1. No category

PHYS2265 Modern Physics/
PHYS2627 Introductory Quantum Physics
2014/15 Semester 1 Homework 5
(Due date: 24 Nov 2014, 5:00 pm)
1. Consider an element having excited states at E1 = 3.6 eV and E2 = 4.6 eV used as a
gas in the Franck-Hertz experiment. Assume that the work functions of the materials
involved cancel out. List all the possible peaks that might be observed with electron
scattering up to an accelerating voltage of V = 18 V.
2. An electron in a hydrogen atom is in the 6f state.
(a) What are the values of the principal quantum number n and orbital quantum
number `?
(b) Compute the energy E of the electron.
~
(c) Compute the magnitude of the electron’s orbital angular momentum L.
(d) Find the possible values of the z-component of the electron’s orbital angular momentum Lz in this situation.
3. The normalized wave function for the hydrogen atom in the state n = 4, ` = 2, and
m` = 2 is
√
r r2 −r/4a0
3
ψ4,2,2 (r, θ, φ) =
12 −
e
sin2 θ e2iφ
√
2
3/2
a
a
0
3072 2πa0
0
where r is the radial coordinate of the electron and a0 = 4π0 ~2 /me e2 is the Bohr
radius.
(a) Sketch the wave function ψ4,2,2 (r, θ, φ) versus r for θ = π/2 and φ = 0.
(b) Sketch the radial probability density P (r) for the hydrogen atom in this state.
(c) Verify that the wave function ψ4,2,2 (r, θ, φ) is normalized.
R∞
(Hint: Use the integral formula 0 xn e−x dx = n! for any non-negative integer n.)
4. The normalized radial wave function for the hydrogen atom in the 3p state is
r
r −r/3a0
4
e
R3,1 (r) = √ 3/2 6 −
a0 a0
81 6a0
where a0 = 4π0 ~2 /me e2 is the Bohr radius.
(a) Calculate the probability P of an electron in this state being found between a0
and 3a0 from the nucleus.
(b) Compute the average distance hri from the nucleus for an electron in this state.
(c) Find the most probable distance rp from the nucleus for an electron in this state.
R∞
(Hint: Use the integral formula 0 xn e−x dx = n! for any non-negative integer n.)
5. The normalized angular wave function for the hydrogen atom in the ` = 2 and m` = −1
state is
Θ2,−1 (θ) = A2,−1 sin θ cos θ
where A2,−1 is the normalization constant.
(a) Verify that Θ2,−1 (θ) is a solution to the polar equation for the case ` = 2 and
m` = −1.
√
(b) Prove that the normalization constant A2,−1 = 15/2.
(c) Find the direction in space where the angular probability density P (θ, φ) for an
electron in this state has the maximum value.
(d) Calculate the probability that an electron in this state being found within 30◦ of
the z-axis, irrespective of the distance r from the nucleus.
2