Homework 9

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q9.1

Use 1st order perturbation theory to evaluate the ground-state energy for a harmonic oscillator that with a cubic term \(ax^3\) added to the potential.

Q9.2

Use the harmonic oscillator solution and 1st order perturbation theory to evaluate the ground-state energy for a harmonic oscillator with an additional \( ax^4\) term.

Q9.3

What is the first-order wavefunction for the potential in Q9.1? (Hint: the infinite sum in the expression of perturbed wavefunctions can be simplified with orthogonality relations).

Q9.4

An electron moving in a conjugated bond framework of a molecule can be viewed as an electron in a box of length \(L\). If an externally applied electric field of strength \(\epsilon\)

\[\vec{F}= \epsilon x\]

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

\[V = \epsilon e x\]

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.

Q9.5

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:

\[ V = V_0 \textrm{ for } \frac{1}{4} l < x < \frac{3}{4} l \]

\[V = 0 \textrm{ for } 0 \leq x \leq \frac{1}{4} l \]

\[V = 0 \textrm{ for } \frac{3}{4} l \leq x \leq l \]

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

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

Q9.6

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

Q9.7

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

Q9.8

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. \(1s(1)2s(2)\)

Q9.9

Normalize this two-electron wavefunction

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

Q9.10

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

Q9.10B

A filled or half-filled s, p, d or f shell has "spherical symmetry" with respect to orbital. Go through the periodic table up to \(Z = 54\) and predict which atomic ground states should have spherically-symmetrical electronic distributions.

Q9.11 (Delayed for Larsen's Class)

Find the spectroscopic terms originating from the following configurations (two electrons in two non-equivalent orbitals in the same atom):

• 1s¹2s¹
• 1s¹3s¹
• 2p¹3p¹
• 2s¹3d¹

Q9.12 (Delayed for Larsen's Class)

Give the electronic configurations and term symbols of the first excited electronic states of the atoms up to Z=10.

Q9.13 (Delayed for Larsen's Class)

Find the spectroscopic term originating from the ground states configuration of the sodium atom and the boron atom.

Q9.14 (Delayed for Larsen's Class)

Find the spin-orbit energy levels of the hydrogen atom with an electron in 2p and 3d orbitals.