Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

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 nite 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 dene 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 , mand 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~?