# Homework 8A

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

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.

1. List three basis that can be used to expand a general 1-D function (Hint: consider Taylor, Fourier and Laplace expansions).
2. Write the mathematical expansion formula for each of the three expansions above
3. What is the definition of a complete basis?
4. Are the three basis systems identified above complete basis for describing 1-D functions?

## Q2

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 $$\langle \Psi |$$).

## Q3

a. 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.

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

b. Is this a good trial wavefunction for this approximation (justify your answer)?

c. Why is this not a good wavefunction?

d. Can you solve this problem both analytically and numerically? Pay careful attention to limits of integration.

## Q4

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.

## Q5

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?

## Q6

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. Feel free to calculate the first order wavefunction. (Optional)

## Q7

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. Feel free to calculate the first order wavefunction. (Optional)

## Q8

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.

## Q9

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 but easier than it looks.)

## Q10

Normalize this two-electron wavefunction

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