QM in 1D position-space

QM of a particle in a harmonic oscillator

A particle in the harmonic oscillator potential

where is a dimensionless variable has ground state

with all stationary states and their energies given by

where and

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

Proof of solutions

The time independent Schrödinger equation is

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

Crucially, have the property that given a solution to the TISE, then also solve the Schrödinger equation:

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

is a First-order linear differential equation with normalized solution

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

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

and hence , implying for .

An alternate representation in terms of Hermite polynomials² is³

Properties

1. The harmonic oscillator potential is a good approximation for many potentials with a minimum at , since .
2. The following general equations for expectation values hold for a stationary state

Proof of 2

Clearly by Integration properties, proving ^Ex. Invoking various Properties of the ladder operators

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

Properties of the ladder operators

9. for

Proof of 1–5, 8–10

Properties 1–3 and 10 follow from

For ^LP4 note

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

From ^LP5 we have

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

hence

thus if and are normalized, and , proving ^LP8 and ^LP9

---

tidy | en | sembr

Footnotes

1. 2018. Introduction to quantum mechanics, §2.3.1, pp. 40ff ↩

2. Normalized so that the highest power of has coëfficient . ↩

3. 2018. Introduction to quantum mechanics, §2.3.2, p. 52 ↩

4. This follows from completeness since the behaviour of matches that predicted by ^P1 for all eigenfunctions, and therefore is the same operator. ↩