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Second quantization
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Second quantization is a way of writing the quantum mechanics of
indistinguishable particles in such a way that it makes statistical
mechanics calculations easy and makes everything look like field theory.
One starts with a distinguished vacuum state |vac> and a family of
annihilation operators a(x) whith their adjoints, the creation
operators a^*(x), satisfying the canonical commutation relations (CCR)
[a(x),a(y)]=[a^*(x),a^*(y)]=0,
[a^(x),a^*(y)]=delta(x-y).
(This is for Bosons; for Fermions one has instead canonical
anticommutation relations, CAR, and everything below gets additional
minus signs in certain places.)
A pure (permutation symmetric) N-particle state with wave function
psi(x_1:N) is written in second quantization as
psi = integral dx_1:N psi(x_1:N) a^*(x_1:N) |vac>,
hence the corresponding density matrix
rho = psi psi^*
takes the form
rho = integral dx_1:N dy_1:N rho(x_1:N,y_1:N),
where rho(x_1:N,y_1:N) is the rank one operator
psi(x_1:N)psi^*(y_1:N)a^*(x_1:N)|vac> = integral dx dy f(x,y)
defines the 1-particle density matrix Rho. The form of f in second
quantization is
f = integral dx dy f(x,y) a^*(x) a(y)
(exercise: check that it has indeed the desired action on an
N-particle state!), hence one has
= integral dx dy f(x,y) .
and comparison with the definition of Rho gives the formula
= = trace a(x) rho a^*(y),
which can therefore be viewed as the definition of the 1-particle
density matrix in second quantization.
Authors who are afraid of integrals write instead similar formulas with
sums in place of integrals and discrete indices in place of the x,y.
Also, one can do the same in momentum space rather than position space,
which amounts to a change of basis but generally leads to
computationally more tractable formulations.