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So, what constitutes a symmetry in 
quantum physics? 

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We've talked about some of this before, 
but here's, here it is to be made 

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explicit. 
So, let's suppose we have a solution to 

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the Schrodinger equation. 
We have a wave function which evolves in 

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accordance with the Schrodinger equation. 
And let's, let's suppose there's an 

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operator, U, that commutes with H. 
So, U U for which minus HU is identically 

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zero everywhere. 
Then, I think you can see that if we 

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apply U to psi on this, so first of all 
the U is to be not time dependent. 

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I mean, there could be some trivial time 
dependence that this one the equitation 

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is still satisfied. 
But we're looking primarily for things 

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that don't, don't change in time. 
so it passes through the time derivative. 

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And then, it also passes through H so 
because of the commutation relationship. 

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So, you can see that now if psi is 
solution, then U psi is also a solution. 

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So this, this means that if we do have a 
symmetry, we can use it to, either to 

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generate different solutions that have 
the same evolution as an underlying wave 

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function. 
Or more significantly we can 

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simultaneously diagonalize U and H, the 
Hamiltonian matrix. 

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So we can, we can, in this, in the time 
independent Schrodinger's equation, we 

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can look for for solutions that are 
eigenfunctions of U. 

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And for the case of case of discrete 
symmetries it's, it's they give a lot of 

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practical guidance on how to construct 
wave functions. 

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It, it's easy to generate functions that 
are eigenfunctions of U. 

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And then that, that provides say, a 
smaller basis for expiration of the the 

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Hamiltonian. 
Why is symmetry useful? 

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Well, the point is, one point is, it 
gives you a way of organizing work that's 

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highly efficient. 
So for example, a graphene is this, this 

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new material relatively new material, 
subject to recent Nobel Prize in Physics. 

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It's a it's a two dimensional network. 
And so if you have a large, if you have a 

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large sheet of graphene, there are many 
electrons, there's many, there's many 

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nuclei. 
if you can focus your solution on solving 

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solving wave functions and have a 
particular symmetry that conforms to the 

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symmetry to the underlying graphene. 
there's a lot less work to do, and you've 

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better predictive capabilities. 
There's, there's some things where you 

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can actually determine what type of what 
type of electronic orbitals being excited 

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by looking at an angular distribution. 
So in a case like this symmetry is widely 

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used to simplify practical calculations 
or analysis of experimental data. 

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Another example, discreet symmetry. 
So this is a schematic of the green laser 

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pointer which is described in an 
accessible form in the co, additional 

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materials in the course. 
And so, this is sort of a hypothetical 

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example, but it's, it's emblematic of 
things that happen in the real world. 

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Let's say you you, you need to design a 
laser that's going to operate a certain 

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wavelength region. 
Say that the telecommunications region, 

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very important area. 
and you know of some sharp line systems 

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that can be used to get lasing action. 
But may be the the specific measurements 

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of the relative transition probabilities 
haven't been made in those systems. 

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So, you know, how do you, so here's a set 
of, here's a set of levels of the 

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Neodymium 3+ ion. 
Now, you'll, in principle, you can have a 

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transition between any two, any of these, 
these levels. 

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But it may be that some of those 
transitions are forbidden, or they're 

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very weak, for reasons of symmetry. 
So appreciating the the so-called 

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selection rules that are dictated by 
symmetry considerations. 

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This is a very important practical 
concern. 

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Okay, now we're going to talk about 
parity. 

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So one might say this is some sense the 
simplest discrete symmetry that's, that's 

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basically, let's say it's a symmetry 
associated with the vacuum. 

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Rather than with a crystal or particular 
molecular structure. 

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and it's the that the parity 
transformation is something that inverts 

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all coordinates of the the particles of 
the system. 

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All the spacial coordinates of the 
particles of a system. 

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All the spacial coordinates of the 
particles of a system. 

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Okay, I hope that was relatively 
straightforward. 

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clearly the potential is unaffected by 
the parity transformation. 

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You change the sign of both things. 
You're taking the absolute value. 

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It's still a difference. 
Now, you do have to think a little bit, 

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perhaps about the kinetic energy 
operator. 

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but I'll just say it this way. 
The kinetic energy is a gradient sorry, 

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the momentum is a gradient. 
I think that was made explicit in the 

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statements of the inline video. 
And so the gradient of a function, with 

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respect to a coordinate, does change sign 
under inversion. 

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But since the Hamiltonian depends upon 
the square of the gradient that's doesn't 

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matter. 
Okay, well, now let's just take a, make a 

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slight modification to this problem, 
which again, corresponds to real thing 

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that one might do in the laboratory. 
So now we, you know, we, let's say we're 

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in the lab, we study that previous 
system. 

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Now, we want to understand it's response 
to an electric field. 

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You know, by the way, the response of the 
hydrogen atom to an electric field was 

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one of the first problems studied by 
Schrodinger in his early papers. 

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So, this is another huge triumph of 
quantum physics. 

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It became the first theory that was able 
to actually predict from first 

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principles, how matter interacted with 
electricity. 

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I mean, that's really huge, right? 
prior to that there were only these 

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effective ideas, things had dialectic 
constants, or they were ideal conductors. 

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But quantum mechanics turns out to give a 
really good account of how electric 

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fields how, how they cause behavior to 
change in material systems. 

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So we're going to follow that model and 
apply an electric field to this system. 

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So, we have the same, the same particle 
interactions going on. 

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But now this is the Ham, this is the 
Hamiltonian for the potential associated 

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with the position of a particle of charge 
Q at position R in electric field. 

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Just the scale or product. 
Okay, so now we're, our parity operator 

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is the same as before, is applied in the 
same way. 

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And my question is, is parity conserved 
in this system? 

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So I hope, I hope it's evident to you 
that, this term here, causes, causes 

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parity to to, to not be conserved in the 
system. 

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Because there is a, an externally applied 
force that has a particular direction. 

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And so that when you you know, when you 
change when you change positions in that 

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force field, you change the energy. 
So, the, it is, it is the case, it's, 

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it's, it's, common place that you can 
break parity by the use by changing 

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making asymmetries in, in potentials. 
but once again as you can see from this, 

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very general Hamiltonian. 
There's sort of a natural hope that 

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parity will be conserved in material 
systems. 

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And it's largely true but not entirely. 
So, as I mentioned in the introduction 

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this is one of the the great experiments 
and theoretical predictions of mid-20th 

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century physics that in beta decay, you 
know, a very fundamental process. 

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There would, there could be 
parity-violating interactions. 

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And this has since become even more 
widely interesting with the and in the 

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context of the standard model. 
Because there are specific predictions 

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about how parity for non-conserving 
interactions would be found even in a 

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spectra of atoms. 
Which are ordinarily thought to be 

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completely dominated by these simple 
electrostatics in quantum mechanics. 

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Arguments that were illustrated in that 
previous model. 

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So, it's a very active field of research 
a lot of opportunities, both for theory 

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and experiment. 
Okay, next time, we're going ahead with 

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angular momentum. 
Much more substantial subject. 

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And one that's going to have a little bit 
more of a mathematical mathematical 

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training for those who like that sort of 
thing.