Wave eigenfunctions of quantum harmonic oscillator

The time-independent Schrödinger equation describing the wave function of the quantum oscillator can be solved to get the corresponding wave eigenfunctions and energy eigenvalues of the Hamiltonian operator. Each eigenfunction describes a stationary state of the quantum mechanical system with the corresponding energy value (eigenvalue of the Hamiltonian). The combination of all eigenfunctions and eigenvalues represent the energy states allowed.

Notation:

1. $\hslash$ (hbar) is hbar.

2. $H_n$ (hermite) is the $n$-th physicists’ Hermite polynomial.

Links:

1. Wikipedia.

wave_function

$n$-th wave eigenfunction (solution of the time-independent Schrödinger equation). See wave_function.

Symbol:

psi

Latex:

$\psi$

Dimension:

1/sqrt(length)

mode_number

Mode number. See nonnegative_number.

Symbol:

N

Latex:

$N$

Dimension:

dimensionless

oscillator_mass

mass of the oscillator.

Symbol:

m

Latex:

$m$

Dimension:

mass

angular_frequency

angular_frequency of the oscillator.

Symbol:

w

Latex:

$\omega$

Dimension:

angle/time

position

position of the oscillator.

Symbol:

x

Latex:

$x$

Dimension:

length

law

psi = (m * w / (pi * hbar))^(1/4) / sqrt(2^N * factorial(N)) * exp(-m * w / (2 * hbar) * x^2) * hermite(N, sqrt(m * w / hbar) * x)

Latex:

$$\psi =\frac{\sqrt[4]{\frac{m\omega }{\pi \hslash }}}{\sqrt{2^{N}N!}}\exp (-\frac{m\omega }{2\hslash }{x}^2)H_N\left(\sqrt{\frac{m\omega }{\hslash }}x\right)$$