Physics 249 Midterm Exam 1
Oct 12th 2012
Constants:
ℎ𝑐 = 1240𝑀𝑒𝑉𝑓𝑚
Energy of the rest mass of the electron: 0.511𝑀𝑒𝑉
Charge over mass ratio of the electron: 1.76x10 11 C/kg
Useful Integrals:
1) An electron of kinetic energy 45keV moves in a circular orbit perpendicular to a magnetic field of
0.325T. Find the radius of the orbit in meters
(15 points)
2) Derive the units of the fine structure constant.
(15 points)
3) Demonstrate that the wave function of the lowest energy level of the particle in a box obeys the
uncertainty relation by calculating the Δ𝑥 and Δp using expectation values.
For the potential use:
V(x) = 0, 0 < x < L
V(x) = infinity x < 0, x > L
(30 points)
4) Explain why the infinite potential well wave function above satisfied the uncertainty principle, but with
slightly larger uncertainty in x and p, while the lowest energy state of the harmonic oscillator satisfies the
uncertainty principle with the minimum possible uncertainty in x and p.
(15 points)
5) For an electron in a finite potential well with L=0.001nm and V=10MeV how many bound state
solutions are there? The conditions for the quantized energy solutions are:
𝑘𝑛 𝐿
𝛼
tan (
)=
2
𝑘𝑛
and
𝑘𝑛 𝐿
𝛼
−cot (
)=
2
𝑘𝑛
(25 points)