Phys 825
homework #1
due Thurs. Sept. 3
1. In SI units the permittivity of free space is 0= 8.85 ×10−12 C2/N·m2. In this
course it will be convenient to set 0= 1. This defines our unit of charge.
(i) Show that in ¯h=c=0= 1 units electric charge is dimensionless.
(ii) In these units, what is the charge of an electron?
2. Consider the following Lagrangian for the vector potential Aµ,
L=−1
4Fµν Fµν
where Fµν =∂µAν−∂νAµis the field strength.1
Vary the corresponding action with respect to Aµ. Show that the principle of
least action gives the two Maxwell equations which aren’t already implied by
writing the field strength in terms of the vector potential.
3. Consider the following Lagrangian for a complex field ψ(t, x).
L=i¯hψ∗∂ψ
∂t −¯h2
2m~
∇ψ∗·~
∇ψ−V(x)ψ∗ψ
What is the equation of motion? Does it look familiar? We’ll have more to say
about this Lagrangian later in the course.
Hint: when you have a complex field, you can vary ψand ψ∗as though they
were independent fields. If you don’t believe this, set ψ=ψ1+iψ2and show
that varying the real and imaginary parts separately leads to the same answer.
1I’m setting c=0=µ0= 1. If you want to restore units I think it should be L=−1
4µ0FµνFµν .