2. 14
CHAPTER MATHEMATICAL PREREQUISITES
(cos 45 , sin 45 ). The dot product of (cos 45 , sin 45 ) and (cos 45 , sin 45 ) is cos2 45
sin2 45 . That is zero, because cos 45 = sin 45 , so the two eigenvectors are orthogonal.
In linear algebra, you would write the relationship r2 = Ar out as:
x2
y2
2.6.3
x
y
11
11
=
=
x+y
x+y
Solution herm-c
Question: Show that the operator 2 is a Hermitian operator, but i is not.
Answer: By denition, 2 corresponds to multiplying by 2, so 2g is simply the function 2g .
Now write the inner product f |2g and see whether it is the same as 2f |g for any f and g :
f |2g =
all x
f 2g dx =
all x
(2f ) g dx = 2f |g
since the complex conjugate does not aect a real number like 2. So 2 is indeed Hermitian.
On the other hand,
f |ig =
all x
f ig dx x
(if =
all ) g dx = if |g
so i is not Hermitian. An operator like i that ips over the sign of an inner product if it is
moved to the other side is called skew-Hermitian. An operator like 2 + i is neither Hermitian
nor skew-Hermitian.
2.6.4
Solution herm-d
Question: Generalize the previous question, by showing that any complex constant c comes
out of the right hand side of an inner product unchanged, but out of the left hand side as its
complex conjugate;
f |cg = c f |g
cf |g = c f |g .
As a result, a number c is only a Hermitian operator if it is real: if c is complex, the two
expressions above are not the same.
Answer: Since constants can be taken out of an integral:
f |cg =
cf |g =
all x
all x
f cg dx = c
(cf ) g dx = c
all x
f g dx = c f |g
all x
f g dx = c f |g .