Homogeneous gravity field time evolution position and momentum operato

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Homework Statement


I am trying to solve Problem 21 from this sheet:
View attachment T2SS14-Ex7.pdf


Homework Equations


The equation describing the time evolution of operators is given in the problem.


The Attempt at a Solution


I have found the commutators of the position and momentum operator with the Hamiltonian.
View attachment Übung 21_3.pdf
I don't know how to do the rest of the exercise.
 
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(a) See equation (5) relating the commutator to the time evolution to get ##\hat X(t)## and ##\hat P(t)##
(b) follows from there.
 
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This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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Thread 'Understanding how to "tack on" the time wiggle factor'

Thread 'Understanding how to "tack on" the time wiggle factor'

The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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