PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 3
Sept. 13, 2004
Due: Sept. 22, 2004
Reading: “Otherevidence” notes, Peebles Ch. 1
1. Quantum Dumbell. A pair of masses m are held a distance d apart by a
massless rigid rod, as shown.
1a) Quantize this system using the Bohr-Ehrenfest quantization condition for
the angular momentum, and find the form of the energy levels En .
1b) We used the rigid dumbell in our discussion of the specific heat of Hydrogen
gas, H2 . Compare your answer for n = 1 in part 1a) with the energy of the
lowest rotational state of the Hydrogen molecule deduced from specific heat
measurements (or alternatively the temperature at which the rotational
states begin contributing to the specific heat), taking m ' mp , the mass
of the proton, and d ' 1A. [Hint–compare with Bohr’s derivation for the
H-atom.]
1c) Why does this calculation work for the H2 atom (in an order of magnitude
sense!) even though we neglected the potential energy due to the Coulomb
forces (technically, van der Waals forces) holding the molecule together?
2. de Broglie wavelength. Estimate the de Broglie wavelengths of
(a) a thermal neutron; assume the neutron is at high temperatures in thermal
equilibrium such that its kinetic energy is about 3kB T /2, where T = 300K
is room temperature, and the neutron mass can be taken to be the same
as that of a proton,
(b) an electron with energy E=1 GeV=109 eV, where the momentum in this
case is related to the energy by the relativistic expression E 2 = p2 c2 +m2 c4 .
1
3. Maple problems.
3a) Wave packet. Consider the 1D wave packet
ψ(x, t) =
Z ∞
−∞
dkf (k)ei(kx−ωt) ,
(1)
with f (k) = 1/[(k − 5)2 + 4] and ω = k 2 . ( Don’t worry about units yet).
i) First plot Re ψ(x, t0 ) vs. x from -5. to 15., with t0 = 0.3. Note you
may need to integrate over a finite range instead of [−∞, ∞].
ii) Then plot Im ψ(x, t0 ) from -5. to 15..
iii) Plot P (x, t0 ) ≡ (Reψ(x, t0 ))2 + (Imψ(x, t0 ))2 from -5. to 15.
iv) Plot P [x, t1 ] over a convenient range, where t1 is any earlier or later
time you can get to work. In which direction is the packet moving?
3b) Animation. “Animate” the function 1/((x − t)2 + 1), i.e. use the animate
command to show a series of sixteen small plots at successive times, with
t between 0 and 4.
2