Key Points:

1. Marginal Distributions Must Agree with Quantum Theory

The joint probability distribution must reproduce the standard quantum probability distributions when integrated over one of the variables:

where ψ(q) and φ(p) are the wavefunctions in position and momentum space, respectively.

2. Fourier Transform Approach

The characteristic function of an observable A is given by:

By analogy, a joint characteristic function for position and momentum can be introduced, leading to a proposed joint probability distribution.

3. Wigner Function and Negativity Issue

One approach is to define a phase-space distribution such as the Wigner function:

However, the Wigner function can take negative values, which prevents it from being interpreted as a genuine probability distribution.

4. Impossibility of a Classical Joint Distribution

Analysis by Cohen and Margenau shows that it is impossible to construct a classical probability distribution P(q, p; ψ) that satisfies all quantum mechanical requirements, particularly those related to operator ordering and the uncertainty principle.

Conclusion

While various attempts have been made to construct joint probability distributions for position and momentum, they either fail to meet quantum consistency conditions or lead to negative probabilities. This demonstrates a fundamental departure of quantum mechanics from classical probability theory.

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