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.