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. $\hslash$ (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 $E_0=\hslash \omega /2$.

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:

$$E_n=(N+\frac{1}{2})\hslash \omega$$