So, what constitutes a symmetry in quantum physics?
We've talked about some of this before, but here's, here it is to be made
explicit. So, let's suppose we have a solution to
the Schrodinger equation. We have a wave function which evolves in
accordance with the Schrodinger equation. And let's, let's suppose there's an
operator, U, that commutes with H. So, U U for which minus HU is identically
zero everywhere. Then, I think you can see that if we
apply U to psi on this, so first of all the U is to be not time dependent.
I mean, there could be some trivial time dependence that this one the equitation
is still satisfied. But we're looking primarily for things
that don't, don't change in time. so it passes through the time derivative.
And then, it also passes through H so because of the commutation relationship.
So, you can see that now if psi is solution, then U psi is also a solution.
So this, this means that if we do have a symmetry, we can use it to, either to
generate different solutions that have the same evolution as an underlying wave
function. Or more significantly we can
simultaneously diagonalize U and H, the Hamiltonian matrix.
So we can, we can, in this, in the time independent Schrodinger's equation, we
can look for for solutions that are eigenfunctions of U.
And for the case of case of discrete symmetries it's, it's they give a lot of
practical guidance on how to construct wave functions.
It, it's easy to generate functions that are eigenfunctions of U.
And then that, that provides say, a smaller basis for expiration of the the
Hamiltonian. Why is symmetry useful?
Well, the point is, one point is, it gives you a way of organizing work that's
highly efficient. So for example, a graphene is this, this
new material relatively new material, subject to recent Nobel Prize in Physics.
It's a it's a two dimensional network. And so if you have a large, if you have a
large sheet of graphene, there are many electrons, there's many, there's many
nuclei. if you can focus your solution on solving
solving wave functions and have a particular symmetry that conforms to the
symmetry to the underlying graphene. there's a lot less work to do, and you've
better predictive capabilities. There's, there's some things where you
can actually determine what type of what type of electronic orbitals being excited
by looking at an angular distribution. So in a case like this symmetry is widely
used to simplify practical calculations or analysis of experimental data.
Another example, discreet symmetry. So this is a schematic of the green laser
pointer which is described in an accessible form in the co, additional
materials in the course. And so, this is sort of a hypothetical
example, but it's, it's emblematic of things that happen in the real world.
Let's say you you, you need to design a laser that's going to operate a certain
wavelength region. Say that the telecommunications region,
very important area. and you know of some sharp line systems
that can be used to get lasing action. But may be the the specific measurements
of the relative transition probabilities haven't been made in those systems.
So, you know, how do you, so here's a set of, here's a set of levels of the
Neodymium 3+ ion. Now, you'll, in principle, you can have a
transition between any two, any of these, these levels.
But it may be that some of those transitions are forbidden, or they're
very weak, for reasons of symmetry. So appreciating the the so-called
selection rules that are dictated by symmetry considerations.
This is a very important practical concern.
Okay, now we're going to talk about parity.
So one might say this is some sense the simplest discrete symmetry that's, that's
basically, let's say it's a symmetry associated with the vacuum.
Rather than with a crystal or particular molecular structure.
and it's the that the parity transformation is something that inverts
all coordinates of the the particles of the system.
All the spacial coordinates of the particles of a system.
Okay, I hope that was relatively straightforward.
clearly the potential is unaffected by the parity transformation.
You change the sign of both things. You're taking the absolute value.
It's still a difference. Now, you do have to think a little bit,
perhaps about the kinetic energy operator.
but I'll just say it this way. The kinetic energy is a gradient sorry,
the momentum is a gradient. I think that was made explicit in the
statements of the inline video. And so the gradient of a function, with
respect to a coordinate, does change sign under inversion.
But since the Hamiltonian depends upon the square of the gradient that's doesn't
matter. Okay, well, now let's just take a, make a
slight modification to this problem, which again, corresponds to real thing
that one might do in the laboratory. So now we, you know, we, let's say we're
in the lab, we study that previous system.
Now, we want to understand it's response to an electric field.
You know, by the way, the response of the hydrogen atom to an electric field was
one of the first problems studied by Schrodinger in his early papers.
So, this is another huge triumph of quantum physics.
It became the first theory that was able to actually predict from first
principles, how matter interacted with electricity.
I mean, that's really huge, right? prior to that there were only these
effective ideas, things had dialectic constants, or they were ideal conductors.
But quantum mechanics turns out to give a really good account of how electric
fields how, how they cause behavior to change in material systems.
So we're going to follow that model and apply an electric field to this system.
So, we have the same, the same particle interactions going on.
But now this is the Ham, this is the Hamiltonian for the potential associated
with the position of a particle of charge Q at position R in electric field.
Just the scale or product. Okay, so now we're, our parity operator
is the same as before, is applied in the same way.
And my question is, is parity conserved in this system?
So I hope, I hope it's evident to you that, this term here, causes, causes
parity to to, to not be conserved in the system.
Because there is a, an externally applied force that has a particular direction.
And so that when you you know, when you change when you change positions in that
force field, you change the energy. So, the, it is, it is the case, it's,
it's, it's, common place that you can break parity by the use by changing
making asymmetries in, in potentials. but once again as you can see from this,
very general Hamiltonian. There's sort of a natural hope that
parity will be conserved in material systems.
And it's largely true but not entirely. So, as I mentioned in the introduction
this is one of the the great experiments and theoretical predictions of mid-20th
century physics that in beta decay, you know, a very fundamental process.
There would, there could be parity-violating interactions.
And this has since become even more widely interesting with the and in the
context of the standard model. Because there are specific predictions
about how parity for non-conserving interactions would be found even in a
spectra of atoms. Which are ordinarily thought to be
completely dominated by these simple electrostatics in quantum mechanics.
Arguments that were illustrated in that previous model.
So, it's a very active field of research a lot of opportunities, both for theory
and experiment. Okay, next time, we're going ahead with
angular momentum. Much more substantial subject.
And one that's going to have a little bit more of a mathematical mathematical
training for those who like that sort of thing.