Energy levels of harmonic oscillator

As opposed to the classical harmonic oscillator, the energy levels of a quantum harmonic oscillator are quantized, meaning that its energy take a value out of a discrete range. These energy levels are equidistant, i.e. the difference between successive energy levels is the same for all levels.

Notation:

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

Notes

1. This means that the energy of a quantum oscillator cannot be zero and the lowest it can be is the zero-point energy \(E_0 = \hbar \omega / 2\).

Links:

1. Wikipedia.

energy_level

Energy of the level corresponding to the mode_number.

Symbol:

E_n

Latex:

\(E_{n}\)

Dimension:

energy

mode_number

Quantum number of oscillator, which is any non-negative integer (\(0, 1, 2, \dots\)). See nonnegative_number.

Symbol:

N

Latex:

\(N\)

Dimension:

dimensionless

angular_frequency

angular_frequency of the oscillator.

Symbol:

w

Latex:

\(\omega\)

Dimension:

angle/time

law

E_n = (N + 1/2) * hbar * w

Latex:

\[E_{n} = \left(N + \frac{1}{2}\right) \hbar \omega\]