PHY-851 QUANTUM MECHANICS I
Homework 2, 30 points
September 10 - 17, 2003
Wave function. Reading: Messiah, Chapters 1-3.
1. /8/ (a) Starting from the Maxwell distribution function of speed of molecules
in a classical gas, find the distribution function of the de Broglie wave
length and the most probable de Broglie wave length of hydrogen molecules
at room temperature.
(b) Many researchers in various laboratories study the Bose-Einstein con-
densation of identical atoms in atomic traps. This phenomenon starts
when, as temperature decreases, the de Broglie wavelength of thermal
motion of the atoms grows up to the average distance between atoms.
Estimate the necessary density of atoms with the mass number A= 100
in the trap, if the condensation temperature is T= 10â7K.
2. /6/ A particle is moving along the x-axis in the potential field U(x) =
α|x|s, where αand sare positive constants. Using the Bohr-Sommerfeld
quantization rule, find the energy spectrum of bound states En.
3. /8/ In systems with free moving charge carriers (metals, plasmas), electric
charge Ze of a nucleus or an impurity is screened by a cloud of free charges
of the opposite sign. The size of the cloud, called the Debye radius rD,
becomes smaller as the density of free carriers increases. The resulting
electrostatic potential (Yukawa potential),
Ï(r) = Ze
reâÎșr, Îș =1
rD
,(1)
in contrast to the pure Coulomb potential (Îșâ0, rDâ â), exponen-
tially falls off at distances greater than rDwhere the system âcenter +
cloudâ looks neutral. It is observed that in hot hydrogen plasmas spectral
lines gradually disappear with the increasing electron density. Explain
this phenomenon by showing (with the aid of the Bohr-Sommerfeld quan-
tization rule) that there is only a finite number of quantum bound states
supported by the screened potential (1).
4. /8/ An electron with energy Emoves over a metallic strip of width athat
can be modelled by a potential well of depth W.
(a) Find the reflection and transmission coefficients.
(b) Find the energy values corresponding to the full transmission (reso-
nances) and explain the wave mechanism of this phenomenon.
(c) For given values of Eand Wfind the width athat gives the maximum
reflection coefficient and explain the wave mechanism for this case.
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