Von Neumann's uniqueness theorem, also known as the Stone-von Neumann theorem, states that any two irreducible representations of the Canonical Commutation Relations (CCR) that satisfy certain regularity conditions are unitarily equivalent. This means that there is a unitary transformation that can map one representation onto the other, preserving the structure of the CCR.
The Canonical Commutation Relations (CCR) are mathematical expressions that describe the fundamental commutation rules between position and momentum operators in quantum mechanics. Specifically, for position operator \( \hat{q} \) and momentum operator \( \hat{p} \), the CCR is given by \([ \hat{q}, \hat{p} ] = i\hbar\), where \(\hbar\) is the reduced Planck constant and \([ \cdot, \cdot ]\) denotes the commutator.
Von Neumann's uniqueness theorem is crucial because it ensures the consistency of quantum mechanics. It guarantees that the mathematical formulation of quantum mechanics through the CCR is unique up to unitary equivalence. This means that different quantum systems described by the same CCR can be related through a unitary transformation, ensuring that physical predictions remain consistent.
The regularity conditions typically refer to the assumptions that the representations of the CCR are irreducible and that the operators involved act on a Hilbert space in a manner consistent with the standard quantum mechanical framework. Specifically, the representations must be strongly continuous, and the operators must be bounded or densely defined.
An example of the application of Von Neumann's uniqueness theorem is in the context of the quantum harmonic oscillator. The position and momentum operators for the harmonic oscillator satisfy the CCR. According to the theorem, any two irreducible representations of these operators that satisfy the regularity conditions will be unitarily equivalent. This ensures that the quantum harmonic oscillator has a unique description up to unitary transformations, which is essential for the consistency of its physical predictions.