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Okay, so in this video we will talk about,
uh, Schrodinger's equation, which is the

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basic equation of motion in quantum
mechanics. Okay. So, um, let's see, you

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all remember the axiom of unitary
evolution, which says that,a quantum

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system evolves by unitary rotation of the
Hilbert space. So, ah, by some unitary

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rotation U. But we can ask, ah, which
unitary rotation ah, do you use? And this

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is the answer that, this is a question
that ah, Schrodinger's equation answers.

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So shouldn't this equation gives us the
quantum equation of motion. Alright, so

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to describe Schrodinger's equation, we, we
must first focus on a particular

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observable, the energy observable, which
is usually denoted by H. And it's, it's

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called the Hamiltonian of the system. So.
The eigenvectors of age are the states of

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definite energy, and the eigenvalues are
the corres-, the energies of the

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corresponding states. So let's just
um, let's just do an example to um, get

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ourselves straight on this. So, so, let's
say that uh, we go back to our example of

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a hydrogen atom and the electron, right,
we think of. The electron is in one of,

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one of several states ah. Um, and the
ground, ground state, which is the lowest

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energy state. The first excited state and
so on up to the. If you, if you limit its

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energy, then maybe there are only K
different orbitals that come into play.

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And so, we have orbitals all the way
through K-1. Now. How did we pick these

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states? Well, these are the states of
definite energy, this is a ground state,

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it's a lowest energy state. That's the
next highest energy state, and so on. And

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of course, you know, by the superposition
principle, in general, the state of the

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system is a superposition over all of
these states. Okay. But, but now, when you

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have a superposition like this, what is
the energy of this state? Well, we don't

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know. It's in a superposition of states of
definite energy. And so, if we were to

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actually write down an, the energy
observable both surface which right have the

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Hamiltonian for the system, where, what
would it look like, the states of definite

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energy of the system happen to be, the
basis state, zero one through K-1.

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And so, and what are their energies? While
may be the energy of the ground state is E

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note, energy of the first excited state is
E one and so on, up to E K minus one. And

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so the Hamiltonian is a diagonal matrix,
in this case, and the eigenstates of this,

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of this Hamiltonian, which are the basis
states of definite energy. And of course

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the general state of the system is a
superposition over these, over these

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eigenstates, it's a superposition over
these basis states, and those are not

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states of definite energy. Here's another
example uh, it's, it's an example of a

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observable we have seen before. It's,
let's say, let's say we call it the

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Hamiltonian of the system. So it happens
to be, be this one. H here which is minus

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half on the diagonal and five over two on
the off-diagonal. Now, this was a, this

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was an observable which had two
eigenstates, plus and minus, and the

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eigenvalue of the state plus was two, and
of minus, was minus three, and so this

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would correspond to saying that, that uh,
that if this is, if this was the

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Hamiltonian of the system, you would have
two states of definite energy, the minus

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state which would have the lower energy
minus three, and the plus state with

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energy of plus two. Okay, so now in terms
of this Hamiltonian, the Shrodinger's

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equation is a differential equation that
tells us how the state psi of t, so, so,

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we, we, we are, we want to know the state
of the system at time t. Which we denote

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by psi of t, we want to know how it
evolves in time. So for instance we might

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be given, we might be given. Psi at times
zero. And the Hamiltonian of the system,

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and we want to know. What's the state of
this system at time t? Say after, after

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one unit of time, or ten units of time.
And, so this is described by, by

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Schrodinger's equation, which is a
differential equation, which says d by dt

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of psi t, is proportional to H times si of
t. So that's a very funny thing. It's a,

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it's a diffe rential equation which
involves an operator h. Okay, so how do we

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even come to grips with this? So we'll see
how to do this um, in this lecture, but

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then we'll, over the next lecture or two,
we'll get more and more of a, of a feel,

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an intuitive feel for what Schrodinger's
equation is, so we'll study from two or

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three different viewpoints. Um, the other
thing here is, there's, there's a very

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important thing here, which is, which is
that it's i times d psi by dt. Where i is, of

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course, uh, the square root of -1. And
this is quite important, because this is

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what leads to the unitary evolution. And
finally, h bar is, um, is a reduced plank

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constant. It's h, uh, h over two pi where
h is a planks constant. Which is a

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physical constant reflecting the size of
energy. Okay, so. So now, how do we

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actually understand Schrodinger's
equation. How do we solve Schrodinger's

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equation? So we want to uh, we are given
psi at time zero, and we want to

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understand what psi at time t looks like.
Well, it turns out that the key to solving

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Schrodinger's equation is understanding
the eigenvectors of h, the states of

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definite energy. So they evolve in a very
nice way. So let's say that. Uh, initial

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state psi, psi at time zero was, was, was
one of the eigenstates. Phi, phi sub J of

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this, uh, of the Hamiltonian. And let's
say that the corresponding eigen value was

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lambda sub J. So this the energy of the,
of the eigen state phi sub J. So then, the

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claim is that, that our state at time t is
just, it's still phi, phi sub j, the same

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eigenstate but with a, with a phase
associated with it, e to the minus I

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lambda sub j t over h-bar. Okay, so, uh,
so I hope everybody, everybody understands

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what, what a, what I mean by phase. It's,
it's a unit complex number, so if we are

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looking in the. In the complex plane, this
is the real axis, and that's the imaginary

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axis, and that's the. Unit circle. This is
a unit circle. This is one. This is i.

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This is minus one, minus i, and If you
look at a number here, angle theta, then

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it's e to the. i. Okay. Alright, so, so
that's, that's um, that's as far as ah,

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this, this phase is concerned. So, what's
this telling us? It's, it's telling us

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that if you start off in an eigenstate of
h. Then, then the state evolves by, by

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just a change of phase. Right, so, so the,
the over all phase in front of these,

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these, these state changes but, but
that's, but that's all happens, and, and

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it, it, the phase rotates at an, at a rate
that's proportional to the energy, So the

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higher the energy, the quicker the, the
phase rotates, the lower the energy, the,

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the slower it rotates. Okay, so, so now,
how do we, how do we see this? Well, since

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phi sub j is a, phi is a, is a, is an
eigenstate of uh, of h. h 5 sub

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j is just lamba j 5 sub j. Okay.
So, so, what, what this means is that the

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change in, uh, the differential change in,
in, in the state, at time, t, is just

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proportional to five sub j itself. And so
what, what, what this implies is that at

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every time. The state is going to still
point in the, in the direction of i sub

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j. The only question is, what's the
amplitude? Right so, so this implies. That

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psi of t. If you start with sine of zero
equal to five sub j, the eigenstate, then

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sine time t must, must look like some
constant, a of t five j. Okay, so

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now, now we can go back and solve our
differential Equation. So we have. i, h

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power. d by dt, of a of t. Pie sub j is
equal to h psi of t which is h

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times a of t. 5j, [inaudible]. This is a constant which
we can pull out. So that's, a of t,

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Lambdas sub j, psi sub j. And psi sub j, of
course, doesn't, there's, there's no time

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variance and so, so what we, what we get
is d a of t, just rearranging by a of t is

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equal to lambda sub j over h-bar, divide
by i, which is minus i. Dt. Okay, so I'm

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just rearranging this. And this is a
simple differential equation. We all know

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how to, how to solve. It's just, uh, it
just gives, uh. a of t is e to the minus i

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lambda jt over h bar. Okay, which is what
we wanted. So now we understand, um, how

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the eigenstate, how an eigenstate of, of
h, evolves under, um, in time. So, what

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Shrodinger's equation tell us is
that, is that, if you start with an eigen

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state of h. Then, the state always points
in the same direction. All that changes is

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the phase. And the phase changes via this,
this precession rule. It, it precesses at

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a rate proportional to lambda j, which is
the energy of that, of that eigenstate.

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So now, of course in general, our starting
state is not going to be an eigenstate,

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it's going to be a superposition of
eigenstate. So we might start with a, with

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a starting state superposition of, of,
over phi sub-j, with amplitude alpha

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sub-j. So then, of course, paralinearity,
this data at time t is just going to be

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the sum over j of. Is always, of course,
sum over J of alpha J. E to the minus I.

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Lambda JT over H bar. Five sub J. So,
basically, each eigen state, or each, each

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state of definite energy. It sort of stays
invariant in time. Except, it precesses,

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right? It, uh, each, each eigen state is
precessing at its own rate. Uh, it marches

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to its own drummer. Which drummer? Well,
it's its energy. And that's, that's the

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basic equation of motion in quantum
mechanics. Okay, so now, we could also try

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to write this out. As an operator. So,
what, what, you know, if you want to know

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how to get from. The state at time zero to
the state at time T. Well there's a,

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there's clearly a linear operator that
tells us how to map from time zero to time

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T. And if you write it out in the, in the
eigen bases of H, it is a diagonal matrix.

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Because each eigen state sort of stays
invariant except for phase changes. And so

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it's a, it's a diagonal matrix with
diagonal entries E to the -I lambda JT

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over H bar. And so. Now this is, this is
clearly a unitary matrix. Right? Uh, if,

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if you, if you call this matrix, if you
call this matrix U. Then clearly U

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satisfies U  U dagger= to U dagger U is
the identity. Okay. So another way you can

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write down this , the axiom of unitary
evolution is you can just say, well.

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Evolution is by schneider equation. If you
solve schneid er equation you get this,

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you know, you, you get unitary
transformation for evolution and time.

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Okay, so sorry, I should call it T off T,
same here. And a, the other thing we can

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do is a. You know, U of T is, is now E to
the minus I H T over H bar. So this is

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just notation. You can think of it as, um,
you know, E to the, um. E to the A is a,

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is a, is a matrix. If, if you write, um,
um, a matrix B equal to E to the A. Then B

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has the same eigen values as A. So, sorry,
B has the same eigenvectors as A. Um.

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Eigenvectors, but the eigen values of B
are, expla, the exponential of eigen

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values of A. So if, if one of the eigen
values of A. Is lambda sub J, then the

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corresponding eigenvalue of A of B. ,, .
Is E to the lambda , okay? So this is

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just, um. You know, you can take it as
notation. You can, um, uh, think about it

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as a, as a expansion. But, um, you know.
Uh, just think of it as, as notation for,

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uh, um, uh, for matrix expo-,
exponentials. Okay, so finally, let's do

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an example. Suppose our hamiltonian is, is
x, the bit, the bit flick, uh, matrix, and

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we start off in the, in, in the zero
state. And what's the state at time T?

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Well, so, we all know the eigenvectors of
x are plus and mi- uh, plus and minus uh,

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with eigenvalues plus one and minus one.
So what we need to do is first write psi

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of zero in the eigenbasis, so, so psi of
zero is one over square root two plus,

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plus one over square root two minus. And
so, psi of t. Is going to be equal to one

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over square root two. And what does plus
evolve to? It evolves to eight to the

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minus I, lambda is one, T over H/. Plus
one O by square root two. E to then minus,

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minus I lambda, lambda in this case is
minus one, so we get a plus. T over H bar

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minus. .