QM of a particle in a harmonic oscillator

A particle in the harmonic oscillator potential

𝑉 (𝑥) = 12 𝑚 𝜔 2 𝑥 2 = 12 𝜔 ℏ 𝜉 2

where $𝜉 = \sqrt 𝑚 𝜔 / ℏ 𝑥$ is a dimensionless variable has ground state

𝜓 0 (𝑥) = (𝑚 𝜔 𝜋 ℏ) 1 / 4 𝑒 - 𝑚 𝜔 𝑥 2 / 2 ℏ = (𝑚 𝜔 𝜋 ℏ) 1 / 4 𝑒 - 𝜉 2 / 2

with all stationary states and their energies given by

𝜓 𝑛 = 1 \sqrt 𝑛! (ˆ 𝑎 +) 𝑛 𝜓 0 𝐸 𝑛 = (𝑛 + 12) ℏ 𝜔

where $𝑛 \in ℕ 0$ and

ˆ 𝑎 \pm = 𝑚 𝜔 ˆ 𝑥 \mp 𝑖 ˆ 𝑝 \sqrt 2 𝑚 𝜔 ℏ = 1 \sqrt 2 (ˆ 𝜉 \mp 𝑑 𝑑 𝜉)

are the so-called ladder operators (see properties below).

Proof of solutions

The time independent Schrödinger equation is

1 2 𝑚 (ˆ 𝑝 2 + (𝑚 𝜔 ˆ 𝑥) 2) 𝜓 = 𝐸 𝜓

which is only normalizable for $𝐸 > 0$ (see). Motivated by finding a “difference of perfect squares” like representation for $ˆ 𝐻$ ,¹ we define the ladder operators given above with the properties listed below. thus the time-independent Schrödinger equation becomes

ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp \pm 12) 𝜓 = 𝐸 𝜓

Crucially, $ˆ 𝑎 \pm$ have the property that given a solution $ˆ 𝐻 𝜓 = 𝐸 𝜓$ to the TISE, then $ˆ 𝑎 \pm 𝜓$ also solve the Schrödinger equation:

ˆ 𝐻 (ˆ 𝑎 \pm 𝜓) = ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp \pm 12) (ˆ 𝑎 \pm 𝜓) = ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp ˆ 𝑎 \pm \pm 12 ˆ 𝑎 \pm) 𝜓 = ℏ 𝜔 ˆ 𝑎 \pm (ˆ 𝑎 \mp ˆ 𝑎 \pm \pm 12) 𝜓 = ˆ 𝑎 \pm ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp \pm 1 \pm 12) 𝜓 = ˆ 𝑎 \pm (ˆ 𝐻 \pm ℏ 𝜔) 𝜓 = ˆ 𝑎 \pm (𝐸 \pm ℏ 𝜔) 𝜓 = (𝐸 \pm ℏ 𝜔) ˆ 𝑎 \pm 𝜓

which also follows from the defining property of Ladder operators. Since successively applying $ˆ 𝑎 -$ lowers energy, and normalizable solutions have nonnegative energy, the sequence must terminate with $ˆ 𝑎 - 𝜓 0 = 0$ . Finding this “bottom rung” amounts to solving the differential equation

𝑑 𝜓 0 𝑑 𝑥 = - 𝑚 𝜔 ℏ 𝑥 𝜓 0

is a First-order linear differential equation with normalized solution

𝜓 0 (𝑥) = (𝑚 𝜔 𝜋 ℏ) 1 / 4 𝑒 - 𝑚 𝜔 𝑥 2 / 2 ℏ

All normalizable solutions must be given by the ladder operators, since otherwise an alternate bottom rung could be found.

Orthonormality

It follows from ^LP8 and ^LP9 that

| 𝜓 𝑛 ⟩ = 1 \sqrt 𝑛! (ˆ 𝑎 +) 𝑛 | 𝜓 0 ⟩

hence the states given above are normalized. Orthogonality is manifest in

𝑛 ⟨ 𝜓 𝑚 | 𝜓 𝑛 ⟩ = ⟨ 𝜓 𝑚 | ˆ 𝑎 + ˆ 𝑎 - | 𝜓 𝑛 ⟩ = ⟨ 𝜓 𝑚 | (ˆ 𝑎 + ˆ 𝑎 -) † | 𝜓 𝑛 ⟩ = 𝑚 ⟨ 𝜓 𝑚 | 𝜓 𝑛 ⟩

and hence $(𝑚 - 𝑛) ⟨ 𝜓 𝑚 | 𝜓 𝑛 ⟩ = 0$ , implying $⟨ 𝜓 𝑚 | 𝜓 𝑛 ⟩ = 0$ for $𝑚 \neq 𝑛$ .

An alternate representation in terms of Hermite polynomials² is³

𝜓 𝑛 (𝑥) = (𝑚 𝜔 𝜋 ℏ) 1 / 4 1 \sqrt 2 𝑛 𝑛! 𝐻 𝑛 (𝜉) 𝑒 - 𝜉 2 / 2

Properties

The harmonic oscillator potential is a good approximation for many potentials with a minimum at $0$ , since $𝑉 (𝑥) = 𝑉 (0) + 𝑉' (0) 𝑥 + 12 𝑉 ″ (0) 𝑥 2 + \dots$ .
The following general equations for expectation values hold for a stationary state $\ket{\psi_{n}}$
- $⟨ ˆ 𝑥 ⟩ = 0$
- $⟨ ˆ 𝑝 ⟩ = 0$
- $⟨ ˆ 𝑥 2 ⟩ = (𝜎 ˆ 𝑥) 2 = 𝐸 𝑛 / 𝑚 𝜔 2 = (𝑛 + 12) ℏ 𝑚 𝜔$
- $⟨ ˆ 𝑝 2 ⟩ = (𝜎 ˆ 𝑝) 2 = 𝑚 𝐸 𝑛 = (𝑛 + 12) 𝑚 ℏ 𝜔$
- $⟨ 𝑉 ⟩ = 12 𝑚 𝜔 2 ⟨ ˆ 𝑥 2 ⟩ = 𝐸 𝑛 2 = (𝑛 + 12) ℏ 𝜔 2$
- $⟨ 𝑇 ⟩ = ⟨ 𝑝 2 ⟩ 2 𝑚 = 𝐸 𝑛 2 = (𝑛 + 12) ℏ 𝜔 2$

Proof of 2

Clearly $⟨ 𝜓 𝑛 | ˆ 𝑥 | 𝜓 𝑛 ⟩ = \int \infty - \infty 𝑥 | 𝜓 𝑛 (𝑥) | 2 𝑑 𝑥 = 0$ by Integration properties, proving ^Ex. Invoking various Properties of the ladder operators

⟨ 𝜓 𝑛 | ˆ 𝑝 | 𝜓 𝑛 ⟩ = 𝑖 \sqrt ℏ 𝑚 𝜔 2 ⟨ 𝜓 𝑛 | (ˆ 𝑎 + - ˆ 𝑎 -) | 𝜓 𝑛 ⟩ = 𝑖 \sqrt ℏ 𝑚 𝜔 2 (⟨ 𝜓 𝑛 | ˆ 𝑎 + | 𝜓 𝑛 ⟩ - ⟨ 𝜓 𝑛 | ˆ 𝑎 - | 𝜓 𝑛 ⟩) = 𝑖 \sqrt ℏ 𝑚 𝜔 2 (\sqrt 𝑛 + 1 ⟨ 𝜓 𝑛 | 𝜓 𝑛 + 1 ⟩ - \sqrt 𝑛 + 1 ⟨ 𝜓 𝑛 + 1 | 𝜓 𝑛 ⟩) = 0 ⟨ 𝜓 𝑛 | ˆ 𝑥 2 | 𝜓 𝑛 ⟩ = ℏ 2 𝑚 𝜔 ⟨ 𝜓 𝑛 | (ˆ 𝑎 + + ˆ 𝑎 -) 2 | 𝜓 𝑛 ⟩ = ℏ 2 𝑚 𝜔 ⟨ 𝜓 𝑛 | ((ˆ 𝑎 +) 2 + (ˆ 𝑎 -) 2 + {ˆ 𝑎 +, ˆ 𝑎 -}) | 𝜓 𝑛 ⟩ = ℏ 2 𝑚 𝜔 (⟨ 𝜓 𝑛 | (ˆ 𝑎 +) 2 | 𝜓 𝑛 ⟩ + ⟨ 𝜓 𝑛 | (ˆ 𝑎 -) 2 | 𝜓 𝑛 ⟩ + 2 ℏ 𝜔 ⟨ 𝜓 𝑛 | ˆ 𝐻 | 𝜓 𝑛 ⟩) = 𝐸 𝑛 𝑚 𝜔 2 = (𝑛 + 12) ℏ 𝑚 𝜔 ⟨ 𝜓 𝑛 | ˆ 𝑝 2 | 𝜓 𝑛 ⟩ = - ℏ 𝑚 𝜔 2 ⟨ 𝜓 𝑛 | (ˆ 𝑎 + - ˆ 𝑎 -) 2 | 𝜓 𝑛 ⟩ = - ℏ 𝑚 𝜔 2 ⟨ 𝜓 𝑛 | ((ˆ 𝑎 +) 2 + (ˆ 𝑎 -) 2 - {ˆ 𝑎 +, ˆ 𝑎 -}) | 𝜓 𝑛 ⟩ = - ℏ 𝑚 𝜔 2 (⟨ 𝜓 𝑛 | (ˆ 𝑎 +) 2 | 𝜓 𝑛 ⟩ + ⟨ 𝜓 𝑛 | (ˆ 𝑎 -) 2 | 𝜓 𝑛 ⟩ - 2 ℏ 𝜔 ⟨ 𝜓 𝑛 | ˆ 𝐻 | 𝜓 𝑛 ⟩) = 𝑚 𝐸 𝑛 = (𝑛 + 12) 𝑚 ℏ 𝜔

proving ^Ep, ^Ex2, and ^Ep2, whence ^EV and ^ET immediately follow.

Properties of the ladder operators

$ˆ 𝑎 \mp ˆ 𝑎 \pm = ˆ 𝐻 ℏ 𝜔 \pm 12$
$[ˆ 𝑎 -, ˆ 𝑎 +] = 1$
$ˆ 𝐻 = ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp \pm 12)$
$[ˆ 𝐻, ˆ 𝑎 \pm] = \pm ℏ 𝜔 ˆ 𝑎 \pm$
$(ˆ 𝑎 \pm) † = ˆ 𝑎 \mp$
$ˆ 𝑥 = \sqrt ℏ 2 𝑚 𝜔 (ˆ 𝑎 + + ˆ 𝑎 -)$
$ˆ 𝑝 = 𝑖 \sqrt ℏ 𝑚 𝜔 2 (ˆ 𝑎 + - ˆ 𝑎 -)$
$ˆ 𝑎 + | 𝜓 𝑛 ⟩ = \sqrt 𝑛 + 1 | 𝜓 𝑛 + 1 ⟩$
$ˆ 𝑎 - | 𝜓 𝑛 ⟩ = \sqrt 𝑛 | 𝜓 𝑛 - 1 ⟩$ for $𝑛 > 0$
${ˆ 𝑎 \pm, ˆ 𝑎 \mp} = 2 ˆ 𝐻 ℏ 𝜔$

Proof of 1–5, 8–10

Properties 1–3 and 10 follow from

ˆ 𝑎 \mp ˆ 𝑎 \pm = 1 2 ℏ 𝑚 𝜔 (ˆ 𝑝 2 + (𝑚 𝜔 ˆ 𝑥) 2) \mp 𝑖 2 ℏ [ˆ 𝑥, ˆ 𝑝] = ˆ 𝐻 ℏ 𝜔 \pm 12

For ^LP4 note

[ˆ 𝐻, ˆ 𝑎 \pm] = [ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp \pm 12), ˆ 𝑎 \pm] = ℏ 𝜔 [ˆ 𝑎 \pm ˆ 𝑎 \mp, ˆ 𝑎 \pm] \pm ℏ 𝜔 [12, ˆ 𝑎 \pm] = ℏ 𝜔 (ˆ 𝑎 \pm ˆ 𝑎 \mp ˆ 𝑎 \pm - ˆ 𝑎 \pm ˆ 𝑎 \pm ˆ 𝑎 \mp) = ℏ 𝜔 ˆ 𝑎 \pm [ˆ 𝑎 \mp, ˆ 𝑎 \pm] = \pm ℏ 𝜔 ˆ 𝑎 \pm

which shows that these are indeed ladder operators, and thus ^LP5 follows from ^P1.⁴

From ^LP5 we have

⟨ 𝜓 𝑛 | (ˆ 𝑎 \pm) † ˆ 𝑎 \pm | 𝜓 𝑛 ⟩ = ⟨ 𝜓 𝑛 | ˆ 𝑎 \mp ˆ 𝑎 \pm | 𝜓 𝑛 ⟩

but from the Schrödinger equation, ^LP3, and the ^energies formula, it follows that

ˆ 𝑎 + ˆ 𝑎 - | 𝜓 𝑛 ⟩ = 𝑛 | 𝜓 𝑛 ⟩, ˆ 𝑎 - ˆ 𝑎 + | 𝜓 𝑛 ⟩ = (𝑛 + 1) | 𝜓 𝑛 ⟩

hence

⟨ 𝜓 𝑛 | (ˆ 𝑎 +) † ˆ 𝑎 + | 𝜓 𝑛 ⟩ = ⟨ 𝜓 𝑛 | ˆ 𝑎 - ˆ 𝑎 + | 𝜓 𝑛 ⟩ = (𝑛 + 1) ⟨ 𝜓 𝑛 | 𝜓 𝑛 ⟩ ⟨ 𝜓 𝑛 | (ˆ 𝑎 -) † ˆ 𝑎 - | 𝜓 𝑛 ⟩ = ⟨ 𝜓 𝑛 | ˆ 𝑎 + ˆ 𝑎 - | 𝜓 𝑛 ⟩ = 𝑛 ⟨ 𝜓 𝑛 | 𝜓 𝑛 ⟩

#state/tidy | #lang/en | #SemBr

2018. Introduction to quantum mechanics, §2.3.1, pp. 40ff ↩
Normalized so that the highest power of $𝜉$ has coëfficient $2 𝑛$ . ↩
2018. Introduction to quantum mechanics, §2.3.2, p. 52 ↩
This follows from completeness since the behaviour of $ˆ 𝑎 -$ matches that predicted by ^P1 for $(ˆ 𝑎 +) †$ all eigenfunctions, and therefore is the same operator. ↩

QM of a particle in a harmonic oscillator

Properties

Properties of the ladder operators

Footnotes