Second Quantization of Conserved Particles - PowerPoint Presentation

Slide1

Second Quantization of Conserved Particles

Electrons, 3He, 4He, etc.

And of Non-Conserved Particles

Phonons, Magnons, Rotons…Slide2

We Found for Non-Conserved Bosons

E.g., Phonons that we can describe the system in terms of canonical coordinates

We can then quantize the systemAnd immediately second quantize via a canonical (preserve algebra) transformSlide3

We create our states out of the vacuum

And describe experiments with Green functionsWith Slide4

Creation of (NC) Particles at x

We could Fourier transform our creation and annihilation operators to describe quantized excitations in space

poetic licenseThis allows us to dispense with single particle (and constructed MP) wave functionsSlide5

We saw, the density goes from

And states are still created from vacuumThese operators can create an N-particle stateWith conjugate

Most significantly, they do what we want to!

Think <x|p>Slide6

That is, they take care of the identical particle statistics for usI.e., the operators must

And the Slater determinant or permanent is automatically encoded in our algebraSlide7

Second Quantization of Conserved Particles

For conserved particles, the introduction of single particle creation and annihilation operators is, if anything, natural

In first quantization, Slide8

Then to second quantize

The density takes the usual form, so an external potential (i.e. scalar potential in E&M)And the kinetic energySlide9

The full interacting Hamiltonian is then

It looks familiar, apart from the two ::, they ensure normal ordering so that the interaction acting on the vacuum gives you zero, as it must. There are no particle to interact in the vacuumCan I do this (i.e. the ::)? Slide10

p42c4Slide11

The Algebra

Where + is for Fermions and – for Bosons

Here 1 and 2 stand for the full set of labels of a particle (location, spin, …)Slide12

Transform between different bases

Suppose we have the r and s bases

WhereI can write (typo)

If this is how the 1ps transform then we use if for operators x or k (n)Slide13

With algebra transforming as

I.e. the transform is canonical. We can transform between the position and discrete basis Where is the nth

wavefunction. If the corresponding destruction operator is justSlide14

Is this algebra right?

It does keep countSinceF [

ab,c]=abc-cab + acb-acb =a{b,c}-{a,c}bB [ab,c]=abc

-cab + acb-acb =a[b,c]+[a,c]b For FermionsEq. 4.22Slide15

It also gives the right particle exchange statistics.

Consider Fermions in the 1,3,4 and 6th one particle states, and then exchange 4 <-> 6

Perfect!Slide16

And the Boson state is appropriately symmetric

3 hand written examples (second L4 file)Slide17

Second Quantized Particle Interactions

The two-particle interaction must be normal ordered so that

Also hw example