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. \(\hbar\) (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:

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