**Quantum Mechanics and Momentum**

Part 7: Plane Waves

Last week, we introduced the time-dependent Schrödinger equation , and we noted that for an energy eigenstate *ψ*_{n}(*x*) with energy *E*_{n} (), we have time-dependent solution , with .

Now, consider the case of a free particle of mass *m*. Here, the potential is everywhere zero, and the energy is given entirely by the kinetic energy. Recalling the (non-relativistic) formula for kinetic energy in terms of momentum, we have . Working in three dimensions, and replacing the classical momentum with the quantum momentum operator , we have , and thus the free particle Hamiltonian:

.

Note that a momentum eigenstate with momentum will also be an energy eigenstate with energy . This can be confirmed by applying to the momentum eigenfunction , where . Now, examining the corresponding solution to the time-dependent Schrödinger equation, we have

.

This, we see, is a wave propagating in the direction of , with (angular) frequency *ω* and (angular) wavenumber . The wavefronts of this wave, the surfaces of constant phase, can be seen to be planes perpendicular to . Thus, this kind of solution is known as a *plane wave*.

We have .

Now, from this we find that the group velocity of our plane wave is

which is just the classical velocity of the particle of mass *m* and momentum *p* (from *p*=*mv*).

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Tags: Free Particle, Friday Physics, Group Velocity, Momentum, physics, Plane Wave, Quantum Mechanics, Schrödinger Equation

This entry was posted on December 12, 2008 at 3:21 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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January 30, 2009 at 12:12 am |

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