
functions of these derivative operators, you
can get fairly complicated operators.
Suppose q and p involves differentiation of
the state vector, this H would be some function
of this derivative operator, perhaps e to
the power d over dx and so on. But they are
not the analogs of the classical equations.
There are no direct analogs of these equations.
As it stands, the equations are going to be
slightly more complicated and we will see
what these equations become. So whatës the
time evolution of a quantum system and how
is it described? Now keeping this in mind,
letís go back and look at what we said about
expectation values and then we are going to
get a hint as to whatís going to happen.
I pointed out that if you took a physical
observable and I asked what is the expectation
value or average value of this observable
at some instant of time in a quantum system
specified by a certain state, then we asked
the following question. The state of the system
psi of t is given to you at some instant of
time and now i ask the following question,
what is A at that instant of time. I explained
yesterday that if you make a measurement on
this system to measure this physical quantity
A, you are guaranteed to get one of the eigenvalues
of the system. Which one it is you canít
say a priory depends on the state of the system.
So now lets assume for simplicity of notation
that A has eigenvalues lambda 1, lambda2,
etc possibly an infinite number. Letís called
them lambda i. Eigenvectors phi 1 phi 2 etc
form a bases set in this Hilbert space of
the system. This is an assumption. There would
be operators for which you may not be able
to justify this but just to set the formalism,
let me assume this for the moment. Itís a
Hermitian operator or a self-adjoint operator.
It has these eigen values and these eigen
vectors. In other words A acting on phi n
is lambda n times phi n and they form an orthonormal
basis. That immediately implies that i can
expand psi of t at any instant of time in
the form summation over n, all the allowed
values of n, some coefficient cn times phi
n.
Now this operator A is perhaps like position,
angular momentum etc. Itís supposed to be
made up of the dynamical variables of the
qís and pís. To start with we assume there
is no explicit time dependence in A. After
all, if I want to measure the position of
a particle, that position x has no explicit
time dependence. It may depend on time after
you solve the equations of motion but there
is no explicit time dependence. On the other
hand, say let me measure x squared + y squared
+ t times z squared divided by some tau.
So i could put in t explicitly and then this
observable becomes explicitly dependent on
t. But letís start by saying A is a not explicitly
time dependent in which case it has some eigen
values and some eigen vectors. These eigen
vectors are found once and for all form a
basis set and i have an expansion of this
kind here. If itís an orthonormal basis,
this is a unique expansion. The time dependence
go into the coefficients. Imagine in three
dimensions, I fix my coordinate axis x, y,
z. these are the unit vectors and I take the
position of a particle which is moving and
I expand that in this basis. Then of course
the components are the ones that carry the
time dependence.
So you would say immediately that r of t = x
of t times ex + y of t times ey + z of t times
ez.
So itís these components and not the unit
vectors that carry the time dependence. In
exactly the same way these components carry
the time dependence here and they would change
from instant to another. To start with, the
basis is fixed. This implies that the probability
amplitude that the state psi of t is in fact
the unit vector phi n is given by this here.
The probability interpretation of quantum
mechanic says that mod mod cn of t whole square
is 
the probability that the system is in a state
phi n at time t. This is a postulate. Now
of course you may start with a state which
is not normalized to unity which presumes
that this state is normalized to unity. Because
you immediately see that if i do a psi of
t bra vector here, then by orthonormality
we know that psi of t psi of t = summation
over n mod cn of t whole squared. This follows
by orthonormality immediately.
Therefore this is a probability provided,
this is unity. But i give an arbitrary state;
there is no reason why it should be normalized
to unity. Just as i can expand this position
vector in terms of some vector along the x
direction, something along the y and something
along the z, it may not be unity.
Only if I make it unit magnitude, can I talk
about the magnitude of this vector as exactly
equal to x squared + y squared + z squared,
squared root. i have to first normalize these
unit vectors. I may not do it all the time.
i may start with an arbitrary state and you
may need to normalize it. You do exactly what
you do in statistical physics. You divide
by the normalization factor which in statistical
mechanics was called the partition function.
Because relative probability of a system having
an energy e in contact with the heat power
that inverse temperature beta was e to the
power minus beta e. But the absolute probability
was e to the power minus beta e divided by
the sum over all these betas. So you have
to keep track of that of the normalization
all the time.
So if not divide by the norm and then you
normalize. This is how you normalize any vector.
You take the vector, divide by its magnitude
and you get a unit vector always. So we will
assume that we have normalized this state.
We will see what happens if you donít normalize
it. If that is the case we can write a formulary
down for the average.
This immediately implies that A average is
equal to the weighted average summed over
n of all the Eigen values of A together with
the corresponding probability. When it is
in the state phi n, the value of A is lambda
n. you are guaranteed this is the value you
will produce and now you know the probability
with which itís in the state phi n. Its mod
cn squared. So itís immediately clear that
this quantity is mod cn of t whole squared
times lambda n. thatís the meaning of the
average. Once you give me a probability distribution,
the average is just the value at each point
in this the probability distribution times
the probability. Incidentally if this were
not true and it were not normalized, you simply
divide by n mod cn of t whole square in general.
To normalize it later, you just divide by
the total probability.
So we have here a formula which says A, which
is the average as a function of t 
and does not explicitly depend on time. So
this is summation lambda n mod cn of t the
whole squared divided by summation n mod cn
of t the whole squared. I would like to go
back and write it in terms of psi to get a
compact formula
cn of t is this. So I take its complex conjugate
and that gives you cn * = 
psi of t phi n because the complex conjugate
of this is the reverse. So letís put that
in. Now the average A of t is psi of t psi
n 
phi n psi of t divided by summation over n
psi of t phi n psi of t. The summation can
be moved in and we write this as psi of t
because that has nothing to do with the summation.
A summation over n lambda n 
phi n phi n psi of t ket vector.
Now this is psi of t summation over n phi
n phi n psi t. now you see the power of this
notation. This thing here is a number and
must have two angular brackets. So 
what I have done is to move this out of the
summation because we know that the sum over
n of these scalar products we will take sum
inside and sum one by one because itís a
linear vector space.
But this 
quantity 
is the 
unit operator. So the denominator becomes
psi of t psi of t, which is a normalization
factor as we expected. But in the numerator,
I canít take phi n phi n out and call it
the unit operator because of the presence
of the lambda ones.
But we also know that A acting on phi n = lambda
n on phi. So we use this relation in reverse.
So the lambda n phi n can be written as A
acting on phi n. But A is a linear operator
and when it acts on a sum of states, it is
as if you can sum those states first and act
with A. So you can move the A out of the bracket
and use the fact that this is a complete set
of states. So you get a psi of t A psi of
t and at the bottom, we get psi of t with
psi of t. this is a fundamental formula. You
have this A sandwiched between two states.
This looks like a diagonal matrix element
in a basis psi.
This is the reason why in quantum mechanics
we say matrix elements are expectation values.
Itís by this little piece of rigmarole that
you know that once you put in the probability
interpretation, it follows that the expectation
value of A apart from that normalization constant
below is in fact the matrix element of A.
Take the state vector psi operate with A.
you get a new state vector. Take the inner
product with the original state and that gives
you the expectation value.
It is very important to notice that A sandwiched
between the states is a very complicated object
here and cannot be taken as something that
can be normalized to one. So the whole purpose
of everything is to find these quantities.
And now this gives us a hint as to what these
equations of motion would look like. Whatever
they are, they should be such that when I
take average values, I recover the formula
here. So we will keep that in mind. Now you
could ask how does the average A of t change
as a function of time, i.e., dA over dt. Since
every element here is a function of t, we
need a rule now for how psi of t changes.
So I will start with that because there are
two ways of doing quantum mechanics at this
stage. There is an active way of doing it
which follows directly from this and its call
the Heisenberg picture.
And there is another way of doing it which
is called the Schrodinger picture which specifies
what the state vector does as a function of
time and these two are completely equivalent
to each other because they will lead to exactly
the same answer for these physical quantities.
That is the consistency check. So without
further ado, let me write the Schrodinger
equation down here to show you how the state
vector changes as a function of time. So we
replaced dynamical variables by operators.
We replaced Poisson brackets by commutators.
Next thing we do is we replace the phase space
by the Hilbert space of the system. We replace
knowledge of the phase space variables by
knowledge of the state vector of the system.
The phase space variables change with time
classically according to Hamiltonís equations
of motion. The state vector changes as a function
of time according to Schrodingerís equation
and that is the input here in the Schrodinger
equation.
Schrˆdinger equation says ih cross d over
dt on psi = H psi of t and itís a postulate.
This is not an eigen value equation 
because this tells you we have to define what
this quantity is but intuitively we know what
it is. Itís the state vector at time t +
delta t subtract the state vector at time
t divide it by delta t and you get some other
vector. This is given by the action of the
Hamiltonian on this state so it takes you
to some other state vector. After all this
is an operator the whole point is this is
an operator and it takes you to some other
state all together. When we discuss eigen
states of this operator, then we will see
that this equation reduces to an eigen value
equation but only for those eigen states.
In general, itís a first order differential
equation but an operator equation for vectors
in a Hilbert space. Our task would be 
to see how to represent this Hamiltonian and
solve physical problems here.
Itís a first order in time. In principle
if you treat this like an ordinary differential
equation this together with initial an initial
condition is needed to solve this equation.
The unknown here is the state vector psi of
t. so you have to specify state at some initial
instant of time. Itís an initial value problem.
So we will assume that we know the state at
some instant of time. We will see later how
we have to specify the states along with initial
condition psi of zero. If this H does not
have explicit time dependence, this is an
autonomous system.
Similarly if this H is a function of just
qís and pís, then it is very much an autonomous
system. There is no explicit time dependence
here in this H and if I change t to (t + delta),
d over dt doesnít change. So itís clear
that this whole thing is time translation
invariant. You could choose any instant as
the initial instant of time, just as you could
for an autonomous system. Now letís find
the solution of Schrˆdinger equation. Then
comes a question of whether it exists and
it can be written in this form. So for the
moment it helps to think of psi as a column
vector in some vector.
So the solution is 
psi of t equal to e to power Ht over ih cross.
So letís put the i up and these results in
e to the power - i Ht over h cross psi of
0. Please notice that you need to have a state
vector at time t and therefore that initial
vector is put on the right hand side and this
operator e to the power minus i Ht over h
cross acts on it from the left. So order starts
mattering here in 
quantum mechanics. Suppose I choose some arbitrary
number t 0. The solution becomes t - t 0 acting
on psi of t 0. This is a formal solution.
Now we know that Hamiltonian evolution preserves
the volume in phase space. That was Liouvilleís
theorem and that was one of the crucial inputs.
So phase space flow in classical dynamics
is like the flow of an incompressible fluid
in real space.
The analog would have to again do with the
probabilistic evolution. So this is just like
saying here that points donít disappear from
phase space. The fluid moves around and this
volume element doesnít change. In exactly
the same way, here too there is a conservation
which operates. The norm will be preserved.
So you can see that psi of t = 
psi of t0 e to the + i H (t - t 0) over h
cross. The i becomes - i but the matrix H
must become H dagger and it acts from the
right. H 
is Hermitian. Itís the Hamiltonian of a system.
So all the systems we are going to look at
is assumed to be described by Hermitian Hamiltonians.
When you include dissipation in quantum mechanics,
then you may need Hamiltonians which are not
Hermitian. The reason is dissipation would
mean the energy is not conserved. Things would
die down as a function of time, etc. But as
long as the eigen values of the Hamiltonian
are real, this can never happen. Damping would
involve imaginary components and for the movement
we look at systems with Hermitian Hamiltonians.
Therefore to write the norm, one can close
oneís eyes and put this on the left and this
on the right and you have psi of t 0 e to
i H (t - t 0) over h cross e to the power
- i H (t - t 0) over h cross psi of t 0.
I have e to the A e to the B = e to the (A+B).
This will be true if A and B commute with
each other. This is a complicated formula
and we are going to use this over and over
again. But e to the A e to power - A = e to
the (A ñA) = e to the power 0= 1. This is
certainly true because A and - A commute with
each other. And 
of course if A commutes with any scalar multiple
of itself and therefore e to the A e to power
minus A is indeed one. And this becomes a
unit operator and therefore this gives us
a preservation of norm. Thatís the analog
of the Liouville theorem which says the volume
element in phase space is preserved. Here
the quantum evolution says the state vector
preserves its norm as time goes along. Thatís
a consequence of the fact that the Hermitian
conjugate of this operator is its inverse.
It came out from that fact that the Hermitian
conjugate of that operator was just the inverse
of that operator.
Let us consider the operator e to the power
- i H (t - t 0) over h cross. Letís call
it something lets call it an operator U of
t, t 0. It depends on t and t 0 and happens
to depend on difference in (t - t 0) in the
simple case. And this operator takes the state
of a system from what it was at time t 0 to
what it was at time t. This operator is called
the evolution operator or the time development
operator. When I take its matrix elements,
I will call it the propagator because thatís
what propagates you from one time to another.
The time development operator has some interesting
properties as its stands. To get U dagger,
I have to take Hermitian conjugate everywhere.
So this becomes + i and this becomes H dagger
and of course these are real numbers and nothing
happens to them. But H dagger is the same
as H and therefore U dagger (t, t 0) = U inverse
(t, t0) or U U dagger is 1, which is the identity
operator. I have just left out the time arguments
for convenience. You call a matrix which has
these properties as unitary. So the time development
operator is unitary. It satisfies some important
properties among which unitarity is first
and foremost. The fact that itís unitary
leads to the conservation of this quantity.
If you identify this with the probability
total probability then it says the conservation
of probability follows from the unitarity
of the time development operator.
So you find a time development operator which
is not unitary. You know itís wrong so immediately
you know that probability is not going to
be conserved. So unlike classical physics
where Liouvilleís theorem doesnít seem to
play much of a role it just happened there
that it was volume preserving evolution. At
least in the elementary treatments it didnít
seem to play a very fundamental role.
In quantum mechanics, on the other hand the
unitarity of the evolution is very important
and you have to keep track of that at all
times. I will mention here that even if H
is time dependent explicitly and the Hamiltonian
changes from instant to instant while remaining
Hermitian, the time development operator is
not given by this formula as you can see even
in elementary differential equations, if this
coefficient becomes a function of time then
this is not exponential is not the solution
you need some e to the power some integral
and so on. The time development operator in
those cases would continue to be unitary.
We are going to look at some problems where
it becomes explicitly time dependent.
For instance I take an atom and I switch an
electric field on and off. Then of course
itís a time dependent Hamiltonian and I need
to know what the time development is. This
operator has the following property.
In time, suppose this is an instant t 0, this
is the instant t 1 and this is the instant
t 2, as time elapses this direction, then
you could ask what is U of t 2 , t 0 equal
to. And its immediately clear from this exponential
structure that it is equal to U of t 2 , t
1 and U of t 1, t 0 in that order. The order
is important. Of course here there is the
Hamiltonian which is time independent so it
doesnít really matter but in general, once
again you have to keep track of this.
U (t 2, t 0) says that the system propagates
from time t 0 to time t 2. The state evolves
from time t 0 to time t 2. If there is an
intermediate time t 1, then this evolution
operator which describes the evolution from
t 0 to t 2. So it goes from t 0 to t 1 first
and then it starts at t 1 and goes to t 2
in this order. You canít interchange these
orders except in the simplest of cases and
this is called a semi-group property. Itís
called a semi-group because you know when
you have a set of elements and you multiply
them together over a time, you get more elements
and that forms a group. But here the order
of multiplication is important. Itís a semi-group
it doesnít happen the other way. I emphasize
this because even when the Hamiltonian is
time dependent, this would still happen. Let
me point out what would happen if the Hamiltonian
is time dependent. Well itís a much harder
problem.
If the Schrˆdinger equation were given as
equal to H of t, psi of t, then the solution
is highly nontrivial. Letís look at it classically.
You have d over dt, some f of t = H of t,
f of t. Suppose f of t is just an ordinary
function, you have an equation of this kind.
If H were a constant, then you write e to
the power Ht and thatís the end of the story.
But whatís the solution now given initial
condition f of zero. f of t = 
e to the power integral 0 to t, dt prime,
H of t prime, f of 0. Thatís the solution.
Of course if H is time independent, this integral
just becomes t. but otherwise this is the
solution to this differential equation.
To verify this solution, you differentiate
and then you have to find the derivate of
this quantity as a function of t. you then
use this famous formula for differentiation
under the integral sign. It is not 
a solution for this problem here and the reason
is this is a summation. So itís like writing
e to power H at one instant and the following
instants and adding them all up. There is
no guarantee because of this property e to
the A, e to the B is not e to the (A + B).
Because H, at one instant of time may not
commute with itself at another instant of
time. Then you immediately run into problems.
So this is not the formal solution. When we
look at time dependent Hamiltonians, I will
point that out. But the evolution still is
unitary. Its still is true that you can write
psi of t as U of t, t 0, psi of t 0 with this
being a unitary operator with the semi-group
property. It is a much more complicated formula
itís called a time ordered exponential and
we will look at that a little later.
So the formal solution to the Schrodinger
equation looks fairly simply but itís not
all that simple because it involves exponentiation
of the Hamiltonian and this is always a nontrivial
task. You have to take a matrix and exponentiate
it which is simple. But if you take a differential
operator and exponentiate it, it becomes much
more complicated. So thatís the basic problem
in quantum physics that you need to find the
exponential of the Hamiltonian. But that is
nothing new because even in classical statistical
physics or in any statically physics, you
have to find e to the power - beta H.
So exponentiation of the Hamiltonian is in
fact the fundamental problem both in equilibrium
statistical physics as well as in quantum
mechanics and it continues to be case in quantum
physics theory. This is the reason for the
mathematical commonality that you need to
exponentiate the Hamiltonian theory. What
I will do next time is to start with this
equation here and show you why the Hamiltonian
plays such a fundamental role itís called
the time infinitesimal generator of time translations
and we will see why it is so fundamental.
Itís analogous to what it does in classical
physics but in quantum physics, there is an
even more fundamental role because it really
controls the entire state of the system as
you can see and we will see why.We still have
to make contact with the classical Hamilton
equations and see what are they are going
to be replaced by. We replaced it by the Schrodinger
equation but I would like to show that for
physical observables there exist differential
equations called the Heisenberg equations
of motion. They are true analogs of the Hamilton
equations of motion. Thank you!
