Chemistry 362
Spring 2017
Dr. Jean M. Standard
February 10, 2017
Name ___________________________________
Physical Chemistry II – Exam 1
Constant
h
! = h / 2π
c
e
me
ε0
R (Rydberg const.)
Value in SI Units
6.62607×10–34 J s
1.05457×10–34 J s
2.99793×108 m/s
1.60217×10–19 C
9.10938×10–31 kg
8.85419×10–12 C2J–1m–1
109737.3 cm–1
1.) (16 points) The idea of quantization is one of the key elements of quantum theory. Explain the
concept of quantization. Then, using one example system, describe at least one property of the
system that exhibits quantization. Make sure to include in your discussion an equation that illustrates
quantization of the property of interest.
2
2.) (17 points) Consider two operators, Â = d / dx and B̂ = x . For the function f (x) = x e
a constant, evaluate the following quantities.
a.) Â B̂ f ( x )
b.) B̂ Â f ( x )
c.) Do the operators  and B̂ commute?
(a yes or no answer is sufficient)
−ax
, where a is
3
3.) (17 points) An electron in a one-dimensional box of width 6.0 Å undergoes a transition from an
initial state with n=2 to some final state. The wavelength of light absorbed in this transition was
determined to be 56.5 nm. Calculate the quantum number of the final state.
[1 Å = 10–10 m; 1 nm = 10–9 m]
4
⎛
x⎞
4.) (17 points) Consider the function ψ (x) = x ⎜1 − ⎟ on the interval 0 ≤ x ≤ L , where L is a
⎝
L⎠
constant. The function is equal to zero outside this region, ψ (x) = 0 for x < 0 and x > L .
a.) Other than normalization, what are the conditions for an acceptable wavefunction? Does the
wavefunction given above meet these conditions? Explain.
b.) Verify whether or not the wavefunction given above on the interval 0 ≤ x ≤ L is an eigenfunction
of the one-dimensional kinetic energy operator, T̂ . If it is an eigenfunction, give the eigenvalue.
5
5.) (17 points) When light with a wavelength of 3000 Å shines on a metal surface, the kinetic energy of
ejected electrons is 2.960×10–19 J. When light with a wavelength of 5400 Å shines on the same metal
surface, the kinetic energy of ejected electrons is 1.362×10–21 J. From this information, determine the
value of Planck's constant (in units of J s). [1 Å = 10–10 m]
6
6.) (16 points)
a.) For the particle in a one-dimensional box of width a, make a sketch of the second excited state
(n=3) wavefunction and its corresponding probability density. How many nodes does this
wavefunction possess? What are their positions?
b.) What is the most probable position for finding the particle in the second excited state (n=3)?
c.) Determine by explicit calculation the probability of finding the particle in the left one third of the
box if the particle is in the second excited state (n=3).
7
Exam 1 Equation List
r =
L = mvr = n!
€
$ me 4 ' $ 1
1'
ΔE = & 2 2 ) & 2 − 2 ) €
n2 (
% 8ε 0 h ( % n1
€
IP = E n=∞ − E n=1
€
λν = c
KE =
€
pˆα
mv
2
€
λ =
α = x, y,z
€
# !2 d 2
&
$−
'
2 + V ( x) ψ ( x) = E ψ( x)
% 2m dx
(
Hˆ = Tˆ + Vˆ
€
€
€
Probability = ψ 2 dτ
€
N =
€
En =
1
[
∫ ψ 2 dτ
]
1/ 2
h
h
=
p
mv
2
d 2ψ ( x )
€4 π ψ ( x )
=
−
dx 2
λ2
€
= hν − φ
d
= − i!
,
dα
m e4
8 ε 02 n 2 h 2
$1
1
1'
= R & 2€− 2 )
λ
n2 (
% n1
Hˆ ψ = Eψ
€
En = −
E photon = hν
€
1
2
ε 0n 2h 2
π m e2
€
∫ ψ 2 dτ
= 1
€
n 2h 2
8mL2
ψn ( x) =
€
$ nπx '
2
sin&
)
% L (
L
8
Useful Integrals
∞
1.
e
∫
−bx
dx =
0
∞
x n e −bx dx =
∫
2.
1
b
0
∞
3.
€
e
∫
−bx 2
0
∞
4.
1
2
0
∞
5.
2 −bx 2
x e
∫
b
1 % π (2
' *
dx =
2 & b)
x e −bx dx =
∫
n!
0
∫ sin
7.
∫ x sin bx
1
2b
1
1 %π ( 2
' *
dx =
4b & b )
2
6.
bx dx =
x
sin2bx
−
2
4b
dx =
sinbx
x cosbx
2 −
b
b
x2
xsin2bx
cos2bx
−
−
2
4
4b
8b
x sin2 bx dx =
8.
∫
9.
∫ sin
3
n +1
bx dx = −
cosbx
sin2 bx + 2
3b
[
]
sin 2bx
sin bx cosbx dx =
2b
10.
∫
11.
∫ cos
12.
∫
x 2 sin 2 bx dx = −
13.
∫
sin α x sin β x dx = 2
bx dx =
x
sin2bx
+
2
4b
x 2 cosbx
2x sinbx
2 cosbx
+ + 2
b
b
b3
sin (α − β )x
sin (α + β )x
− 2(α − β )
2(α + β )