PHY4604–Introduction to Quantum Mechanics
Fall 2004
Practice Test 3
November 22, 2004
These problems are similar but not identical to the actual test. One or two parts will
actually show up.
1. Short answer.
• Recall the Bohr energy levels of the Hydrogen atom are
En =
m
2h̄2 n2
³
e2
4π²0
´2
(1)
Compare the wavelengths of the 2p → 1s transitions in
(a) hydrogen
(b) deuterium (mass of nucleus 2× that of H.
(c) positronium.
• An electron is in the ground state of tritium, a form of heavy hydrogen
where the nucleus has two neutrons in addition to a proton. A nuclear
reaction now changes the nucleus instantaneously to 3 He, i.e. two protons
and a neutron. Calculate the probability that theqelectron remains in the
ground state of 3 He. You may need ψ100 = (2/ 4πa30 ) exp −r/a0 , with
a0 = h̄2 /(mZe2 ), where Z is the nuclear charge.
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• An electron moving in the Coulomb field of a proton is in a state described
by the wave function
√
1
[4ψ100 + 3ψ211 − ψ210 + 10ψ21−1 ]
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(2)
(a) What is the expectation value of the energy?
(b) What is the expectation value of L̂2 ?
(c) What is the expectation value of L̂z ?
• How large would a constant magnetic field have to be to split two H-atom
states which are degenerate in zero field by an amount so as to maximally
absorb light of wavelength λ?
• For two particles a and b such that `a = 1 and `b = 1, argue that it must
be true that
|` = 2, m = 1i = α|ma = 1, mb = 0i + β|ma = 0, mb = 1i,
where `, m are the quantum numbers corresponding to total angular momentum Lq= La + Lb , and find the coefficients α and β. Hint: use
L± |`mi = `(` + 1) − m(m ± 1)|`m ± 1i.
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2. Angular momentum 1. Consider an angular momentum 1 system.
(a) What are the possible eigenvalues of L2 and Lz corresponding to the eigenvectors |` = 1, mi?
(b) In the basis where the eigenvectors |` = 1, mi of the operator L̂z are given
by (1, 0, 0), (0, 1, 0) and (0, 0, 1), construct the matrix representation of
the operator L̂x (Hint: you will need to calculate the matrix elements
hm|L̂x |m0 i.)
(c) Find the eigenvectors of L̂x .
(d) If a system is prepared in the state vector


1
1 
|ψi = √  4 
,
26 −3
what is the probability that a measurement of L̂x yields the value 0?
3
(3)
3. Pauli principle.
Consider two electrons described by the Hamiltonian
H=
where
p̂21
p̂2
+ 2 + V (x1 ) + V (x2 )
2m 2m


 ∞
x < −a/2
V (x) = 0 −a/2 ≤ x ≤ a/2


∞ x > a/2
(4)
(5)
Assume both electrons are in same spin state.
(a) What is the lowest (ground state) energy?
(b) What is the energy eigenfunction for this ground state?
(c) What is the energy and the wave function of the first excited state (still
with equal-spin condition!)?
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