PHY 4604 Fall 2008 – Practice Questions for Exam 1
1. In each figure (a)–(d) below, the dashed line in the upper part represents an infinite
square-well potential that vanishes everywhere between the hard walls denoted by the
upward-pointing arrows. The solid line shows a modified potential V (x) having the
same hard-wall positions. The magnitude of V (x) inside the well has a maximum value
about one-third of the ground-state energy.
Using qualitative reasoning rather than detailed calculation, sketch on the lower part
of each figure both the ground-state wave function and the first excited-state wave
function of V (x). Use the corresponding wave functions of the infinite square well
(dashed lines) as a guide to the vertical scale.
(a)
(b)
(c)
(d)
2. A particle moves in the one-dimensional potential V (x) shown below. It is in its ground
state, which has the energy E indicated by the dashed line in the figure. Sketch the
wave function.
V(x)
E
x
3. A one-dimensional step potential takes the form
0
for x < 0,
V (x) =
for x > 0.
V0 > 0
A particle of mass m approaches the step from the left with an energy E = V0 .
(a) Find the wave function describing this situation. Your solution should contain
just one unknown parameter.
(b) Find the probability density and the probability current (i) in the region x < 0,
and (ii) in the region x > 0.
(c) Calculate the particle’s reflection and transmission coefficients.