ABSTRACT
GREAT EXPECTATIONS In Chapter 5, I introduced the expectation value-the average of a set of measurement results taken from a collection of systems in the same state. A straightforward calculation of the expectation value takes the following form, with ˆ O being an operator representing a measurement of a speci c physical variable and φ〉 the state of each system in the collection:
〈 ˆ O 〉 = 〈$ ˆ O$〉 If we choose a basis {i〉} , we can expand φ〉 and 〈φ
SUM SUM
i i j j a i a j( ) ( )*
and plug the expansions into our formula for the expectation value:
O a a j O i
j i j i SUM SUM * ( )
To take the next step, we need to remember that ai = 〈iφ〉 and aj* = 〈φj〉, which can be used to replace the a’s in the formula giving
〈 ˆ O 〉 = SU j M (SU
i M [〈 j 〉〈i 〉〈 j ˆ Oi 〉]) = SU
j M (SU
i M [〈i 〉〈 j 〉〈j ˆ Oi 〉])
Where in the last step, all I have done is change the order of the terms in the square bracket.
