Fall 2024 Homework Assignments
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No late homework accepted after Monday, Dec. 23
This first homework is to test your knowledge of basic calculus which will be needed throughout this course. It is intended to give you an idea about how well your math background is in relation to this course. If you are experiencing major problems in carrying out these operations, then you have defficiencies in the knowledge prerequisite to this course. Chances are you will have trouble in following this course, and will not be able to perform well in the exams. I suggest you do a quick review of the relevant material.
Normalize the following wave functions and find the corresponding expectation
values <x>, <x2>, <p> and <p2>.
{ A (1 + x/a) exp(ikox) for -a < x < 0
(a) ψ(x) = { A exp(-x/a) for x > 0
{ 0 otherwise
(b) ψ(x) = A exp[-(x/a)2 + i qo x3 ]
Hint: ∫ exp(-α x2) x2n dx = (-∂/∂α)n ∫ exp(-α x2) dx
where a, ko and qo are constants. What are the units of qo?
A particle is confined into a one dimensional box with the potential energy
{ 0 for 0 < x < a
V(x) = {
{ ∞ otherwise
(a) Find the normalized eigenfunctions un(x) and their energies En.
(b) Given that at time t=0 the wavefunction of the particle is
{ A sin(3πx/a) for 2a/3 < x < a
ψ(x,0) = {
{ 0 otherwise.
Normalize ψ(x,0) and evaluate the coefficients cn in the expansion
ψ(x,0) = ∑n cn un(x)
using the orthonormality of the eigenfunctions.
(c) Using a computer program, plot |ψ(x,t)|2 at times corresponding to
t=0, t=πħ/(80 E1), t=πħ/(40 E1), t=πħ/(20 E1), and t=πħ/(8 E1).
(Hint: Use unitless scaled variables such as x/a for distance and tE1/ħ for time
so that you will not need to put in physical quantities for these.)
Submit the computer code as well as your plots as a part of the homework.
A particle with energy E is incident from x = -∞ onto a potential energy region
{ 0 for x < 0
V(x) = { α δ(x) at x = 0 (with α > 0)
{ Vo for x > 0 (with Vo > 0)
Consider separately the cases when E < Vo and E > Vo
(a) What are the forms of the time-independent eigenfunctions for x<0 and x>0 ?
(b) Write down the boundary conditions at x=0.
(c) Solve these equations to obtain the transmission and reflection coefficients.
(d) Make a rough plot of the transmission coefficient as a function of E.
While solving the quantum mechanical oscillator problem, we used the operators a+ = (x/xo - i P/po) and a- = (x/xo + i P/po) . Note that we can use these relations to construct x and P operators from a+ and a-: x = xo (a+ + a-)/2 and P = i po (a+ - a-)/2 . One can then use these relations to find expectation values of products of operators very easily. For example, <x>n = (xo/2) ∫ un* (a+ + a-) un dx = 0 because the eigenfunctions un form an orthonormal set, so ∫ un* un±1 dx would give zero. Expand the operators given below in terms of the a± operators, and evaluate their expectation values for the state un: (a) x2 (What is the value of expectation value <kx2/2>n for the oscillator?) (b) P2 (What is the value of expectation value <P2/2m>n for the oscillator?) (c) xP + Px (d) x3 (e) x4 (f) x5 Note that when you expand these quantities, only terms with equal numbers of a+ and a- operators will give non-zero results, so you do not need to consider other terms.
(1) (a) Prove the following identities, where † stands for the Hermitian adjoint operation:
(i) (AB)† = B† A†
(ii) (a+)† = a- (where a± are the raising/lowering operators of the harmonic oscillator problem)
(b) Are the following operators Hermitian?
(i) XP
(ii) XP + PX
(iii) a+ and a-
(iv) a+a-
(v) a+ + a-
(vi) i(a+ - a-)
(2) Given the following inner products:
<x|un> = √(2/a) sin(nπx/a) for 0<x<a and 0 otherwise
<x|k> = exp(ikx)/√(2π)
<k|ψ> = A exp[-(k-ko)2xo2]/√(2π) where A, ko and xo are constants
Find the following inner products:
(a) <k|un>
(b) <ψ|x>
(c) <un|ψ> (Write down an integral corresponding to this inner product, but do not attempt to evaluate it.)
A function of a matrix can be determined easily if it is diagonal:
If
| a1 0 0 ⋯ 0 | | a1k 0 0 ⋯ 0 |
| 0 a2 0 ⋯ 0 | | 0 a2k 0 ⋯ 0 |
A = | 0 0 a3 ⋯ 0 | so that Ak = | 0 0 a3k ⋯ 0 |
| ⋯ | | ⋯ |
| 0 ⋯ 0 an| | 0 ⋯ 0 ank|
then
| f(a1) 0 0 ⋯ 0 |
| 0 f(a2) 0 ⋯ 0 |
f(A) = | 0 0 f(a3) ⋯ 0 |
| ⋯ |
| 0 ⋯ 0 f(an) |
If the matrix is not diagonal, it can be transformed into a basis which will make it diagonal.
The matrix will be diagonal in a basis formed by its eigenvectors:
A = ∑i ∑j |ai><ai| A |aj><aj| = ∑i |ai> ai <ai| because <ai| A |aj> = ai δij
so that
f(A) = ∑i |ai> f(ai) <ai| (*)
(1) In class, we worked out the eigenvalues ai and the corresponding eigenvectors |ai> for the Hamiltonian
| 1 i√2 |
H = | | ε
| -i√2 2 |
(a) Find exp(-itH/ℏ) using the (*) equation above.
| 1 |
(b) Calculate exp(-itH/ℏ) |Ψ(0)> with |Ψ(0)> = | | and obtain the result we had in class.
| 0 |
| 0 1 0 |
(2) Find sin | 1 0 1 |
| 0 1 0 |
Consider a three dimensional Harmonic oscillator with the Hamiltonian
H = P2/2m + k r2/2 with P2 = Px2 + Py2 + Pz2 and r2 = x2 + y2 + z2
(a) Carry out a separation in the variables x, y, and z to determine
the energy levels. List the lowest 5 energy levels and their degeneracies.
(b) Find the expectation value of the angular momentum operator Lz in
the state (u100(x,y,z) + i u010(x,y,z))/√2. Where
uklm(x,y,z) = uk(x)ul(y)um(z) with uk representing the k'th eigenfunction
of the one dimensional oscillator.
(c) Repeat part (b) for the angular momentum operator Lx
(d) Repeat part (b) for the ground state u000(x,y,z).
1. Find the matrix representations of the operators S2, Sz, S+, S-, Sx and Sy for the s=1 spin system.
Use the basis which diagonalizes The S2 and Sz operators. i.e., use their eigenvectors.
(A useful notation for this case are the kets |↑⟩ , |0⟩ and |↓⟩. )
2. (a) An experiment is carried out on an s=1 particle and x-component of its angular momentum was determined to be +ℏ.
What are the possibilities and the corresponding probabilities for a subsequent measurement of the z-component
of the angular momentum?
(b) Repeat the previous part if the result of the Sx measurement resulted with zero angular momentum.
(c) Comment on the results of parts (a) and (b).
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
Please name the file in the format:
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.