Fall 2023 Homework Assignments
This first homework is to test your knowledge of basic calculus which will be needed throughout 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.
Find the expectation values <x>, <x2>, <p>, <p2> and <K>, where x, p and K are the position, momentum and the kinetic energy of a particle with the following wavefunctions: (a) A exp(-|x|/xo) (b) A exp(-|x|/xo + ikx) (c) A exp(-|x-a|/xo) where A, xo, k and a are constants with appropriate units. Normalize all wavefunctions using the normalization constant A. Warning: Be careful when differentiating discontinious functions!
A particle is confined to a one dimensional box with infinite walls:
{ 0 if 0 < x < a
V(x) = {
{ ∞ othervise
It is known that the wavefunction at time t=0 is given by
{ A sin(4πx/a) if 0 < x < a/2
ψ(x,0) = {
{ 0 otherwise.
(a) Determine A, assuming it is real and positive
(b) Expand ψ(x,0) in terms of the energy eigenfunctions of the system.
i.e. find the coefficients cn in the expansion
ψ(x,0) = Σn cn un(x) .
(Note that this a Fourier - sine expansion. Use the ortogonality of the sine
functions. Obtain a general expression for cn.)
(c) Write down the expression for ψ(x,t) .
(d) Keeping a sufficient number terms in the series for accuracy, provide computer plots
of |ψ(x,t)|2 for time values t=0, (E1 t/ħ)=π/16, (E1 t/ħ)=π/8, (E1 t/ħ)=π/4,
(E1 t/ħ)=π/2, and (E1 t/ħ)=2π where E1 is the ground state energy of the system.
Find the momentum-space wavefunctions g(k) corresponding to the wavefunctions below: (a) A exp(-|x|/xo) (b) A exp(-|x|/xo + ikox) (c) A exp(-|x-a|/xo) where A, xo, ko and a are constants with appropriate units. Normalize all wavefunctions using the normalization constant A.
(a) A particle is in a superposition of the ground state and the first excited state of
a one dimensional harmonic oscillator:
ψ(x,0) = (uo + u1) / √2
Find the following quantities as a function of time:
<x> <x2> <p> <p2> Δx Δp
(b) Repeat the problem above when the particle is in a superposition of the ground state
and the second excited state of the one dimensional harmonic oscillator:
ψ(x,0) = (uo + u2) / √2
Why do you get qualitatively different results for the two cases?
(Give a description of what is happening to the wavefunction,
not a description of the mathematical operation!)
(c) Find the expectation value <x4> when a particle
is in the n'th eigenstate of the harmonic oscillator.
(a) Given that
<x|u> = A exp(-|x/xo|) and <k|f> = δ(k-ko)
(where A, xo and ko are constants).
Find <u|f> . Hint: Remember that <x|k> = exp(ikx)/√2π .
(b) The matrix representation of an operator is given as
( 1 i )
A = ( )
(-i 1 )
in a basis |ui>. i.e. <ui|A|uj> = Aij as given above.
(i) Find the eigenvalues aj and the eigenvectors |aj> of this matrix.
(Note that the elements you find in this |ui> basis are <ui|aj>.)
(ii) What is the matrix A in this basis - i.e. with elements <ai|A|aj> ?
(iii) What is the matrix exp(A) in the same basis - i.e. with elements <ai|exp(A)|aj> ?
(iv) What is the matrix exp(A) in the original basis |ui> - i.e. with elements <ui|exp(A)|uj> ?
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).
(a) Find the matrix representation of the angular momentum operator Ly (in the basis formed
by the eigenvectors |l,m> of the operators L2 and Lz ) for l=1.
(b) Find the eigenvalues and eigenvectors of this matrix.
(c) Ly component of the angular momentum of a particle is to be measured. Calculate the
probabilities of possible values of the if
(i) the particle is known to be in state |1 1> (i.e. l=1, m=mz=1 )
(ii) the particle is known to be in state |1 0> (i.e. l=1, m=mz=0 )
Consider a spin 3/2 particle.
(a) Find the matrices corresponding to Sz, Sx and Sy.
(c) You should be able to guess what the eigenvalues of these matrices are.
Find the corresponding eigenvectors.
(b) A measurement is to be carried out to determine the x-component of the spin of this particle,
when it is known to be in the |3/2 1/2 > state. Find the possibilities and their probabilities.
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.