Homework 7A (Koski)

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Section: _____________________________

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Q1

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

Q2

The most probable radius for an electron in the 1s wavefunction is called the Bohr radius (\(a_o\)) and is 52.9 pm. What is the most probable radius for an electron in the 2s wavefunction?

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. 2s
2. \(2p_o\)
3. \(2p_{\pm 1}\)
4. 3s
5. \(3d_o\)
6. \(3d_{\pm 1}\)

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

Evaluate the following commutators:

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

Q7

1. What is the average value of the kinetic energy for the \( \psi_{320} \) wavefunction?
2. What is the average value of the potential energy for the \( \psi_{320} \) wavefunction?
3. What is the relationship between the kinetic and potential energies in terms of magnitude (Hint: Virial Theorem)?

Q8

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

1. \( \psi_{320} (r,\theta, \phi ) \)
2. \( \psi_{3s} (r,\theta, \phi ) \)
3. \( \psi_{2p_y} (r,\theta, \phi ) \)

Q9

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. \( \psi_{3p_x} (r,\theta, \phi ) \)

Q10

Use Slater's rules to calculate \(Z_{eff}\) and \(Z\) for

1. The valence electron of the Neon atom
2. The the innermost electron of Beryllium atom
3. The \(4s^2\) electrons of the I atom
4. The outermost electron of the row 2 elemental atom with the largest effective nuclear charge

Q11

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.

Q12

How would use the variational method approximation in Q11 to determined the energy of the next highest eigenstate for the particle in a box with \(n=2\)?

Q13

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?

Q14

A basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.

• List three basis that can be used to expand a general 1-D function (Hint: consider Taylor, Fourier and Laplace expansions).
• Write the mathematical expansion formula for each of the three expansions above

Q15

What is the definition of a complete basis? Are the three basis systems identified above complete basis for describing 1-D functions?

Q16

What are the following dot products for the particle in a box (where \(| \psi \rangle\) is an eigenstates of the Hamiltonian with Principle Quantum \(n\))?

1. \(\langle \psi_{n=1} | \psi_{n=1} \rangle \)
2. \(\langle \psi_{n=1} | \psi_{n=2} \rangle \)
3. \(\langle \psi_{n=2} | \psi_{n=1} \rangle \)
4. \(\langle \psi_{n=2} | \psi_{n=2} \rangle \)

Q17

The wavefunction can be expanded into the complete set of basis of eigenstates of the Hamiltonian:

\[| \Psi \rangle =\sum_i c_i | \phi_i \rangle \]

What is the general expression of the off diagonal (\(i \neq j\)) and diagonal (\(i = j\)) matrix elements for the Hamiltonian in the basis set of its eigenstates?

\[H_{ij} = \langle \phi_i | \hat{H} | \phi_j \rangle \]

(hint: Apply the Hamiltonian \(\hat{H}\) on this arbitrary wavefuction \(| \Psi \rangle\) and its bra version \(\langle \Psi | \)).

Q18 (Optional)

Estimate the ground state energy and wavefunction for a particle in a box using the variational method with the following trial wavefunction, where N is the normalization constant and \(\beta\) is a variational parameter that should be minimized.

\[ | \psi \rangle = N \exp(-\beta x^2)\]

1. Is this a good trial wavefunction for this approximation (justify your answer)?
2. Why is this not a good wavefunction?
3. Can you solve this problem both analytically and numerically? Pay careful attention to limits of integration.