Homework 7A

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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

Evaluate the following commutators:

\( \left[ \hat{L_z}, \hat{L_x}\right] \)
\(\left[\hat{L_y}, \left[ \hat{L_z}, \hat{L_x}\right]\right] \)
\(\left[\hat{L_y}, \left[ \hat{L^2_z}, \hat{L_x}\right]\right] \)
\( \left[\hat{L_z}, \left[ \hat{L^2}, \hat{L_x}\right]\right] \)
\( \hat{L_x}\left[ \hat{L_z}, \hat{L_y}\right] \)
\( \hat{L^2}\left[ \hat{L_z}, \hat{L_y}\right] \)

Q2

Show that the hydrogenic wavefunctions \( \psi_{1s} \) and \( \psi_{2s} \) are mutually orthogonal and normalized.

Q3

Determine the probability of finding an electron between each of the following radii for each given orbital. Using your favorite plotting tool or program, plot the probability as a function of radius. Identify each requested region.

Orbital	0 - \(a_o\)	0 - 168 pm	168 - 508 pm	508 - 1,000,000,000 pm
1s
2s
3s

Q4

Calculate the average radial expectation value \( \langle r \rangle \) of electrons in the following hydrogenic orbitals (as a function of \(Z\)).

1s
\(2p_z\)
2s

Hint: This requires volume integration! Do not forget the \(4\pi r^2 \) spherical volume element.

Q5

Calculate the uncertainty of the radius of the electron in the 2s wavefunction (i.e., \( \langle \Delta r \rangle \) ).

Q6

What is the average value of the kinetic energy for the \( \psi_{210} \) wavefunction?
What is the average value of the potential energy for the \( \psi_{210} \) wavefunction?
What is the relationship between the kinetic and potential energies in terms of magnitude (Hint: Virial Theorem)?

Q7

How many angular and radial nodes are there for the following hydrogenic orbitals/wavefunctions?

\( \psi_{210} (r,\theta, \phi ) \)
\( \psi_{1s} (r,\theta, \phi ) \)
\( \psi_{2p_z} (r,\theta, \phi ) \)

Q8

For the following H orbitals, locate the radial and angular nodes.

\( \psi_{1s} (r,\theta, \phi ) \)
\( \psi_{2s} (r,\theta, \phi ) \)
\( \psi_{2p_x} (r,\theta, \phi ) \)
\( \psi_{2p_y} (r,\theta, \phi ) \)
\( \psi_{2p_z} (r,\theta, \phi ) \)
\( \psi_{3s} (r,\theta, \phi ) \)
\( \psi_{3p_x} (r,\theta, \phi ) \)

Q9

Evaluate the trial energy of the unnormalized trial function

\[ | \varphi \rangle = x(L−x) \rangle \]

to estimate the ground state energy for a particle in a one-dimensional box of length \(L\). (Hint: If you want the general pain, you can find this answer in this paper, but it is not necessary). You are not minimizing a parameter in this question.

Q10

Consider the "quartic oscillator" with the following Hamiltonian

\[ \hat{H} = \dfrac{1}{2} \dfrac{d^2}{dx^2} + \dfrac{1}{2} x^4\]

What is the zero point energy of this this system determined with the variational method approximation using the unnormalized trial wavefunction \[ | \varphi \rangle = e^{-\dfrac{1}{2} \alpha (x-x_o)^2} \]
What is the value of \(\alpha\) for the trial wavefunction used in this approximation?
How accurate would this wavefunction be in estimating the zero point energy of the harmonic oscillator?