Second Quantization of Conserved Particles Electrons, 3He, 4He, etc. And of Non-Conserved Particles...
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Transcript of Second Quantization of Conserved Particles Electrons, 3He, 4He, etc. And of Non-Conserved Particles...
Second Quantization of Conserved Particles
Electrons, 3He, 4He, etc.
And of Non-Conserved ParticlesPhonons, Magnons, Rotons…
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 system
• And immediately second quantize via a canonical (preserve algebra) transform
• We create our states out of the vacuum
• And describe experiments with Green functions
• With
Creation of (NC) Particles at x
• We could Fourier transform our creation and annihilation operators to describe quantized excitations in space poetic license
• This allows us to dispense with single particle (and constructed MP) wave functions
• We saw, the density goes from
• And states are still created from vacuum
• These operators can create an N-particle state
• With conjugate
• Most significantly, they do what we want to!
Think <x|p>
• That is, they take care of the identical particle statistics for us
• I.e., the operators must
• And the Slater determinant or permanent is automatically encoded in our algebra
Second Quantization of Conserved Particles
• For conserved particles, the introduction of single particle creation and annihilation operators is, if anything, natural
• In first quantization,
• Then to second quantize
• The density takes the usual form, so an external potential (i.e. scalar potential in E&M)
• And the kinetic energy
• 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 vacuum
• Can I do this (i.e. the ::)?
p42c4
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, …)
Transform between different bases
• Suppose we have the r and s bases
• Where• I can write (typo)
• If this is how the 1ps transform then we use if for operators
x or k (n)
• 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 just
Is this algebra right?
• It does keep count• Since– F [ab,c]=abc-cab + acb-acb =a{b,c}-{a,c}b– B [ab,c]=abc-cab + acb-acb =a[b,c]+[a,c]b
– • For Fermions
Eq. 4.22
• 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!
• And the Boson state is appropriately symmetric
• 3 hand written examples (second L4 file)
Second Quantized Particle Interactions
• The two-particle interaction must be normal ordered so that
• Also hw example