PHY 6645 Fall 2002 – Homework 6
Due by 5 p.m. on Friday, October 25. Partial credit will be available for solutions
submitted by 5 p.m. on Monday, October 28.
Answer all questions. To gain maximum credit you should explain your reasoning and show
all working. Please write neatly and include your name on the front page of your answers.
1. Shankar Exercises 7.3.4 and 7.4.1.
Note that in the first exercise, you should use the real-space wave functions to derive
the matrix elements of X and P , while in the second exercise, you should use only the
algebraic properties of the creation and annihilation operators and the eigenstates of
the number operator.
2. Shankar Exercise 7.3.7.
3. For the energy eigenstates with n = 0, 1, and 2, compute the probability that the
momentum p of a harmonic oscillator in state n has a value greater in magnitude than
the largest possible momentum of a classical harmonic oscillator of the same energy.
4. For each of the eigenstates n of the harmonic oscillator listed below, generate a single
graph, with y as the horizontal axis (extending over the range 0 ≤ y ≤ ymax ), plotting
vertically three quantities (clearly labeled): (i) ψn (y); (ii) P1,QM (y), where Pw,QM (y) is
the probability of finding the oscillator at a dimensionless coordinate y 0 in the range
y − w/2 < y 0 < y + w/2; (iii) P1,CM (y), the corresponding probability for a classical
harmonic oscillator of the same energy. Note that Pw,CM (y) may be nonzero even if y
lies outside the classical turning points, so long as y − w/2 or y + w/2 is classically
accessible.
You should use the same vertical scale for (ii) and (iii), but feel free to use a different
vertical scale for (i) if it makes the plot clearer.
Generate such a graph for (a) n = 0, ymax = 4; (b) n = 1, ymax = 4; (c) n = 8,
ymax = 7; (d) n = 9, ymax = 7.
To answer this question, you will need to produce some sort of computer program. You
may use any general-purpose programming language, or a higher-level programming
environment such as Mathematica, Matlab, or Maple. You may choose to represent
the wave function as a set of numerical values calculated at a sufficiently fine grid of
y points, or as an algebraic function of y which you can manipulate symbolically. The
main restrictions are that you should develop your own solution to the problem, not
merely copy someone else’s, and you should not use any predefined software for generating the harmonic oscillator wave functions ψn (y). (It is acceptable to use standard
library functions for the Hermite polynomials.)