- Homework 1: (Due Monday, September 28)
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.
First Homework
- Homework 2: (Due Monday, Oct 5)
Given the wavefunction of a particle at time t=0:
Ψ(x,0) = A x exp[-(x/xo)2 + ikox]
Find the following quantities:
(a) The normalization constant A.
(b) < x >
(c) < x2 >
(d) < p >
(e) < p2 >
(f) Δx Δp
- Homework 3: (Due Wednesday, October 14)
A particle in a one -dimensional infinite square-box potential of width \(a\) is known to be in the state
\(\Psi(x,0) = A [3u_1(x) + 4u_3(x)] \) at time \(t=0\). The functions \(u_n\) are the eigenstates of the
system.
Find
- the normalization constant \(A\),
- \( \Psi(x,t) \),
- <\(x\)>\((t)\),
- <\(x^2\)>\((t)\).
How would you qualitatively describe the development of the wavefunction?
- Homework 4: (Due Monday, October 26)
Consider the wavefunction \( \Psi(x)=A \exp[-(x/x_o)^2 +ik_ox]\).
- Obtain the momentum-space wavefunction \(g(k)\).
- Find the expectation value <\(p\)> using \(g(k)\).
- Find the expectation value <\(p^2\)> using \(g(k)\).
- Homework 5: (Due Monday, November 9)
For a particle in a one-dimensional harmonic oscillator potential,
- (a) calculate the expectation value of the following operators in the ground state:
- (i) \(XP + PX\)
- (ii) \(XPX\)
- (iiii) \(XP^2X\)
- (b) repeat part (a) for \(X\) interchanged with \(P\).
- Homework 6: (Due Monday, November 16)
Consider a modified harmonic oscillator with the Hamiltonian \(H=P^2/2m + kx^2/2 + \alpha x\) where the term
\(\alpha x\) corresponds to the action of an additional constant force.
- Define two operators \(b_+\) and \(b_-\) so that the Hamiltonian can be written in the
form \( H=\hbar\omega_c (b_+ b_- + 1/2) + V_o\). What is the physical significance of \(V_o\)?
- Using these operators, find the energy levels of this oscillator.
- Using the fact that the ground state cannot be lowered to another eigenstate, write down the corresponding
differential equation and use it to find the ground state eigenfunction. Interpret the result and compare it
to the properties of the classical oscillator with the same Hamiltonian.
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:
surname-ID-HW#.pdf (e.g. yalabik-34534534-HW1.pdf)
Also, please make sure you include the submission date on the first page so that we can account for early/late submissions.
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.