Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #2
Due Wednesday, September 3 at 2pm
Problem 2.1
: Fun with statistics. (20 points)
To do this problem, while you may use a calculator, spreadsheet or mathematical software
to do simple algebraic operations like adding and dividing, don’t use any built-in statistics
functions.
A box contains 18 small items, of various lengths. The distribution of lengths in cm of
the set of objects is
3
,
3
,
3
,
3
,
4
,
6
,
6
,
6
,
8
,
8
,
8
,
8
,
9
,
9
,
9
,
11
,
11
,
11
.
(1)
a) What is the probability that an object chosen at random from the box will have length 8
cm, assuming there is an equal probability of selecting any one object?
b) What is the average length
h
L
i
(also called the
expectation value of length
) of an object
in the box?
c) What is the probability that an object chosen at random from the box will have length
h
L
i
, assuming there is an equal probability of selecting any one object?
d) What is the average of the square of the length
h
L
2
i
of an object in the box?
e) Use your previous results to compute the standard deviation
σ
for the length of an item
drawn from the box.
f) What is the probability that an object chosen at random from the box will have length in
the range
h
L
i±
σ
, assuming there is an equal probability of selecting any one object? Based
on what you know about standard deviations, does this seem reasonable?
This problem is to give you practice and intuition with probability distributions, and to
think about the diﬀerences between probabilities on the one hand, and averages/expectation
values on the other. Quantum mechanics forces us to think in probabilistic terms, so we’d
better get used to it!
Problem 2.2