Quantum Mechanics for Mathematicians

a†pj|···,npj− 1 ,npj,npj+1,···〉=

√

npj+ 1|···,npj− 1 ,npj+ 1,npj+1,···〉

The occupation numbernpjis the eigenvalue of the operatora(pj)†a(pj) and
takes values 0, 1 , 2 ,···,∞. It has a physical interpretation as the number of
particles in the state with momentumpj (recall that such momentum values
are discretized in units of^2 Lπ, and in the interval [−Λ,Λ]). The state with all
occupation numbers equal to zero is denoted

|···, 0 , 0 , 0 ,···〉=| 0 〉

and called the “vacuum” state.
Observables that can be built out of the annihilation and creation operators
include

• The total number operator

N̂=

∑

k

a(pk)†a(pk) (36.8)

which will have as eigenvalues the total number of particles

N̂|···,npj− 1 ,npj,npj+1,···〉= (

∑

k

npk)|···,npj− 1 ,npj,npj+1,···〉

• The momentum operator

P̂=

∑

k

pka(pk)†a(pk) (36.9)

with eigenvalues the total momentum of the multi-particle system.

P̂|···,npj− 1 ,npj,npj+1,···〉= (

∑

k

npkpk)|···,npj− 1 ,npj,npj+1,···〉

• The Hamiltonian
  Ĥ=

∑

k

p^2 k

2 m

a(pk)†a(pk) (36.10)

which has eigenvalues the total energy

Ĥ|···,npj− 1 ,npj,npj+1,···〉=

(

∑

k

npk

p^2 k

2 m

)

|···,npj− 1 ,npj,npj+1,···〉

With ultraviolet and infrared cutoffs in place, the possible values ofpjare
of a finite numberDwhich is also the complex dimension ofH 1. The Hamil-
tonian operator is the standard harmonic oscillator Hamiltonian, with different
frequencies

ωj=

p^2 j

2 m