The John James Polarization Formula

It is well-known that not only the expectation values, but the matrix elements of an observable in quantum theory are observable in principle, because they can be decomposed as linear combinations of four expectation values by an application of the polarization formula:

 <ψ, φ>  =  ¼ ||ψ + φ||2  –  ¼ ||ψ – φ||2  –  ¼ i ||ψ + i φ||2  +  ¼ i ||ψ – i φ||2

In a first-year graduate level quantum mechanics course I taught at the University of Michigan in Winter, 1975, I assigned as a homework problem to find a polarization formula, without mentioning that one should expect four terms. A very bright student, John V. James, came up with a three-term formula, which was not well-known:

 <ψ, φ>  =  ½ ||ψ + φ||2  –  ¼(1 + i) ||ψ + i φ||2  –  ¼(1 – i) ||ψ – i φ||2