Second Quantization of
Conserved Particles
Electrons, 3He, 4He, etc.
And of Non-Conserved Particles
Phonons, 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
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
Eq. 4.22
–
• For Fermions
• 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 Quantized Particle Interactions
• The two-particle interaction must be normal
ordered so that
• Also
hw example