Sphere Table
Attached, you will find a table showing different representations of physical
quantities associated with a quantum particle confined to a sphere. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be.
pdf link for the Table
or doc link for the Table
QM Sphere with Time Dependence
Consider a quantum particle on a sphere. At t = 0, the particle is in state:\[\left|{\psi(t=0)}\right\rangle =\frac{1}{\sqrt{2}}\left(\left|{2,0}\right\rangle +\left|{1,0}\right\rangle \right)\]
Calculate the following quantities for some later time, \(t>0\), and identify whether each quantity is time-dependent.
\(\left|{\psi(t)}\right\rangle \).
\(\langle L_z\rangle\)
\(\mathcal{P}(L^2=6\hbar^2)\)
The probability that the particle can be found in the “southern” hemisphere.
The probability that the particle can still (at the time \(t\)) be found in the state
\[\left|{\psi}\right\rangle =\frac{1}{\sqrt{2}}\left(\left|{2,0}\right\rangle +\left|{1,0}\right\rangle \right)\]
Confidence Rating
After solving each problem on the assignment, indicate your answers to the following questions for each problem. Answer for the problem as a whole, even if the problem has multiple parts.
Question Confidence How confident are you that you are interpreting the problem the way the instructor intends?
For the rest of the questions, assume you have interpreted the problem correctly
Problem Confidence How confident are you that you could independently come up with a correct solution process to a similar problem on a future problem set?
Answer Confidence How confident are you that your final answer to this question is correct (not solution process)?
Makes Sense To what degree do you understand how your answer fits (or does not fit) the physical or mathematical situation of the problem?
