Unraveling QM Postulate 3: From Origin to Implications

[hasan_researc]

Hey, I am a second year physicist, and I have only recently been introduced to the postulates of QM. I am still trying hard to understand how the original inventor's of QM came up with the third postulate.

Postulate 3 says the classical combination of dynamical variables gives the correct combination
of operators in quantum mechanics.

Why/how did they make up this postulate?
What are its implications?

Combination of dynamics variables (e.g. p) gives the correct combination of operators (e.g. ?) in QM? ::

 

[arkajad]

Usually these are quantities of the form f(x)+f(p) or of the form x p_y - that is sums of functions of commuting variables. There is no ambiguity there and the algebra of commuting quantum observables is isomorphic to the algebra of the corresponding classical functions on the phase space. Taking sums is justified by the fact that the expectation value of a sum is a sum of expectation values.

Ocassionally (bur it is rare) it happens that we have to find a quantum operator that corresponds to a classical quantity like xp_x. Then we have a problem because the corresponding quantum operator is not Hermitian. The simplest way out is to symmetrize and take \frac12 (xp_x+p_x x) as the corresponding quantum observable. Sometimes that works, sometimes not. But these are, as I said, rare cases.

 

[martinbn]

There is also Weyl Quantization. Using Fourier inversion you don't have to worry about non-commuting operators.

http://en.wikipedia.org/wiki/Weyl_quantization"

 

[arkajad]

There is also Weyl Quantization. Using Fourier inversion you don't have to worry about non-commuting operators.

http://en.wikipedia.org/wiki/Weyl_quantization"

There are many different quantizations. Once you choose one - the rest follows from your choice. Weyl's quantization is defective in the sense that it does not respect positivity - which seems to have a physical meaning. Other quantizations may respect positivity but dot respect something else.