Welcome back everyone. I'm Charles Clark.
This is Exploring Quantum Physics, and we're going to begin a set of lectures
today on the phenomenon of symmetry in quantum physics.
It's a very important and pervasive concept and we'll try to give you a lot
of interesting examples to think about. So, here we go.
The forms of symmetry we'll be looking about fall into two classes a sa-, things
that are associated with a single particle, or a, you know, a, an, an
independent system those are like And then they also, you know, they have
the implications for many particle systems as well.
So parity, angular momentum. we'll return to the hydrogen atom and
look at the four, the so called four dimensional symmetry that comes from the
quantum mechanical treatment of the fun a lens factor.
Then there'll be a break for about a week in which Ian Applebaum who's a professor
here at the University of Maryland and an expert in spintronics, is going to be
giving a detailed account of electron spin, which is a fabulous topic.
we turn for brief discussion of things like isobaric spin, time reversal, and
gauge invariance. And then, we'll conclude with discussions
of some of the interesting effects of symmetry in many particle systems.
It's a very important topic. So modern work on ultra-cold gasses has
largely been made possible by the phenomenon of Bose-Eistein condensation.
And the reason they're all around. Is because the the Fermi gas, Fermi-Dirac
gas. the generic Fermi-Dirac gas has a very
high pressure. Basically, electrons are prohibited from
getting close to each other by the Pauli exclusion principle.
So that, that has a, a major effect on our world.
[INAUDIBLE] we'll just see and think about it.
Now, the first thing on this list may seem a little bit strange to some of you,
parity. It's a really important concept, but it's
not really deserved. In fact, as I'll show you from some of
the examples, parity is one of the most useful of the symmetries in in, well the
study of ordinary matter from the standpoint of electronic structure and
other things Because it's it, it works so well.
It's easy to apply and it often produces a tremendous simplification in the
difficulty of a problem. However it was, there was a theory
developed by these two young men. Chen Ning Yang and Tsung Dao Lee.
doctor Lee was only only thirty one years old when he won the Nobel prize for this
work. They predicted.
That, parity would be violated in some weak, decays due to weak interaction.
I'll just say a little bit about that in the next slide.
But this was sort of the, first of all, caused a, it was a downfall of a
principle that had never been known to fail and was widely used and.
You know, almost naturally expected in quantum mechanics.
And it it, it led, it was the beginning of what's now called the electroweak
theory, where we understand the electrici-, electricity and magnetism is
coupled to weak interactions. Now the thing is, the parity
transformation. Is the what you get when you invert all
the coordinates in a system. So you might wonder, well, exactly how
can you do that? and if the, the way it's done is a little
bit indirect. But, I mean, I'll, I'll show that on the
next slide. So once again, sometimes these various
concepts, parity, angle, momentum, spin, are sort of linked together.
And you have to understand several of them before the whole picture falls into
place. But this, this is just a photograph of
the experiment that was done. At the National Bureau of Standards,
1956. And here is a, a schematic of what was
observed. So, what was observed was the, the
preferential decay of a radioactive nucleus by Beta emission.
In one direction which was defined by the, the spin of the nucleus.
And this is actually something that if parity is, is violated, if parity is
conserved, can't possibly happen. There would have to be you know, on the
average, an equal distribution of particles, of beta beta rays coming off
on both sides of the nucleus. And that was, that was the asymmetry
observed/g, and certainly a clear ,uh, sign that parity was not conserved in
that interaction. here's a picture of one of the really
great physicists of the twentieth century in my opinion.
he won the Noble Prize in Physics a little bit later in life than some of
these other fellows. But note the citation here makes
particular, particular reference to his. His discovery and application of sym,
symmetry principles. And he was, in large part, responsible
for making symmetry sort of a, a theme of quantum physics.
So Angular momentum in particular, is really a grand organizing principle of
quantum physics. We'll see why that is in the next, in the
next several lectures. But basically, it's it's extremely
important and it, it reflects a rotational symmetry of a system, or the
isotropy of space. And Wigner helped create a number of
things that made the theory possible. Angular momentum algebra.
This Wigner-Eckert theorem is very important for understanding things like
the whether particles can have permanent magnetic moments or electric dipolar
moments of that sort. Isobaric spin which unified the
description of the neutron and proton as two states of a common underlying
particle turned out to be a terrific idea for rational understanding of the nuclear
structure. some other things that have finally time
reversal. So Wigner is the person who.
Really first worked out the systematic theory of time reversal in quantum
mechanics systems. Now, okay, parity seems, I mean, I hope
it seems like somewhat exotic. Think how do you, how do you manage to
reverse turn something completely inside out?
but time reversal. how do we make time go backwards anyway?
Well, there will be, we'll discuss that later.
But first let's, let's just look at a, a simple example of time reversal, which
has to do with motion of a charged particle in a magnetic field.
So I hope that some of you, got to the point where you realized what this
changing. Of the velocity does to a motion in a
magnetic field. It causes a displacement of the orbit.
So, we've been thinking about this in the context of a sort of an atomtronics
application, and there's a wonderful device used in microwave engineering.
When you want to let's say you have an input, an input channel and an output,
and you're delivering high power loads, and you, you want, you want to avoid the
problem of back reflection, of something coming from that, the channel you're
pumping, coming back in to your input and then destroying the apparatus.
It's a problem that often happens in power engineering.
So, this the, this device which is known [INAUDIBLE] just called a circulator, and
it's a time-reversal non-invariant device.
And, wha, actually if you make up a system where you have a magnetic field in
this region, and you send an electron beam.
In this is what you get. So the, when the beam comes from this
input port, it goes to the output port. Any part that's reflected, when it comes
back, it has a time-reversed velocity that the input beam had here.
But that means, it's reflected to the next port, where you could have a load
dump or something like that. Like that.
So this is, this is an example of the use of time-reversal noninvariance which
requires the use of a magnetic medium, in the case of the micro-circulator.
Wolfgang Pauli, another giant of 20th century physics his Nobel citation you
know, consists in its entirety. For of, recognizing the discovery of the
exclusion principle, called the Pauli Principle.
This was a revolutionary concept because it was an intrinsically many particle
phenomenon, and the indication of a simple symmetry rule that applies to all.
well, as we now know, all particles with half integral spin, including the
electron So, that, that those constitute sort of
half the particles in the world. The others have integral spins.
Those, those are called bosons. But, this exclusion principle is really a
symmetry principle because what it says is.
That the wave function of two particles, including all variables, such as the
space, the spin, or any sort of internal degree of freedom, changes sign when you
exchange the particles. So this is really like a, a reflection
that involves exchange of the position of two particles.
And, a reflection in a, in a plane. So if we have a, we have a particle one
here, and a particle two there. Then it's a reflection in the plane, that
connects the, connects the two particles. And so that, that means for example, the
wave function. Must vanish on the midpoint.
And this is why the Pauli Exclusion Principle in, increases the energy of a
system of particles compared to what it would be if the exclusion principle
weren't valid, because it forces nodes in the wave function in between the
particles. So that means they're, they're more
localized and they had certain principles gives them higher energies.
So indeed, this Pauli principle is, is often is the most important thing in the
problem. And it can set the qualitative ordering
of states. In a manner that's, that's not totally
independent from the particular interactions that are at play but it sets
the stage and they're things that really can't happen because of the Pauli
Principle that, otherwise might be quite possible.
So to conclude just mention a phenomena that's not going to be discussed at any,
at any length in this course, but it's, you know, evidence for how symmetry sort
of dominates thinking in physics. So here is the, the three laureates
recognized for what is now known as The Standard Model, this fabulously
successful. Modical of fundamental particles.
Now it's pretty complicated, you know, they're fundamental, there's like, I
don't know, there's 50 of these particles.
so it isn't simple but it is based, it is based on a gauge symmetry.
So it is very much a a theory whose structure is, is, you know, at least in
form, and, and in many respects, dominated by a type of symmetry that
exists. So you'll, you'll see more of this, I
think, when Victor gives a lecture on topological insulators.
You'll see that there are new developments in Gannett's matter,
physics, that have, that have to do with gauge symmetries becoming available in
those systems. It's also true for ultra cold atoms.