Question

To generate the excited states for the quantum harmonic oscillator, one repeatedly applies
the raising operator ˆa+ to the ground state, increasing the energy by ~ω with each step:
ψn = An(ˆa+)
nψ0(x) with En = (n +
1
2
)~ω
where An is the normalization constant and
aˆ± ≡
1
√
2~mω
(∓ipˆ+ mωxˆ).
Given that the normalized ground state wave function is
ψ0(x) = mω
π~
1/4
e
− mω
2~
x
2
,
show that the first excited state is (don’t normalize)
ψ1(x) = A
0
1
mω
π~
1/4
r
2mω
~

Let Lzf = µf, with Lz the z-component (operator) of the orbital angular momentum, µ
an eigenvalue and f an eigenfunction. Show that L+ ≡ Lx + iLy is a raising operator.
Hence, explain why L+ is called a raising operator.

Answer 1

From equation (3), we infer that when L+ acts on f, it raises the eigen value by one unit. So this is the reason, L+ is called a raising operator.