Joint Probability Distributions in Ballentine’s Statistical Interpretation of Quantum Mechanics
Joint Probability Distributions in Quantum Mechanics
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.
Leave a Reply