Chemistry 120A, Spring 2009
Midterm 1 Solutions
February 20, 2009
Problems
Question 1: Multiple choice problems: 7
×
3 = 21 points
1. At a finite potential step, the 2nd derivative of the wavefunction
∂
Ψ
∂x
, exhibits:
(a) continuity
(b) a finite step
(c) an infinite step
B
2. All else constant, a smaller force constant for the harmonic oscillator will cause the
wavelength corresponding to the energy difference between levels to:
(a) increase
(b) decrease
(c) stay the same
A
3. Can a particle in quantum mechanics pass through a potential barrier that is smaller
than its kinetic energy?
(a) always
(b) sometimes
(c) never
B
4. Which of the following is the quantity

φ
ih
ψ

?
(a) a number
(b) a ket
(c) an operator
C
5. Which of the following is the quantity
h
φ

ψ
i
?
(a) a number
(b) a ket
(c) an operator
A
6. All else being equal, if one compares two wavefunctions, the one with the smaller
curvature
∂
2
Ψ
∂x
2
will have an energy which is:
(a) higher
(b) lower
(c) either is possible
B
7. Which of the following has the longest de Broglie wavelength?
(a) a 1 eV He atom
(b) a 1 eV photon
(c) a 1 eV electron
B
1
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Question 2: (15 points) Consider applying the Heisenberg uncertainty principle, Δ
p
Δ
x
≥
~
2
, to the
problem of a particle in a onedimensional box, with box length
a
.
(a) (5 points) Estimate the uncertainties that are expected in the position (from box size) and
momentum (from Heisenberg), Δ
x
and Δ
p
(1) Because the particle is
in
the box, we know that
Δ
x
≤
a
(Any answer in the range 0
.
1
a
≤
Δ
x
≤
a
is acceptable)
(2) Then, from the Heisenberg uncertainty principle,
Δ
p
Δ
x
≥
~
2
⇒
Δ
p
≥
~
2Δ
x
⇒
Δ
p
≥
~
2
a
(b) (5 points) From your estimate of Δ
p
estimate the lowest possible energy of the system.
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 Spring '09
 HEADGORDON
 Kinetic Energy

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