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For a summary see the HyperPhysics website starting with , scrolling down to the and and finally clicking on the .

To get some understanding of the motion of a free particle in the quantum mechanical sense, the PhET interactive simulation may be used.

Plane wave

Run the simulation and select constant for the Potential on the top right. Select plane wave for the Electron Wave Function form and tick the Show energy values and Show reflection and transmission probabilities boxes.

Use the mouse to change the total energy (green line) and/or the potential energy (purple). Observe the effect on the amplitude and on the frequency of the wave.

Observe the imaginary part, the magnitude and the phase of the wave by checking the associated boxes in turn.

Wave packet

Now select wave packet for the Electron Wave Function form. Note that the width of the wave packet slowly widens (disperses). This is wave-packet spreading; no matter how narrow the initial wavefunction, a Schrödinger wave eventually fills all of space. The growth is a reflection of the momentum uncertainty — the wavepacket is confined to a finite width (s, say) and so has a momentum which is uncertain by the amount  \( { \hbar  \over 2 \sigma} \) corresponding to an uncertainty in velocity of  \( { \hbar  \over 2 \sigma m} \) .Thus the spread in the particle's location increases linearly with time.

Use the mouse to change the average total energy and/or the potential energy. Observe the effect on the amplitude and on the frequency of the wave.

Observe the imaginary part, the magnitude and the phase of the wave by checking the associated boxes in turn.