Okay so um, some of you might be wondering
why should this equation has the
particular form it does. So why does
energy play such an important role in, in
the evolution of the quantum system. Okay
so. Um, so this video is sort of a brief
video to address this concern, um. It,
it's mostly, uh, it's mostly to point you
in some direction if you are interested in
pursuing this any further. Okay. So, so
just to recall, um. You, uh, the
Hamiltonian of the system is the energy
observable, um. Its eigenstates are the
states of definite energy, uh, etcetera.
And Shrodinger's equation says that the evolution of
the system is intimately tied to the
Hamiltonian. Right? It's a differential
equation which says that, um, which
governs the, the evolution of the system.
And it, um, you know, it intimately ties
the change in the state to, to the action
of the Hamiltonian on the, on the, on the
state. So where does all this come from?
So, it, it turns out that the answer lies
in, in this beautiful connection between
symmetry and conserve quantities
in, in, in quantum mechanics. And, this,
uh, goes back to work by, uh, Annie . Um,
who is, uh, who is regarded as, uh, as,
um, the greatest female mathematician in,
in, in history. The greatest, uh, woman
in, in, in mathematics in all of history.
Okay, so, so let's see how we can derive
the form of Shrodinger's equation. So the. The first
thing, uh, we, we observe is that from the
axiom of unitary evolution. We can derive
with a little bit of work that, that, uh,
the time evolution operator must look like
E to the -IMT for some hermitian operator
M. Okay, so for some emission. Okay, so
let's take this for granted. So now what
they're really trying to understand is
why. Is m equal to the Hamiltonian. Okay,
and we don't want to prove that m must be
equal to the Hamiltonian, we want to sort
of argue that it must. Right, so, so
we'll, we'll provide a justification which
is going to be actually quite compelling.
Okay, so what this relies on is a, is a
theorem, which says, if. E is any
observable. Corre sponding to some, some
physical quantity so if A measures some
physical quantity that's conserved.
Meaning, for example, E corresponds to
momentum or angular momentum or energy.
It's some, some quantity that is conserved
over time. Then, a must commute with,
with M. Meaning that E times M equal
to M times A. Right? Now I hope you
realize that matrices, matrix
multiplication in general, doesn't
commute. So, for example, if we have, if
you look at our operators X, the bit-flip
operator. And Z, the phase flip operator.
Right? Then what's, what's X times Z?
Well, so its a, zero one, one zero times,
one minus one zero, zero, which is what
you get zero here. Minus one. One. Zero.
And what's z times x? It's one minus one,
zero, zero. Times zero, one, one, zero.
Which is what? Zero. One. Minus 1-0. Right
which are, these are not equal. So in
general, if you're given two general
operators, they will not commute. If A
corresponds to any conserve quantity, then
it must commute with the time of, you, you
know, this, this [inaudible] operator M
which occurs in, in this unitary
evolution. Now, lets this see why, why
does this follow, so lets sketch out a
proof. So. So lets, lets let psi be the
state of the system at, st some time. And
lets I prime. B equal to psi, B the state
which is, which is E to the minus IMT.
Psi will be the state after an
infinitesimal interval of time T. So, now
what does it mean to say that A is a
conserved quantity. A is conserved
quantity means that, that if you
look at, the expected value of A under
psi. So psi A psi, this must be equal to
psi prime, a psi prime. Okay, but what's,
what's psi prime is psi , psi , psi prime
will, psi prime is U psi , so it's, its U
psi. And then A U dagger psi. Okay, but
this, this, this equation hold for all
psi . Since this hold for all psi , this
implies that in fact A must be equal to U
dagger A U. Right? And so. U-dagger au is,
is of course just um, um, e to the imt a
to minus imt, which if you, you, if you
expand out with a Taylor expansion, this,
this, is, this is to a first order equal
to one plus IMT, times A, times one minus
IMT. Remembering that T is very small, so
higher order terms don't, don't really,
don't really matter here. And this
further is about A + I T times M A minus
A M. And so since A is equal to this, it,
it tells us that this must equal to zero.
Right here I'm ignoring the, or the t
squared term, which comes from this
product here because t is turning to zero.
And so, now, if this, if this is zero,
then that tells us that ma=am, which is,
is, which is, really worth saying that a
must commute with m. Okay, okay, so what
have we learned so far? What we've learned
is that first, first what, what we said is
that just from the fact that we have
unitary evolution. It follows with a
little bit of work which we won't go in
to, that U must be of the form E to the
minus I M T. That M is emission and
the second thing we know is that if E is
conserved then E must commute with M.
Okay . So now, there are various
quantities that might be conserved. There
are things like momentum or angular
momentum . But, one can argue that these
are accidental con, conserved quantities.
The, you know this are conservation
relations which may or may not hold. So
for example conservation of momentum well
you know it may or may not, momentum may
or may not be conserved depending upon,
for example if you, if you have a change
in potential energy then momentum is not
conserved. Okay? Energy is always
conserved and remember we are working
non-relativisticly, so, so energy is
always conserved. Okay, so, so, so there,
there must be some intrinsic reason so, so
what we, what we, are saying is there must
be some intrinsic reason  why M, and H commute.
And what we are going to reason is that,
in fact, the most natural intrinsic reason
why H and M commute is that M is some
multiple of H. Right? So H equal to, say.
Right, this is, this is what they're
really claiming, that there's some
constant, which turns out to be h-bar such
that our energy operator, the Hamiltonian
h is h-bar times m, where m is the
operator which, which describes the
evolution of the system, and that's,
that's what they're claiming. Okay, so,
maybe, maybe, let me actually go through
it quickly. Uh, you can read about it in
the notes if you're interested, but um,
let's, let's just go with it. Uh, um, so,
so the rough form of reasoning is that,
well, in order for M and H to commute
intrinsically, well, you know, it, uh,
the, the one way we can imagine it is if,
if, uh, if H is actually some function of
M, some polynomial in M, general. And
then, the next thing to show is that. In
fact, there are other physical
considerations. So this is, you know, this
was to make sure that H and M actually
commute. But the next, the next thing to
show is that, in fact, there are physical
considerations which imply that if H is
some function of M, this function must
necessarily be linear. Okay and that,
that has to do with saying. Well, suppose
that our system consists of two pieces,
two disjoint pieces, one and two. So, this
has an associated omission operator m1.
This has an associated omission operator m2. This has
a Hamiltonian h1. This has a Hamiltonian
h2. Well then the, then the energy of
the, of the composite system, if our, if
our system consists of these, these two
disjoined pieces, the energy is clearly H1
plus H2. But on the other hand, the operator
associated with this, the hermitian
operator must also be M1 plus M2. And so, H
must be = to F of M1 plus HM2. But we are
claiming H, but, but on the other hand H
one is, is F of M one, H two is F of M
two, and so F of M one plus M two equal
to f of m1, less f of m2. And so that
tells us that f must be a linear function.
Okay, this, this is maybe too much for
most of you, but, I just added it in for
those of you who are interested. Um, it's,
it's certainly, uh, not in any way
required for, anything that follows. Okay.