# Operators and expectation values.

I'm stuck on a question in atkins molecular quantum mechanics 4e (self test 1.9).

If (Af)* = -Af, show that <A> = 0 for any real function f.

I think you are expected to use the completeness relation sum,s { |s><s| = 1.

I'm sure the answer is simple but I'm stumped.

Last edited:

tom.stoer
What are A and f?

I'm assuming A is any real or complex operator for which the relation holds and f is any real function.

tom.stoer
And what is <f> ?

In QM <f> would be the expectation value of an operator f sandwitched between two states. But I guess you have something in mind like ∫ dx f(x)

Sorry, my mistake. Need to show that the expectation value of the operator A is zero. <f|A|f> = 0

tom.stoer
Ok: If

$$(Af)^\ast = -Af$$

show that

$$\langle f|A|f\rangle = 0$$

for any real function f.

Yes.

Does that make sense?

tom.stoer
I think something is missing.

What I can show immediately is

$$a = \int dx \, fAf = \int dx \, f (-Af)^\ast = -a^\ast$$

and therefore that a must be purely imaginary.

Note that I ommited the range of integration which must be symmetric [-L,L]

tom.stoer
As I said, something is missing. In the context of Atkins 1.9 only hermitean operators a considered.

1) so by the singe line above $\text{Re}\,a = 0$
2) by hermiticity $\text{Im}\,a = 0$

Therefore $a=0$

vela
Staff Emeritus