- For completeness, assume〈n|ψ〉= 0 for alln. The expression for the|n〉
as Hermite polynomials times a Gaussian then implies that
∫
F(q)e−
q^2(^2) ψ(q)dq= 0
for all polynomialsF(q). Computing the Fourier transform ofψ(q)e−
q^2
2
gives
∫
e−ikqe−
q^2
(^2) ψ(q)dq=
∫ ∑∞
j=0(−ikq)j
j!
e−q^2(^2) ψ(q)dq= 0
Soψ(q)e−
q^2
(^2) has Fourier transform 0 and must be 0 itself. Alternatively,
one can invoke the spectral theorem for the self-adjoint operatorH, which
guarantees that its eigenvectors form a complete and orthonormal set.
Since in this representation the number operatorN=a†asatisfies
Nzn=z
d
dz
zn=nzn
the monomials inzdiagonalize the number and energy operators, so one has
zn
√
n!
for the normalized energy eigenstate of energy~ω(n+^12 ).
Note that we are here taking the state spaceFto include infinite linear
combinations of the states|n〉, as long as the Bargmann-Fock norm is finite.
We will sometimes want to restrict to the subspace of finite linear combinations
of the|n〉, which we will denoteFfin. This is the spaceC[z] of polynomials,
andFis its completion for the Bargmann-Fock norm.
22.4 Quantization by annihilation and creation operators
The introduction of annihilation and creation operators involves allowing linear
combinations of position and momentum operators with complex coefficients.
These can be thought of as giving a Lie algebra representation ofh 3 ⊗C, the
complexified Heisenberg Lie algebra. This is the Lie algebra of complex poly-
nomials of degree zero and one on phase spaceM, with a basis 1,z,z. One
has
h 3 ⊗C= (M⊗C)⊕C
with
z=1
√
2
(q−ip), z=1
√
2
(q+ip)