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.