- #1
tetris11
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Homework Statement
|O> = k |R1> + 1/9 |R2>
a) Find k if |O> has already been normalized, and b) then the expectation value.
The Attempt at a Solution
a)
To Normalise:
|(|O>)|2 = (1/9 |R2> + k |R1>).(1/9 |R2> - k|R1>) = 1/81|R2>2 - k2|R1>2 = 1
I just assumed that |k| = (1-(1/81))0.5, but I can't justify why. Is this answer correct?
b)
Expectation value:
Operator A of the system D
<D|A|D> = ∫D*AD dt = is the expectation value, where:
A|n> = λn|n>
|D> = Σ Cn |Qn>
<D| = Σ Cn* <Qn|
So: <D|A|D> = Σ |Cn|2 λn
So in the case of O:
<O|A|O> = ΣΣ [(1/9<R2| - k<R1|).A.(1/9|R2> + k|R1>)]
= ΣΣ[(1/81<R2|R2> + k/9<R2|R1> - k/9<R1|R2> - k2<R1|R1>)].A
= ? Help