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492
Quantum Mechanics
Q41.7
Consider Figure 41.8, (a) and (b) in the text. In the square well with infinitely high walls, the
particle’s simplest wave function has strict nodes separated by the length
L
of the well. The particle’s
wavelength is 2
L
, its momentum
h
L
2
, and its energy
p
m
h
mL
2
2
2
2
8
=
. Now in the well with walls of only
finite height, the wave function has nonzero amplitude at the walls. The wavelength is longer. The
particle’s momentum in its ground state is smaller, and its energy is less.
Q41.8
Quantum mechanically, the lowest kinetic energy possible for any bound particle is greater than
zero. The following is a proof: If its minimum energy were zero, then the particle could have zero
momentum and zero uncertainty in its momentum. At the same time, the uncertainty in its position
would not be infinite, but equal to the width of the region in which it is restricted to stay. In such a
case, the uncertainty product
∆∆
xp
x
would be zero, violating the uncertainty principle. This
contradiction proves that the minimum energy of the particle is not zero. Any harmonic oscillator
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 Fall '11
 Staff
 Physics, mechanics

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