Homework 7A
- Page ID
- 92321
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.