Quantum Harmonic Oscillator Simulator

ℏω = 1.00
V(x) Potential Well
|ψ(x,t)|² Probability Density
Re(ψ) Real Component
Classical Probability P_cl(x)
0.00s
1.0x
1.00
Preset Quantum Configurations

Nodes and quantum interference in the saved oscillator

Read the explanation

The saved quantum tool uses dimensionless mass and Planck constant equal to one. At frequency one, its ground wavefunction is pi to the negative one quarter times exponential negative x squared over two. Squaring the magnitude gives probability density pi to the negative one half times exponential negative x squared. At the center this is about point five six four. At x equal to one it falls to point two zero eight. The curve is density, not the probability of one exact point. An interval probability requires integrating its area. The first excited state multiplies the ground spatial wavefunction by square root of two times x. Its density is therefore two x squared times the ground density. The center becomes a node with exactly zero density, while x equal to one has density about point four one five. Each eigenstate acquires a complex phase in time. That rotates real and imaginary components but preserves the density of a single eigenstate. The saved model uses energy n plus one half when frequency is one. For equal zero-phase amplitudes in states zero and one, the source normalizes both coefficients to one over square root of two. At x equal to one, their spatial amplitudes are about point four five six and point six four four. At time zero constructive interference gives density about point six zero five. At time pi, the relative phase reverses and density falls to about point zero one seven eight. These bars are computed from that same source superposition, not a claim that a particle follows a classical trajectory.

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