Quantum Mechanics for Mathematicians

0

~ω

2 ~ω

3 ~ω

Energy

| 0 , 0 〉

| 1 , 0 〉

| 2 , 0 〉

| 3 , 0 〉

| 0 , 1 〉

| 1 , 1 〉

| 2 , 1 〉

Q+

Q+

Q+

Q−

Q−

Q−

Figure 33.1: Energy eigenstates in the supersymmetric oscillator.

Computing anticommutators using the CCR and CAR for the bosonic and
fermionic operators (and the fact that the bosonic operators commute with the
fermionic ones since they act on different factors of the tensor product), one
finds that
Q^2 +=Q^2 −= 0

and
(Q++Q−)^2 = [Q+,Q−]+=H
One could instead work with self-adjoint combinations

Q 1 =Q++Q−, Q 2 =

1

i

(Q+−Q−)

which satisfy
[Q 1 ,Q 2 ]+= 0, Q^21 =Q^22 =H (33.1)
The HamiltonianHis a square of the self-adjoint operatorQ++Q−, and
this fact alone tells us that the energy eigenvalues will be non-negative. It also
tells us that energy eigenstates of non-zero energy will come in pairs

|ψ〉, (Q++Q−)|ψ〉

with the same energy. To find states of zero energy (there will just be one,
| 0 , 0 〉), instead of trying to solve the equationH| 0 〉= 0 for| 0 〉, one can look for