Okay, so in this video we will talk about,
uh, Schrodinger's equation, which is the
basic equation of motion in quantum
mechanics. Okay. So, um, let's see, you
all remember the axiom of unitary
evolution, which says that,a quantum
system evolves by unitary rotation of the
Hilbert space. So, ah, by some unitary
rotation U. But we can ask, ah, which
unitary rotation ah, do you use? And this
is the answer that, this is a question
that ah, Schrodinger's equation answers.
So shouldn't this equation gives us the
quantum equation of motion. Alright, so
to describe Schrodinger's equation, we, we
must first focus on a particular
observable, the energy observable, which
is usually denoted by H. And it's, it's
called the Hamiltonian of the system. So.
The eigenvectors of age are the states of
definite energy, and the eigenvalues are
the corres-, the energies of the
corresponding states. So let's just
um, let's just do an example to um, get
ourselves straight on this. So, so, let's
say that uh, we go back to our example of
a hydrogen atom and the electron, right,
we think of. The electron is in one of,
one of several states ah. Um, and the
ground, ground state, which is the lowest
energy state. The first excited state and
so on up to the. If you, if you limit its
energy, then maybe there are only K
different orbitals that come into play.
And so, we have orbitals all the way
through K-1. Now. How did we pick these
states? Well, these are the states of
definite energy, this is a ground state,
it's a lowest energy state. That's the
next highest energy state, and so on. And
of course, you know, by the superposition
principle, in general, the state of the
system is a superposition over all of
these states. Okay. But, but now, when you
have a superposition like this, what is
the energy of this state? Well, we don't
know. It's in a superposition of states of
definite energy. And so, if we were to
actually write down an, the energy
observable both surface which right have the
Hamiltonian for the system, where, what
would it look like, the states of definite
energy of the system happen to be, the
basis state, zero one through K-1.
And so, and what are their energies? While
may be the energy of the ground state is E
note, energy of the first excited state is
E one and so on, up to E K minus one. And
so the Hamiltonian is a diagonal matrix,
in this case, and the eigenstates of this,
of this Hamiltonian, which are the basis
states of definite energy. And of course
the general state of the system is a
superposition over these, over these
eigenstates, it's a superposition over
these basis states, and those are not
states of definite energy. Here's another
example uh, it's, it's an example of a
observable we have seen before. It's,
let's say, let's say we call it the
Hamiltonian of the system. So it happens
to be, be this one. H here which is minus
half on the diagonal and five over two on
the off-diagonal. Now, this was a, this
was an observable which had two
eigenstates, plus and minus, and the
eigenvalue of the state plus was two, and
of minus, was minus three, and so this
would correspond to saying that, that uh,
that if this is, if this was the
Hamiltonian of the system, you would have
two states of definite energy, the minus
state which would have the lower energy
minus three, and the plus state with
energy of plus two. Okay, so now in terms
of this Hamiltonian, the Shrodinger's
equation is a differential equation that
tells us how the state psi of t, so, so,
we, we, we are, we want to know the state
of the system at time t. Which we denote
by psi of t, we want to know how it
evolves in time. So for instance we might
be given, we might be given. Psi at times
zero. And the Hamiltonian of the system,
and we want to know. What's the state of
this system at time t? Say after, after
one unit of time, or ten units of time.
And, so this is described by, by
Schrodinger's equation, which is a
differential equation, which says d by dt
of psi t, is proportional to H times si of
t. So that's a very funny thing. It's a,
it's a diffe rential equation which
involves an operator h. Okay, so how do we
even come to grips with this? So we'll see
how to do this um, in this lecture, but
then we'll, over the next lecture or two,
we'll get more and more of a, of a feel,
an intuitive feel for what Schrodinger's
equation is, so we'll study from two or
three different viewpoints. Um, the other
thing here is, there's, there's a very
important thing here, which is, which is
that it's i times d psi by dt. Where i is, of
course, uh, the square root of -1. And
this is quite important, because this is
what leads to the unitary evolution. And
finally, h bar is, um, is a reduced plank
constant. It's h, uh, h over two pi where
h is a planks constant. Which is a
physical constant reflecting the size of
energy. Okay, so. So now, how do we
actually understand Schrodinger's
equation. How do we solve Schrodinger's
equation? So we want to uh, we are given
psi at time zero, and we want to
understand what psi at time t looks like.
Well, it turns out that the key to solving
Schrodinger's equation is understanding
the eigenvectors of h, the states of
definite energy. So they evolve in a very
nice way. So let's say that. Uh, initial
state psi, psi at time zero was, was, was
one of the eigenstates. Phi, phi sub J of
this, uh, of the Hamiltonian. And let's
say that the corresponding eigen value was
lambda sub J. So this the energy of the,
of the eigen state phi sub J. So then, the
claim is that, that our state at time t is
just, it's still phi, phi sub j, the same
eigenstate but with a, with a phase
associated with it, e to the minus I
lambda sub j t over h-bar. Okay, so, uh,
so I hope everybody, everybody understands
what, what a, what I mean by phase. It's,
it's a unit complex number, so if we are
looking in the. In the complex plane, this
is the real axis, and that's the imaginary
axis, and that's the. Unit circle. This is
a unit circle. This is one. This is i.
This is minus one, minus i, and If you
look at a number here, angle theta, then
it's e to the. i. Okay. Alright, so, so
that's, that's um, that's as far as ah,
this, this phase is concerned. So, what's
this telling us? It's, it's telling us
that if you start off in an eigenstate of
h. Then, then the state evolves by, by
just a change of phase. Right, so, so the,
the over all phase in front of these,
these, these state changes but, but
that's, but that's all happens, and, and
it, it, the phase rotates at an, at a rate
that's proportional to the energy, So the
higher the energy, the quicker the, the
phase rotates, the lower the energy, the,
the slower it rotates. Okay, so, so now,
how do we, how do we see this? Well, since
phi sub j is a, phi is a, is a, is an
eigenstate of uh, of h. h 5 sub
j is just lamba j 5 sub j. Okay.
So, so, what, what this means is that the
change in, uh, the differential change in,
in, in the state, at time, t, is just
proportional to five sub j itself. And so
what, what, what this implies is that at
every time. The state is going to still
point in the, in the direction of i sub
j. The only question is, what's the
amplitude? Right so, so this implies. That
psi of t. If you start with sine of zero
equal to five sub j, the eigenstate, then
sine time t must, must look like some
constant, a of t five j. Okay, so
now, now we can go back and solve our
differential Equation. So we have. i, h
power. d by dt, of a of t. Pie sub j is
equal to h psi of t which is h
times a of t. 5j, [inaudible]. This is a constant which
we can pull out. So that's, a of t,
Lambdas sub j, psi sub j. And psi sub j, of
course, doesn't, there's, there's no time
variance and so, so what we, what we get
is d a of t, just rearranging by a of t is
equal to lambda sub j over h-bar, divide
by i, which is minus i. Dt. Okay, so I'm
just rearranging this. And this is a
simple differential equation. We all know
how to, how to solve. It's just, uh, it
just gives, uh. a of t is e to the minus i
lambda jt over h bar. Okay, which is what
we wanted. So now we understand, um, how
the eigenstate, how an eigenstate of, of
h, evolves under, um, in time. So, what
Shrodinger's equation tell us is
that, is that, if you start with an eigen
state of h. Then, the state always points
in the same direction. All that changes is
the phase. And the phase changes via this,
this precession rule. It, it precesses at
a rate proportional to lambda j, which is
the energy of that, of that eigenstate.
So now, of course in general, our starting
state is not going to be an eigenstate,
it's going to be a superposition of
eigenstate. So we might start with a, with
a starting state superposition of, of,
over phi sub-j, with amplitude alpha
sub-j. So then, of course, paralinearity,
this data at time t is just going to be
the sum over j of. Is always, of course,
sum over J of alpha J. E to the minus I.
Lambda JT over H bar. Five sub J. So,
basically, each eigen state, or each, each
state of definite energy. It sort of stays
invariant in time. Except, it precesses,
right? It, uh, each, each eigen state is
precessing at its own rate. Uh, it marches
to its own drummer. Which drummer? Well,
it's its energy. And that's, that's the
basic equation of motion in quantum
mechanics. Okay, so now, we could also try
to write this out. As an operator. So,
what, what, you know, if you want to know
how to get from. The state at time zero to
the state at time T. Well there's a,
there's clearly a linear operator that
tells us how to map from time zero to time
T. And if you write it out in the, in the
eigen bases of H, it is a diagonal matrix.
Because each eigen state sort of stays
invariant except for phase changes. And so
it's a, it's a diagonal matrix with
diagonal entries E to the -I lambda JT
over H bar. And so. Now this is, this is
clearly a unitary matrix. Right? Uh, if,
if you, if you call this matrix, if you
call this matrix U. Then clearly U
satisfies U  U dagger= to U dagger U is
the identity. Okay. So another way you can
write down this , the axiom of unitary
evolution is you can just say, well.
Evolution is by schneider equation. If you
solve schneid er equation you get this,
you know, you, you get unitary
transformation for evolution and time.
Okay, so sorry, I should call it T off T,
same here. And a, the other thing we can
do is a. You know, U of T is, is now E to
the minus I H T over H bar. So this is
just notation. You can think of it as, um,
you know, E to the, um. E to the A is a,
is a, is a matrix. If, if you write, um,
um, a matrix B equal to E to the A. Then B
has the same eigen values as A. So, sorry,
B has the same eigenvectors as A. Um.
Eigenvectors, but the eigen values of B
are, expla, the exponential of eigen
values of A. So if, if one of the eigen
values of A. Is lambda sub J, then the
corresponding eigenvalue of A of B. ,, .
Is E to the lambda , okay? So this is
just, um. You know, you can take it as
notation. You can, um, uh, think about it
as a, as a expansion. But, um, you know.
Uh, just think of it as, as notation for,
uh, um, uh, for matrix expo-,
exponentials. Okay, so finally, let's do
an example. Suppose our hamiltonian is, is
x, the bit, the bit flick, uh, matrix, and
we start off in the, in, in the zero
state. And what's the state at time T?
Well, so, we all know the eigenvectors of
x are plus and mi- uh, plus and minus uh,
with eigenvalues plus one and minus one.
So what we need to do is first write psi
of zero in the eigenbasis, so, so psi of
zero is one over square root two plus,
plus one over square root two minus. And
so, psi of t. Is going to be equal to one
over square root two. And what does plus
evolve to? It evolves to eight to the
minus I, lambda is one, T over H/. Plus
one O by square root two. E to then minus,
minus I lambda, lambda in this case is
minus one, so we get a plus. T over H bar
minus. .