Physics 301
Homework Problems
Wavefunctions and Uncertainty
Page 1
Just as a reminder, on all homeworks this semester, please show your work and explain your reasoning. I
will grade for clarity of explanation at least as much as I do for mere “correctness of final answer”!
1)
A
ABCD is a square. M is the midpoint of BC and N is the midpoint of
CD. A point is selected at random in the square. Calculate the
probability that the point lies in the triangle MCN.
B
M
2)
D
Two balanced dice are rolled. Let X be the sum of the two dice.
N
C
(a) Obtain the probability distribution of X (i.e., what are the possible values for X and the
probability for obtaining each value?).
(b) What is the probability for obtaining X ≥ 8?
(c) What is the average value of X? (Use mathematics, and not symmetry arguments.)
3)
An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with
no electrons falling outside of this interval.
(a) Draw a graph of ψ ( x )
2
for these electrons. Provide numerical scales on both axes.
(b) What is the probability that an electron will land within the interval 0.79 cm to 0.81 cm?
(c) If 106 electrons are detected, how many will be detected in the interval 0.79 cm to 0.81 cm?
(d) What is the probability density at x = 0.80 cm?
4)
The figure below shows the wave function of a particle confined between x = 0 nm and x = 1.0
nm. The wave function is zero outside this region.
ψ( )
(
)
(a) Determine the value of the constant c, as defined in the figure.
2
(b) Use Mathematica to create a graph of the probability density P ( x ) = ψ ( x ) .
(c) Calculate the probability of finding the particle in the interval 0.6 nm ≤ x ≤ 0.8 nm.
(d) Calculate the expectation value for the particle.
Physics 301
5)
Homework Problems
Wavefunctions and Uncertainty
Page 2
Consider the electron wave function
⎧
2
⎪ c 1− x
ψ ( x) = ⎨
⎪ 0
⎩
x ≤ 1cm
x ≥ 1cm
where x is in cm.
(a) Determine the normalization constant c.
(b) Use Mathematica to create a graph of ψ ( x ) over the interval –2 cm ≤ x ≤ 2 cm.
(c) Use Mathematica to create a graph of ψ ( x )
2
over the interval –2 cm ≤ x ≤ 2 cm.
(d) If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?
6)
Consider the electron wave function
⎧
⎛ 2π
⎪ csin ⎜
⎝ L
ψ ( x) = ⎨
⎪ 0
⎩
⎞
x⎟
⎠
0≤x≤L
x < 0 or x > L
(a) Determine the normalization constant c. Your answer will be in terms of L.
(b) Use Mathematica to create a graph of ψ ( x ) over the interval –L ≤ x ≤ 2L.
(c) Use Mathematica to create a graph of ψ ( x )
2
over the interval –L ≤ x ≤ 2L.
(d) What is the probability that an electron is in the interval 0 ≤ x ≤ L/3?
7)
A pulse of light is created by the superposition of many waves that span the frequency range
f0 − 12 Δf ≤ f0 ≤ f0 + 12 Δf , where f0 is called the center frequency of the pulse. Laser technology
can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs
(1 fs = 1 femtosecond = 10–15 s).
(a) What is the center frequency of this pulse of light?
(b) How many cycles, or oscillations, are completed during the 6.0 fs pulse?
(c) What range of frequencies must be superimposed to create this pulse?
(d) What is the spatial length of the laser pulse as it travels through space?
(e) Draw a snapshot graph of this wave packet.