Discuss the formulation used by Shimony and that of chapter 3.

In understanding quantum mechanics the key will be to visualize or understand a given state by looking at it through states of interest.

Above we find many equivalent ways of writing down the state .

In particular we might be interested in knowing whether the state has observable A or B when one is given that the state has components , . The states , are not orthogonal but we will require that and are normalized and orthogonal.

The first step is to normalize the state.

As long as we are considering the state in terms of orthogonal states we can formulate the normalization as the sum. Let us assume that A and B were chosen so that

Then =.

In terms of the states . One can see that they are not orthogonal but that is okay. One can also see that the probability of the state and the probability of the state do not necessarily add to 1.

Since we have chosen the values of A and B to properly normalize the state

and looking at

Clearly if the coefficients are real then all the cross terms cancel. However if there are complex coefficients these terms are not necessarily zero. For example, examine

where delta defined above tells us a phase difference between the amount in state A between and . The states overlap in the subspace A but are not in phase. Therefore they can interfere. [Same as the two slit problem where going through slit 1 and going through slit 2 have amplitudes to hit screen that overlap and interfere.]