Half-harmonic quantum oscillator

The half-harmonic quantum oscillator

Simulation

Challenges

Wave function $ψ_{1} (x)$

$x$

Potential energy $V (x)$

$x$

Harmonic oscillator: $V (x) = \frac{1}{2} m ω^{2} x^{2}$ for all $x$

Half-harmonic oscillator: $V (x) = \frac{1}{2} m ω^{2} x^{2}$ for $x > 0$ ; $V = \infty$ for $x \leq 0$

Quantum number $n$ = 1

The graphs show you the spatial parts of the energy eigenfunction or the probability density and the potential energy $V (x)$ of either a one-dimensional quantum harmonic oscillator (parabolic $V (x)$ for all $x$ ) or a half-harmonic oscillator (parabolic $V (x)$ only for positive $x$ and an impenetrable wall at $x \leq 0$ where $V$ goes to infinity). Press the ? buttons for more information. Then try the challenges in the Challenges tab!

Main controls

Quantum number $n$ = 1

Angular frequency $ω$

low

high

$ψ (x)$ graph

$| ψ (x) |^{2}$ graph

$E_{1} = ℏ ω (1 + \frac{1}{2}) = \frac{3}{2} ℏ ω$

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By what factor does the normalization constant of the probability density of the half-harmonic oscillator differ compared with the harmonic oscillator?

greater by a factor of 2

greater by a factor of $\sqrt{2}$

smaller by a factor of $\sqrt{2}$

smaller by a factor of 2