Quantum oscillator

The quantum harmonic oscillator

Simulation

Challenges

Wave function $ψ_{0}$

$x$

Potential energy $V (x)$

$x$

The graphs show you the spatial part of the energy eigenfunction $ψ_{n} (x)$ or the probability density $| ψ_{n} (x) |^{2}$ and the potential energy $V (x) = \frac{1}{2} m ω^{2} x^{2}$ of a particle (mass $m$ ) confined to a one-dimensional harmonic oscillator potential. Press the ? buttons for more information. Then try the challenges in the Challenges tab!

Main controls

Quantum number $n$ = 0

$ψ_{n} (x)$ graph

Show energy

$E_{0} = ℏ ω (0 + \frac{1}{2}) = \frac{1}{2} ℏ ω$

$| ψ_{n} (x) |^{2}$ graph

Show classical probability density Show classical turning points

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What is the spacing between adjacent energy levels ( $E_{n}$ and $E_{n + 1}$ ) for the quantum harmonic oscillator?

$0.5 ℏ ω$

$ℏ ω$

$n ℏ ω$

$(2 n + 1) ℏ ω$