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 Incoming links: Nondegeneracy propagated by an oscillator ladder

Let

ψ

E_{n}

and put

γ = 2 m ℏ ω

. The lowering norm formula gives, inductively,

∥ (a_{-})^{j} ψ ∥^{2} = γ^{j} \frac{n !}{( n - j )!} ∥ ψ ∥^{2}, 0 \leq j \leq n .

In particular

(a_{-})^{n} ψ

is a nonzero ground state. By the assumed nondegeneracy, it equals

c ψ_{0}

for some

c \neq = 0

It remains to show that raising this vector recovers

ψ

, rather than merely another state at the same energy. On an energy eigenstate at

E_{n}

a_{+} a_{-} ψ = γn ψ

. Iterating the factorization down the ladder yields

(a_{+})^{n} (a_{-})^{n} ψ = γ^{n} n! ψ .

Applying

(a_{+})^{n}

(a_{-})^{n} ψ = c ψ_{0}

therefore proves

ψ = \frac{c}{( 2 m ℏ ω ) ^{n} n !} (a_{+})^{n} ψ_{0} .

For

n = 0

this is the original ground-state assumption. This proves nondegeneracy propagated by an oscillator ladder: every energy eigenspace is one-dimensional if the ground state is nondegenerate.