492 Quantum MechanicsQ41.7Consider Figure 41.8, (a) and (b) in the text. In the square well with infinitely high walls, theparticle’s simplest wave function has strict nodes separated by the length Lof the well. The particle’swavelength is 2L, its momentum hL2, and its energy pmhmL22228=. Now in the well with walls of onlyfinite height, the wave function has nonzero amplitude at the walls. The wavelength is longer. Theparticle’s momentum in its ground state is smaller, and its energy is less.Q41.8Quantum mechanically, the lowest kinetic energy possible for any bound particle is greater thanzero. The following is a proof: If its minimum energy were zero, then the particle could have zeromomentum and zero uncertainty in its momentum. At the same time, the uncertainty in its positionwould not be infinite, but equal to the width of the region in which it is restricted to stay. In such acase, the uncertainty product ∆∆xpxwould be zero, violating the uncertainty principle. Thiscontradiction proves that the minimum energy of the particle is not zero. Any harmonic oscillator
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