Exam 2
Chem 4502 - Spring 2000 - Prof. Darrin York
80 points
Constants:
h (Planck’s constant) = 6.6262 × 10−34 J·s = 4.1352 × 10−15 eV·s
~ = h/(2π)
c (speed of light) = 2.9979 × 108 m·s−1
NA (Avogadro’s number) = 6.022 × 10−23 mol−1
β (Bohr magneton) = 9.274 × 10−24 J·T−1
Units:
1
1
1
1
eV = 1.6022 × 10−19 J
amu = 1.66056 × 10−27 kg
Å= 10−10 m
N = 1 J·m−1 = 1 kg·m·s−2
Hydrogen-like atom with atomic number Z:
En = −
Z 2 e2
(no magnetic field)
8π0 a0 n2
ψnlm = Ylm (θ, φ)Rnl (r)
3/2
3/2
1
Z
1
Z
−Zr/a0
0
e
, and ψ1s = ψ100 = √
Y0 = √ , R10 = 2
e−Zr/a0
a0
π a0
4π
Possibly useful Integrals:
Z
∞
rn e−αr dr =
0
b
Z
2 −αr
r e
0
∞
Z
2 −αr
r e
b
2
2
dr = 3 − 3
α
α
2
dr = 3
α
n!
αn+1
b2 α2 −αb
1 + bα +
e
2
b2 α2 −αb
1 + bα +
e
2
(1)
(2)
1. (10 pts) Hydrogen.
What is the probability of finding a 1s electron of hydrogen (Z = 1)
within 2a0 of the nucleus, where a0 is the Bohr radius?
ANSWER:
Z 2a0
Z
2
2
r |Rnl (r)| dr =
3
Z
4
r2 e−2Zr/a0 dr
a
0
0
0
Let α = a20 , and since Z = 1,
Z
4 2a0 2 −αr
4 2
2
4a20 α2 −2a0 α
... = 3
r e
dr = 3
−
1 + 2a0 α +
e
a0 0
a0 α3 α3
2
4 a3 = 3 0 1 − 13e−4 = 1 − 13e−4 = 0.762
a0 4
2a0
2. (10 pts) Virial Theorem.
a) Write the equation for the quantum mechanical virial theorem and use
it to determine the kinetic energy of the hydrogen-like atom in terms of the
potential energy.
ANSWER:
∂V
∂V
∂V
<x
+y
+z
>= 2 < K̂ >
∂x
∂y
∂z
or n < V̂ >= 2 < K̂ > or V̂ = Crn
b) Use the virial theorem to determine the value of the kinetic energy for
the ground state of the hydrogen -like atom. You can use the expression for
the total energy of the hydrogen-like atom given on the front page.
ANSWER:
1
From a) we have < K̂ >= − < V̂ >,
2
1
Since < E >=< K̂ > + < V̂ >, < K̂ >= − < E > − < K̂ >
2
Z 2 e2
So, < K̂ >= − < E >=
for n = 1
8π0 a0
3. (10 pts) Variational principle.
Consider the trial wave function φ = e−αr , where α is a parameter. State the quantum mechanical variational principle (either by a formula or in
words), and use it (i.e. solve for α) to estimate the ground state energy and
wave function for the hydrogen-like atom (show your work).
~2 2
Ze2
Ĥ = − ∇ −
2µ
4π0 r
Some useful integrals are:
Z
e2 a 0
Ze2
e−αr Ĥe−αr d3 r =
−
80 α 40 α2
(3)
(4)
where a0 is the Bohr radius, and
Z
e−2αr d3 r =
φ∗ Ĥφ dτ
E[φ] = R ∗
φ φdτ
R
π
α3
Ze2 α3
e2 a 0
−
80 α 40 α2 π
=
=
e2 a0 α2 Ze2 α
−
80 π
4π0
dE
e2 a0 α
Ze2
=
−
=0
dα
4π0
4π0
α=
Z
a0
−Z 2 e2
Emin =
80 a0 π
(5)
4. (10 pts) Perturbation theory.
a) Given the Hamiltonian for a 1-dimensional quartic oscillator:
Ĥ = −
~2 d 2
+ cx4
2
2µ dx
(6)
(1)
What is the formula for the first order energy correction En based on the
(0)
wave function ψn for the harmonic oscillator Hamiltonian with potential
energy (1/2)kx2 ? (Make sure you correctly identify Ĥ (1) - you don’t have to
calculate any integrals, just write the formula).
ANSWER:
Z
1
(1)
En = ψn(0)∗ Ĥ (1) ψn(0) dτ , with Ĥ (1) = cx4 − kx2 .
2
b) What is the formula for the first order correction to the wave function
(identify all terms)?
(1)
ψn
ANSWER:
Z
(1)
X
Ĥkn
(0)
(1)
(0)∗
(1)
ψn =
ψ , with Ĥkn = ψk H (1) ψn(0) dτ .
(0)
(0) k
k6=n En − Ek
5. (10 pts) Many-electron systems.
a) Write the Hamiltonian for an N-electron atom with nuclear charge Z.
Identify the kinetic energy, nuclear attraction, and electron-electron repulsion
terms. Use either atomic units or SI units.
Ĥ =
N
X
|i
N
N
N
1 2 X Z XX 1
− ∇i −
+
2
ri
rij
i
{z } | {z } | i j>i
{z }
kin. ener.
nuc. attr.
elec. repul.
b) Which of the following are valid wave functions in terms of the spinorbitals ψ(x) according to the Pauli principle?
a.
ψ1 (x1 ) ψ2 (x1 ) · · · ψN (x1 )
ψ1 (x2 ) ψ2 (x2 ) · · · ψN (x2 )
Ψ = ..
..
..
.
.
.
ψ1 (xN ) ψ2 (xN ) · · · ψN (xN )
b. Ψ = ψ1 (x1 )ψ2 (x2 )
c. Ψ = ψ1 (x1 )ψ1 (x2 ) − ψ2 (x1 )ψ2 (x2 )
d. Ψ = ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 )
only A and D are valid
6. (10 pts) Term symbols.
Give the ground state term symbols for
a) phosphorus [Ne]3s2 3p3
↑ ↑↑
−1 0 1
ms =
3
3
→ S=
2
2
m` = 0 → L = 0 → S
3
3
3
J = 0 + , . . . , 0 − =
2
2
2
4
S3
2
2
3
b) vanadium [Ar]4s 3d
ms =
Shell is <
1
2
3
3
→ S=
2
2
m` = 3 → L = 3 → F
9 7 5 3
3
3
J = 3 + , . . . , 3 − = , , ,
2
2
2 2 2 2
filled ∴ smallest J is the ground state.
4
F3
2
7. (10 pts) Atomic spectra.
Some of the electronic states of sodium are designated by the term symbols:
1
S1/2 , 2 P3/2 , 2 P1/2 , 2 D3/2 , 2 F5/2
What are the allowed transitions between these states according to the
selection rules for atomic spectra?
SELECTION RULES:
∆L = ±1
∆J = 0, ±1, but J = 0 6−→ J = 0
∆S = 0
ALLOWED TRANSITIONS:
2
P 3 −→2 D 3 because ∆S = 0, ∆L = 1, ∆J = 0
2
2
P 1 −→2 D 3 because ∆S = 0, ∆L = 1, ∆J = 1
2
2
2
2
D 3 −→2 F 5 because ∆S = 0, ∆L = 1, ∆J = 1
2
2
8. (10 pts) Electronic states.
a) How many total electronic states are possible for a 4f 12 electronic
configuration?
G!
14 · 13 · 12 · ...1
=
N ! (G − N )!
12 · 11 · ...1 (2 · 1)
14 · 13
=
= 91 states
2
b) How many electronic states are associated with a 3 D term symbol (how
many sets of magnetic quantum numbers ml and spin quantum numbers ms )?
(2l + 1) (2s + 1) = (2 · 2 + 1) (2 · 1 + 1)
= 5 · 3 = 15 states
(4 pts) Extra credit.
True/False
Every linear combination of solutions of the time-independent Schrödinger
equation is also a solution of this equation.
FALSE
Every linear combination of solutions of the time-dependent Schrödinger
equation is also a solution of this equation.
TRUE