Homework 7A
- Page ID
- 92321
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
- 1s
- 2s
- 2p
- 3d
- 5f
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.