The Hidden Variables section in Ballentine’s Statistical Interpretation of Quantum Mechanics examines the possibility of supplementing quantum mechanics with additional parameters (hidden variables) that determine the outcome of individual measurements, rather than relying on probabilistic quantum states.

Von Neumann’s Theorem

Von Neumann’s theorem aimed to show that no hidden-variable theory could reproduce all the statistical predictions of quantum mechanics. His proof relied on the assumption that expectation values should be additive:

This condition holds for quantum ensembles but assumes it should also apply to hidden-variable models.

Key Issues in Von Neumann’s Proof

• Noncommuting Observables: Quantum mechanics includes observables that do not commute, such as position q and momentum p. Von Neumann’s theorem does not properly account for these cases.
• Strong Assumption of Linearity: Expectation value linearity does not necessarily hold in hidden-variable models.
• Implication: If Von Neumann’s assumptions were correct, quantum mechanics would be the only possible theory, ruling out hidden variables.

Bell’s Rebuttal

John Bell revisited Von Neumann’s proof and identified its flaws. He pointed out that the assumption:

is not valid for hidden-variable theories since it assumes quantum mechanical averages apply to individual measurements.

Bell’s Key Arguments

• Hidden-Variable Theories Do Exist: Bohmian mechanics (Bohm, 1952) reproduces all quantum statistical predictions.
• Misinterpretation of Linearity: Expectation values of noncommuting variables need not sum linearly.
• Constructing a Working Hidden-Variable Model: Bell provided counterexamples demonstrating that hidden variables could exist.

Bell’s Theorem: A Stronger No-Go Result

While Bell criticized Von Neumann’s proof, he later formulated Bell’s theorem, which provided a stronger argument against local hidden-variable theories. His theorem is based on Bell inequalities,

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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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1. Fundamental Differences

Bell’s Theorem: Demonstrates that no local hidden variable theory can fully explain quantum correlations observed in entangled systems. It is tested through inequalities (e.g., CHSH inequality), and experimental violations indicate nonlocality.

Laws of Thermodynamics: Govern energy, entropy, and equilibrium in macroscopic systems, ensuring that physical processes obey conservation laws and the increase of entropy.

2. Possible Connections

Though these domains are distinct, there are areas where quantum mechanics and thermodynamics interact:

a) Entanglement and the Second Law of Thermodynamics

• The Second Law states that entropy (disorder) never decreases in an isolated system.
• Entanglement generates quantum correlations that can be viewed as a resource.
• Using entanglement in thermodynamic processes is still constrained by the Second Law.

b) Information Theory and the Second Law

• Landauer’s Principle: Erasing information in a classical system requires energy dissipation (k_B T ln 2 per bit).
• Quantum correlations from Bell experiments involve information transfer in ways that challenge classical assumptions.
• Some interpretations suggest that entanglement might provide resources for thermodynamic efficiency beyond classical limits.

c) Quantum Thermodynamics

• Modern research examines thermodynamic cycles using entangled states.
• Bell inequalities can be used to study non-equilibrium thermodynamics.
• The Jarzynski equality and fluctuation theorems have quantum analogs that connect measurement, entropy, and energy exchanges.

Conclusion

While Bell’s theorem itself is not a thermodynamic statement, its implications for nonlocality and quantum information have inspired discussions about the foundations of thermodynamics in quantum systems. Future quantum technologies (quantum engines, quantum heat baths) might use entanglement in ways that challenge our classical understanding of energy and entropy.

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Wave Continuity in Matrix Formulation of Quantum Mechanics?

In the matrix formulation of quantum mechanics (developed primarily by Heisenberg), the concept that replaces
analytic continuity of the wave function is the algebraic structure and consistency of operators in Hilbert space.
While the Schrödinger formulation relies on the analytic properties of the wave function, the matrix formulation emphasizes the following key features:

• Hermiticity and Self-Adjoint Operators:Physical observables in the matrix formulation are represented by Hermitian operators (self-adjoint matrices), ensuring real eigenvalues
  corresponding to measurable quantities. The algebra of these operators must be consistent and preserve the physical requirements of the theory.
• Commutation Relations:The fundamental relationships between observables, such as the canonical commutation relations
  [ ˆx, ˆp ] = iℏ, play a central role. These relations ensure the internal consistency of quantum mechanics
  and replace the need for analytic continuity of the wave function.
• Unitary Evolution:In the Schrödinger picture, the time evolution of the wave function must be continuous and differentiable, governed by the Schrödinger equation.
  In the matrix formulation, the time evolution is encoded in the unitary evolution of state vectors or density matrices in Hilbert space, satisfying
  ˆU(t)ˆU†(t) = ˆI.
• Spectrum and Eigenstates:The spectrum of the operators (eigenvalues) and their corresponding eigenstates provide the quantum mechanical predictions. These eigenvalues and
  eigenvectors are well-defined algebraically, independent of the notion of analytic continuity.
• Matrix Element Consistency:The elements of matrices in this formulation encode transition amplitudes between quantum states, and their consistency is guaranteed by the
  mathematical framework of linear algebra and Hilbert space theory.

In summary, while analytic continuity ensures the smoothness and well-defined behavior of the wave function in the Schrödinger picture, the matrix formulation
relies on the consistency of operator algebra, the structure of commutation relations, and the properties of Hilbert space. This shift reflects the abstract
algebraic nature of the matrix formulation, which avoids the explicit reliance on continuous functions.

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The wave picture and the matrix

All the Useful Math

Analytic continuation of wave funtions, analytic functions – all these conceptes developed over hundreds of years, are useful in the wave function picture.  In fact, without these, appropriate  energy levels (or any other measurable results) cannot be derived.

So what happens to all these constructs when we abandon this wave picture (in favor of the matrix picture)?

The Stark Contrast

These two  pictures could not be more different – mathematically or physically speaking.

One supports – or at least ALLOWS, determinism in physical laws – whereas the other picture completely eliminates it.

So – What’s the research topic?

Are there experiments that support ONLY one of these two pictures? i.e. the results will be different based on which method was used to derive the results?

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Or is there a broader based representation of images?

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The Schrodinger Wave ISN’T REAL

Since the Schrodinger wave is simply an informational wave, how does the AMPLITUDE relate to any kind of energy?

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One cannot perform multiple measurements on the same set of particles, since the first experiment puts the particles in their eigenstates.

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