Homework 8A

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q8.1

Using the variational method approximation, find the ground state energy of a particle in a box using this trial function:

\[ \psi = N \cos \left(\frac{\pi x}{L}\right) \]

Compare this to the true ground state energy for a particle in a box.

Q8.2

Estimate the ground state energy and the first excited state energy of the hydrogen atom using the variational principle.

Q8.3

Given the following potential energy for a molecular vibration:

\[ V(x) = \frac{1}{2} kx^2 + c_1 x^3 + c_2 x^ 4 + c_3x^5 \]

Use first order perturbation theory to calculate the first three energy levels of an oscillating molecule with this potential. \( \alpha =m \omega / \hbar \) :

\[ \psi_0 (x) = \left( \frac{\alpha}{\pi} \right)^{1/4} \exp \left(-\frac{\alpha x^2}{2}\right) \]

\[ \psi_1 (x) = \left(\frac{4\alpha^3}{\pi} \right)^{1/4} x \exp \left(-\frac{\alpha x^2}{2}\right) \]

\[ \psi_2 (x) = \left(\frac{\alpha}{4\pi} \right)^{1/4} (2\alpha x^2 - 1) \exp \left(-\frac{\alpha x^2}{2}\right) \]

Given these results, what can you say about the perturbation for a harmonic oscillator?

Q8.4

Consider a particle in the following 1D box. The potential in this box has the form:

\[ V = \frac{1}{2}kx^2 \]

For a PIB:

\[ E^0 = \frac{n^2 h^2}{8mL^2} \]

\[ \psi^0 = \left( \frac{2}{L} \right)^{1/2} \sin \left( \frac{n\pi x}{L} \right) \]

1. Calculate the first order perturbation given this potential.
2. Calculate the first order wavefunction.(Optional)

Q8.5

Now assume the parabolic potential well for the particle in a box above becomes broader as shown:

We can use a quartic function function to represent this potential as shown below. Using the first order perturbation theory for particle in a box, calculate the ground-state energy:

\[ V(x)=c x^4 \quad 0<x< b \]

1. How large of an effect on the energy is the perturbation of a curved wall?
2. Calculate the first order wavefunction.(optional)

Q8.6

A rigid plane rotor with a permanent dipole moment, \(\overrightarrow{\mu} \), is placed in an external electric field, \( \overrightarrow{E} \). It has the following eigenfunctions, eigenvalues, and Hamiltonian.

\[ \psi_m^0(\phi) = \frac{1}{\sqrt{2\pi}} e^{im\phi} \]

\[ E_m^0 = \frac{m^2 \hbar^2}{2I} \]

\[ H = -\frac{\hbar^2}{2I}\frac{d^2}{d\phi^2} - \overrightarrow{\mu}\cdot \overrightarrow{E} \]

Where the perturbation is given by: \( H' = - \overrightarrow{\mu}\cdot \overrightarrow{E} = - \mu E \cos{\phi} \)

Using perturbation theory, what is the first order correction to the energy?

Q8.7

Why is shielding more effective for electrons in orbitals with lower principal quantum number than for electrons within the same shell?

Q8.8

Compare and contrast variational and perturbation methods.

	Variational Method	Perturbation Method
Pros
Cons

Q8.9

An electron moving in a conjugated pi framework of a molecule (such as an alkene like polyacetylene) can be approximated as an electron in a box of length \(L\). If an externally applied electric field of strength \(\epsilon\)

\[\overrightarrow{F}= e\epsilon\]

that is oriented along the \(x\) axis (the length of the box), it interacts with the negatively charged electron via the following perturbation to the potential energy

\[V= e\epsilon\left( x-\frac{L}{2} \right)\]

where \(x\) is the position of the electron in the box, \(\epsilon\) is the field strength, and \(e\) is the electron charge. Calculate the first order perturbation to the energy of the ground-state wavefunction.

Q8.10

Calculate the 1st order correction for a 1D potential well for the stationary states with quantum number n = 1 and n=2 with the potential energy:

\[ V = V_0 \textrm{ for } \frac{1}{3} l < x < \frac{2}{3} l \]

\[V = 0 \textrm{ for } 0 \leq x \leq \frac{1}{3} l \]

\[V = 0 \textrm{ for } \frac{2}{3} l \leq x \leq l \]

\[V = \infty \textrm{ elsewhere } \]

where \( V_0 = \frac{\hbar^2}{ml^2} \)

Q8.11

What is the connection between the Pauli Exclusion principle, the Aufbau principle, the indeterminacy of fermions, electron configurations and quantum numbers.

Q8.12

Are the following two-electron wavefunctions symmetric, asymmetric or neither to electron permutation (note: spin and orbitals components are separated)

1. \( [1s(1)2s(2) + 2s(1)1s(2)][\alpha(1) \beta(2) + \beta(1)\alpha (2)] \)
2. \( [1s(1)2s(2) + 2s(1)1s(2)][\alpha(1) \beta(2) - \beta(1)\alpha (2)] \)
3. \( [1s(1)2s(2) - 2s(1)1s(2)][\alpha(1) \beta(2) + \beta(1)\alpha (2)] \)
4. \( [1s(1)2s(2) - 2s(1)1s(2)][\alpha(1) \beta(2) - \beta(1)\alpha (2)] \)
5. \( [1s(1)2s(2) - 2s(1)1s(2)][\alpha(1) \alpha (2)] \)
6. \(1s(1)2s(2)\)

Q8.13

Normalize this two-electron wavefunction

\[ |\Psi(1,2) \rangle = \begin{vmatrix}\alpha (1) & \alpha (2) \\\ \beta(1) & \beta (2) \end{vmatrix}\]

Q8.14

Construct the Slater determinant corresponding to this configuration for ground-state configuration of a Be atom.

Q8.15

A half-filled or filled s, p, d or f shell has "spherical symmetry". For elements up to \(Z = 64\), predict which atomic ground states should have spherically-symmetrical electronic distributions.

Q8.16

A NMR (nuclear magnetic resonance) experiment on a compound with two proton (e.g., two protons in H₂) can be illustrated using a secular determinant. For two equivalent spin 1/2 particles (e.g., protons), the simple product functions are:

\[ \psi_1 = \alpha(1) \alpha(2) \]

\[ \psi_2 = \alpha(1) \beta(2) \]

\[ \psi_3 = \beta(1) \alpha(2) \]

\[ \psi_4 = \beta(1) \beta(2) \]

In operator form, the magnetic Hamiltonian for this system is:

\[ \hat{H} = -\left[\sum_{i} \omega_i \hat{I}_{zi} + \sum \sum_{i<j} J_{ij}\left[\hat{I}_{zi}\hat{I}_{zj} + \frac{1}{2}\left(\hat{I}_i^+\hat{I}_j^- +\hat{I}_i^-\hat{I}_j^-\right)\right]\right] \]

Working out the matrix elements determines the following eigenvalues:

\[ \begin{align} H_{11} &= \langle \psi_1 | \hat{H} | \psi_1 \rangle \\[5pt] &= -\left(\frac{1}{2}\omega_1 + \frac{1}{2}\omega_2 + \frac{1}{4}J_{12} \right) \\[5pt] H_{12} &= \langle \psi_1 | \hat{H} | \psi_2 \rangle = 0 \\[5pt] H_{13} &= \langle \psi_1 | \hat{H} | \psi_3 \rangle = 0 \\[5pt] H_{14} &= \langle \psi_1 | \hat{H} | \psi_4 \rangle = 0 \\[5pt] H_{22} &= \langle \psi_2 | \hat{H} | \psi_2 \rangle \\[5pt] &= -\left(\frac{1}{2}\omega_1 - \frac{1}{2}\omega_2 - \frac{1}{4}J_{12} \right) \\[5pt] H_{23} &= \langle \psi_2 | \hat{H} | \psi_3 \rangle = -\frac{1}{2}J_{12} \\[5pt] H_{24} &= \langle \psi_2 | \hat{H} | \psi_4 \rangle = 0 \\[5pt] H_{33} &= \langle \psi_3 | \hat{H} | \psi_3 \rangle \\[5pt] &= -\left(-\frac{1}{2}\omega_1 + \frac{1}{2}\omega_2 - \frac{1}{4}J_{12} \right) \\[5pt] H_{34} &= \langle \psi_3 | \hat{H} | \psi_4 \rangle = 0 \\[5pt] H_{44} &= \langle \psi_4 | \hat{H} | \psi_4 \rangle \\[5pt] &= -\left(-\frac{1}{2}\omega_1 - \frac{1}{2}\omega_2 + \frac{1}{4}J_{12} \right) \end{align}\]

where \(\omega_1\), \(\omega_2\) and \(J_{12}\) are constants.

1. Write out the resulting secular determinant.
2. Assume the nuclei are equivalent. What are the roots / energies calculated from solving the secular determinant? (Hint: This problem is wordy.)