PhyslibAlpha.QuantumMechanics.HarmonicOscillator.Basic

The quantum harmonic oscillator

i. Overview

The harmonic oscillator is one of the most important examples in non-relativistic quantum mechanics. It describes a particle of mass `m` subject to a positive-definite quadratic potential in `d` dimensions.

- `Basic.lean` : Properties of the potential, definition of isotropic oscillators, kinetic, potential and Hamiltonian operators. - `LadderOperators.lean` : Definitions of the raising/lowering/number operators and their algebraic properties.

ii. Key results

`potentialFunction_apply` : the potential function, expanded to `½m · ∑ᵢ ωᵢ²xᵢ²`.
`potentialOperator_isSelfAdjoint` : the potential operator is self-adjoint.

iii. Table of contents

- A. Basic properties - A.1. Positive mass - A.2. Positive natural frequencies - B. Characteristic lengths - C. The quadratic potential function - C.1. Positive-definite matrix - C.2. Quadratic form - C.3. Potential function - D. Isotropic oscillators - E. Hilbert space - F. Operators - E.1. Kinetic energy - E.2. Potential energy - E.3. Hamiltonian - G. As a quantum system

iv. References

A. Basic properties

A.1. Positive mass

A.2. Positive natural frequencies

B. Characteristic lengths

C. The quadratic potential function

C.1. Positive-definite matrix

C.2. Quadratic form

C.3. Potential function

D. Isotropic oscillators

E. Hilbert space

F. Operators

F.1. Kinetic energy

F.2. Potential energy

F.3. Hamiltonian

G. As a quantum system

2 declarations

theorem

Expansion of the potential function $V(\mathbf{x}) = \frac{1}{2} m \sum_i \omega_i^2 x_i^2$

#potentialFunction_apply

For a $d$ -dimensional quantum harmonic oscillator with mass $m$ and natural frequencies $\omega_i$ (for $i \in \{1, \dots, d\}$ ), the potential function $V$ evaluated at a position vector $\mathbf{x}$ is given by: $V(\mathbf{x}) = \frac{1}{2} m \sum_{i=1}^d \omega_i^2 x_i^2$ where $x_i$ denotes the $i$ -th coordinate of the position vector $\mathbf{x}$ .