Quantum harmonic oscillator equation

The Schrödinger equation for the quantum simple harmonic oscillator governs the wave function of the quantum oscillator.

Notation:

1. \(\hbar\) (hbar) is hbar.

Links:

1. Physics LibreTexts, formula 7.6.4.

position

position of the particle.

Symbol:

x

Latex:

\(x\)

Dimension:

length

wave_function

wave_function of the oscillating particle.

Symbol:

psi(x)

Latex:

\(\psi{\left(x \right)}\)

Dimension:

1/sqrt(length)

particle_mass

mass of the particle.

Symbol:

m

Latex:

\(m\)

Dimension:

mass

particle_energy

energy of the particle.

Symbol:

E

Latex:

\(E\)

Dimension:

energy

angular_frequency

angular_frequency of the oscillations.

Symbol:

w

Latex:

\(\omega\)

Dimension:

angle/time

law

-hbar^2 / (2 * m) * Derivative(psi(x), (x, 2)) + m * w^2 / 2 * x^2 * psi(x) = E * psi(x)

Latex:

\[- \frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi{\left(x \right)} + \frac{m \omega^{2}}{2} x^{2} \psi{\left(x \right)} = E \psi{\left(x \right)}\]