PHY4605–Introduction to Quantum Mechanics II
Spring 2004
Test 2
March 17, 2004
No other materials or calculators allowed. If you can’t do one part of a problem,
solve subsequent parts in terms of unknown answer–define clearly. All parts 10 pts.,
max=70. Problem 1 required, attempt 1 of remaining 3 problems; circle which ones
you want graded.
Possibly helpful formulae and constants
Z π/2
−π/2
dy y cos y sin 2y = 8/9
δ|ni =
Z a/2
nπx
a
cos2
=
a
2
−a/2
Z a/2
nπx
a
sin2
=
a
2
−a/2
iθ
2
|e − 1| = 4 sin2 θ/2
me4 1
En =
2h̄2 n2
H = p̂2 /2m + mω 2 x2 /2
µ
¶
1
En = h̄ω n +
2
Hψ = Eψ
∂ψ
Hψ = ih̄
∂t
h̄2 2
H=−
∇ + V (r)
2m
δE (1) = hn|V̂ |ni
δE (2) =
m6=n
En − Em
1
Y00 = √
4π
s
3
Y10 =
cos θ
4π
s
3
sin θe±iφ
8π
Ã
!
∂
∂
L̂± = h̄e±iφ ± + i cot θ
∂θ
∂φ
hφ|H|φi
≥ E0
hφ|φi
∂
L̂z = −ih̄
∂φ
L̂± = L̂x ± iL̂y
Y1±1 = ∓
q
L̂+ ψ`m = h̄ `(` + 1) − m(m + 1)ψ`m+1
q
L̂− ψ`m = h̄ `(` + 1) − m(m − 1)ψ`m−1
X |hn|V̂ |mi|2
m6=n
X hm|V̂ |ni|mi
L̂2 = L̂21 + L̂22 + 2L̂1z L̂2z +
En − Em
+L̂1+ L̂2− + L̂1− L̂2+
1
1. Short Answer. Must attempt (only) 3 of 6. Circle answers to be graded.
(a) Given the exact normalized eigenfunctions ψn of a quantum system, Hψn =
En ψn , prove the variational principle, i.e. show that for any normalized φ,
hφ|H|φi ≥ E0 , where E0 is the exact ground state energy.
(b) A chemistry book lists the electronic configuration of an N atom in its
ground state as 1s2 2s2 2p3 and states that the wave function for this state is
simply the product of hydrogenic orbitals corresponding to the 7 electrons
in the levels designated. Criticize this statement.
(c) An atomic state is referred to by its spectroscopic term, 3 P2 . What are
its quantum numbers S, L, J? Using hydrogenic wavefunctions ψn`m and
2-particle spin states |SMS i, write down an appropriate wave function for
2 electrons which might correspond to this term.
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(d) Explain the physical origin of fine structure and hyperfine structure in
Hydrogen.
(e) Angular momentum. Given two angular momenta L1 and L2 , with eigenvalues of the operators L21 and L22 given by 6h̄2 and 15h̄2 /4, what are
the allowed possible eigenvalues of the total angular momentum operator
squared L2 = (L1 + L2 )2 ?
(f) H2 molecule. Sketch the effective potential for the two protons as a function of the distance between them for the ground state and first excited
state. In the same picture, sketch the vibrational levels and explain (do
not calculate) how you would estimate them.
3
2. Ring molecule. In calculating electronic structure of ring-shaped molecules,
one is led to consider simplified models. For example, take two identical fermions
of mass m and spin 1/2, moving on a ring of radius a, with spins constrained
to point up. The particles can interact through a potential of form
V (φ1 , φ2 ) = A cos(φ1 − φ2 )
(1)
where φ1 and φ2 are angular positions of the two particles.
First assume A = 0.
(a) Find all allowed energy levels of this system. (Hint: the only dynamics in
this system are the constrained motion (kinetic energy) of a particle going
in a circle. The Hamiltonian may therefore be written as a rigid rotator
H = L2z /2mr2 ).
(b) Find the ground and 1st excited state wave functions.
(c) State the degeneracies of the ground and 1st excited states.
Now assume A is nonzero but weak, A ¿ h̄2 /(2ma2 ).
(d) Find the energies and degeneracies of the states in part c) to first order in
A.
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3. H − Ion. Consider an H − ion, which is a hydrogen atom with two electrons.
(a) Write down the Hamiltonian for this system. You may assume that the
proton is infinitely heavy.
(b) Show that, if you neglect the Coulomb interaction between the electrons,
the Hamiltonian separates into equations for each electron.
(c) Continue to neglect the Coulomb interaction between the electrons, and
assume that both electrons are in the ground (n = 1) state. Write a valid
wavefunction Ψ1s1s for the electrons, including specification of the possible
spin states. Do the same for one electron in a 1s state and the other in a
2s state, Ψ1s2s .
(d) Including the Coulomb interaction between electrons, would you expect
spin triplet states to be lower or higher in energy than spin singlet states
with the same orbital quantum numbers? Why?
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4. Angular momentum.
(a) Sketch the angular and radial forms of the hydrogenic wavefunctions φn`m
for the cases (n = 2, ` = 1, m) and (n = 3, ` = 1, m). Make sure you make
a separate sketch for each m value.
(b) Given eigenstates of angular two independent angular momenta L1 and L2
designated by |`1 , m1 , `2 , m2 i, consider the state
1
1
1
|Ψi = √ |2, 2, 1, −1i + √ |2, 1, 1, 0i − √ |2, 0, 1, 1i.
3
6
2
(2)
Show that this state is an eigenstate of total Lz ≡ L1z + L2z with m = 1.
(c) Show that |ψi is an eigenstate of total L ≡ L1 + L2 with L = 2.
(d) Use the lowering operator to find the eigenstate of L2 and Lz with L = 2
and m = 0 (call it |` = 2, m = 0i) in terms of the |`1 , m1 , `2 , m2 i basis.
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