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. .