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Section 10.1: Classical Particles and Wave Packets in an Infinite Well
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We are going to study the one-dimensional infinite square well problem and use it to illustrate many quantum-mechanical principles. First consider a classical particle confined to a box of length L by an infinite potential well:
V = ∞ x ≤ 0 , V = 0 0 < x < L , V = ∞ x ≥ L . (10.1)
Classically, we would try to use the relationship between the force and the potential energy: Fx = −dV/dx and then get x(t) and v(t) from Fx = max. However, because the equation above is such a badly-behaved potential energy function, this is impossible to do. Thankfully, determining the trajectory of the particle is relatively straightforward. The classical particle moves freely at a constant velocity (since its potential energy is constant) until it elastically collides with a boundary wall, in which case its velocity changes sign and the particle continues on its way with the exact same speed moving in the opposite direction. This motion is shown in the animation. Think about how you would describe the particle's position and momentum as a function of time. Restart.
Another similar classical situation is that of a wave packet on an idealized string of length L with its ends fixed. Wave packet motion of this type is also shown in the animation.1 Note how the wave packet moves and that as it encounters the end of the string it reflects.
1This wave is dispersionless. Waves with dispersion are covered in Section 5.11.