(b) We have (1/2)(π+π) = 0, but (1/2)π+ (1/2)π=π.
Page 1008, problem 4:
To do anything useful with these expressions describing units, we need to be able to talk about
things like dividing meters by seconds to get meters per second. Thus “addition” needs to be
multiplication, which corresponds to adding the exponents. Scalar “multiplication” actually
has to be exponentiation, e.g., “multiplying” units of meters by the scalar 2 should give square
meters.
Page 1013, problem 20:
〈Ψ|OEΨ〉=〈c∗ +c′∗ |c + 4c′ 〉
=c∗c〈 | 〉+ 4c∗c′〈 | 〉+ 4c′∗c〈 | 〉+ 4c′∗c′〈 | 〉The second and third terms vanish, because and are distinguishable. Because and
are normalized, the result reduces to
c∗c+ 4c′∗c′=|c|^2 + 4|c′|^2 =1
2
(1 + 4) = 2.5.
Answers to self-checks
Answers to self-checks for chapter 0
Page 15:
If only he has the special powers, then his results can never be reproduced.
Page 17:
They would have had to weigh the rays, or check for a loss of weight in the object from which
they were have emitted. (For technical reasons, this was not a measurement they could actually
do, hence the opportunity for disagreement.)
Page 25:
A dictionary might define “strong” as “possessing powerful muscles,” but that’s not an oper-
ational definition, because it doesn’t say how to measure strength numerically. One possible
operational definition would be the number of pounds a person can bench press.
Page 27:
A microsecond is 1000 times longer than a nanosecond, so it would seem like 1000 seconds, or
about 20 minutes.
Page 28:
Exponents have to do with multiplication, not addition. The first line should be 100 times
longer than the second, not just twice as long.
Page 31:
The various estimates differ by 5 to 10 million. The CIA’s estimate includes a ridiculous number
of gratuitous significant figures. Does the CIA understand that every day, people in are born
in, die in, immigrate to, and emigrate from Nigeria?
Page 32:
(1) 4; (2) 2; (3) 2