Quantum Mechanics for Mathematicians

−^32 ~ω

−^12 ~ω

0

1

2 ~ω

3

2 ~ω

5

2 ~ω

7

2 ~ω

Energy

Bosonic

| 0 , 0 , 0 〉

| 1 , 0 , 0 〉, | 0 , 1 , 0 〉, | 0 , 0 , 1 〉

| 1 , 1 , 0 〉, | 1 , 0 , 1 〉, | 0 , 1 , 1 〉,

| 2 , 0 , 0 〉, | 0 , 2 , 0 〉, | 0 , 0 , 2 〉

Fermionic

| 0 , 0 , 0 〉

| 1 , 0 , 0 〉, | 0 , 1 , 0 〉, | 0 , 0 , 1 〉

| 1 , 1 , 0 〉, | 1 , 0 , 1 〉, | 0 , 1 , 1 〉

| 1 , 1 , 1 〉

Figure 27.1:N= 3 oscillator energy eigenstates.

shows the pattern of states and their energy levels for the bosonic and fermionic
cases. In the bosonic case the lowest energy state is at positive energy and
there are an infinite number of states of ever increasing energy. In the fermionic
case the lowest energy state is at negative energy, with the pattern of energy
eigenvalues of the finite number of states symmetric about the zero energy level.
Just as in the bosonic case, we can consider quadratic combinations of cre-
ation and annihilation operators of the form

UA′ =

∑

j,k

a†FjAjkaFk

and we have

Theorem 27.1.ForA∈gl(d,C)adbydcomplex matrix one has

[UA′,UA′′] =U[A,A′]