PHY 4604 Fall 2010 – Comments Concerning Homework 1
The mean and median scores out of 40 (taking into account late penalties) were 30 and 32.5,
respectively. The highest score was 34, and only five students received less than 25.
Comments specific to Homework 1:
1. Question 1 stated that E might be complex. Thus, it was not justified to assume (as
did most students) that | exp(iEt/~)|2 = 1. A careful calculation (see the solution
distributed in class) shows that the normalization factor A varies with time except in
the special case where E is real.
2. The algebra for Question 2(a) is much simpler if one works with terms of the form eiα
as long as possible before applying Euler’s formula. Replacing eiα by cos α + i sin α at
the start of the calculation of the probability current quadruples the number of terms
that must be considered. Despite the lengthy algebra, most students got Questions
2(a) and 2(b) completely correct.
3. To gain full credit on Questions 2(c)–(e), it was necessary to list all the ways that the
stated conditions could be achieved. There were two answers for (c) and (d), and five
for (e). Most students found one answer for each part, and lost 6 points accordingly.
4. Most students demonstrated mastery of the material covered in Questions 3 and 4, and
gained full points for these questions.
More general comments:
1. Whenever one is asked to calculate a physical quantity, it is important to check that
the final answer has the correct physical dimensions. For instance, a probabilty must
be a pure number, an position uncertainty must have dimensions of length, and a
momentum expectation value must have dimensions of momentum ≡ [mass] [length]
[time]−1 . Any quantities that are added and subtracted must have the same dimensions,
so one should never see a combination such as “(1+L)”, where L is a length. It is useful
in quantum mechanics to remember that ~ has dimensions of [length] [momentum] ≡
[mass] [length]2 [time]−1 .
In this course, errors of dimension will be regarded as more serious (and may receive
a greater points deduction ) than mistakes in purely numerical factors.
2. In many physics questions, the results of part (a) are key inputs to part (b), and so
on. If the answer to (a) is wrong, then it usually follows that the answer to (b) is also
incorrect. In this course, the general grading philosopy is that an error should result in
just one deduction at the point where it occurs, and not produce additional deductions
if the erroneous results are correctly manipulated in later parts of the question. When
full credit is given for an incorrect answer, you may see the grading notation “ecf”
meaning “error carried forward.” However, additional deductions may apply for later
parts of the question if (i) further errors are made, or (ii) the final result for a later
part is clearly unphysical, e.g., a probability is negative or a quantity has the wrong
physical dimensions.