# Homework 7A


Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

Evaluate the following commutators:

1. $$\left[ \hat{L_x}, \hat{L_y}\right]$$
2. $$\left[\hat{L_z}, \left[ \hat{L_y}, \hat{L_x}\right]\right]$$
3. $$\left[\hat{L_x}, \left[ \hat{L^2_y}, \hat{L_z}\right]\right]$$
4. $$\left[\hat{L_y}, \left[ \hat{L^2}, \hat{L_z}\right]\right]$$
5. $$\hat{L_z}\left[ \hat{L_x}, \hat{L_y}\right]$$
6. $$\hat{L^2}\left[ \hat{L_x}, \hat{L_y}\right]$$

## 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 $$Z$$).

1. 1s
2. $$2p_z$$
3. 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 1s wavefunction (i.e., $$\langle \Delta r \rangle$$ ).

## Q6

1. What is the average value of the kinetic energy for the $$\psi_{210}$$ wavefunction?
2. What is the average value of the potential energy for the $$\psi_{210}$$ wavefunction?
3. 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?

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

## Q8

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

1. $$\psi_{1s} (r,\theta, \phi )$$
2. $$\psi_{2s} (r,\theta, \phi )$$
3. $$\psi_{2p_x} (r,\theta, \phi )$$
4. $$\psi_{2p_y} (r,\theta, \phi )$$
5. $$\psi_{2p_z} (r,\theta, \phi )$$
6. $$\psi_{3s} (r,\theta, \phi )$$
7. $$\psi_{3p_x} (r,\theta, \phi )$$

## 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

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

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

Homework 7A is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.