#7A Homework

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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q7.1 (repeat from last week since it is important)

What is the orbital angular momentum of an electron in the following orbitals

How many angular and radial nodes exist for the wavefunctions described by the above states?

Q7.2

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

The valence electron of the Neon atom
The the innermost electron of Lithium atom
The \(4s^2\) electrons of the Br atom
The outermost electron of the row 2 elemental atom with the largest effective nuclear charge

Q7.3

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.

Q7.4

How would use the variational method approximation in Q2 to determined the energy of the next highest eigenstate for the particle in a box with \(n=2\)?

Q7.5

Consider the "quartic oscillator" with the following Hamiltonian

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

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

Q7.6

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.
Write the mathematical expansion formula for each of the three expansions above

Q7.7

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

Q7.8

What are the following dot products for the particle in a box?

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

Q7.9

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 \(\langle \Psi | \)).

Q7.10 (Delayed to next HW)

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.

Q7.11 (Delayed to next HW)

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.

Q7.12 (Delayed to next HW)

What is the wavefunction for the potential in Q7.10? (Hint: the infinite sum in the expression of perturbed wavefunctions can be simplified with orthogonality relations).

Q7.13 (Optional. Try it, if you dare!)

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.

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

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