PHYS 221: Practice Exam Problem – Part 1 Spring 2022
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Physics 221 Spring 2022 – Document #16: Practice Exam Problems – Part 1
PHYS 221: Practice Exam Problem – Part 1
2022
Problem 1. A finite vs. an infinite well
The finite-well stationary bound states are (somewhat) complicated, but we can reason about them without writing down all the algebra.
a. Sketch a finite well and sketch ψ(x) for the lowest bound state. (Label it ψ 1F Point out where your sketch obeys various constraints and/or boundary conditions.
b. We will compare this to the n=1 state in an infinite well of the same width. Sketch the infinite well and then add this wave function. (Label it ψ1I )
c. Does ψ1I have a higher energy or a lower energy than ψ1I Explain your reasoning.
c. We attempt to measure the momentum of ψ1I ; we will get a range of answers with uncer- tainty ∆p1I . We do the same for ψ1F and get a range ∆p1F Which is larger, ∆p1I or ∆p1F . Explain how you know this?
Problem 2. A Mixed State Hydrogen Atom
Suppose we define the wave function for a hydrogen atom as a mixed state:
ψmixed = 1p2 ψ 100" + 12 ψ 100# + 12 ψ210#
where the four indices correspond to the four regular quantum numbers n, ℓ , mℓ and ms .
a. Suppose we define the ground state energy of the hydrogen atom as E1 = 13.6 eV. If we
b. What is the expectation value of the measurement of the orbital angular momentum of the system?
c. What are all of the possible measurements of jJ~j, the magnitude of the total angular momen- tum of this system where J~ + S~?
2022-04-09