StoneVonNeumannTheorem

The Stone-von Neumann theorem is a fundamental result in the mathematical formulation of quantum mechanics, particularly in the context of quantum harmonic oscillators. This theorem is significant not only because it guarantees the uniqueness of the representation of the Canonical Commutation Relations (CCR), but also because it provides a method to construct all possible representations of the CCR.

The theorem is articulated as follows: given a pair of self-adjoint operators $\hat{p}$ and $\hat{q}$ on a Hilbert space satisfying the CCR,

$$[\hat{q}, \hat{p}] = i \hbar \mathbb{I}$$

we can define the unitary operators $U$ and $V$ by

$$U(a) = e^{ia \hat{p} / \hbar} \quad V(b) = e^{ib \hat{q} / \hbar}$$

These operators satisfy the Weyl relations, which are a stronger form of the CCR:

$$U(a) V(b) = e^{iab} V(b) U(a)$$

The Stone-von Neumann theorem then asserts that every irreducible unitary representation of the Weyl relations is unitarily equivalent to the representation generated by $U$ and $V$.

This construction is instrumental for understanding the quantum harmonic oscillator because the Hamiltonian operator of the quantum harmonic oscillator can be expressed in terms of $U$ and $V$. The Stone-von Neumann theorem informs us that all possible representations of the quantum harmonic oscillator are unitarily equivalent.

In essence, the Stone-von Neumann theorem is crucial not only for its assurance of the uniqueness of the representation of the CCR, but also for enabling the construction of all possible representations of the CCR, thereby providing a deeper understanding of the algebraic structure of quantum mechanics, especially in the study of quantum harmonic oscillators.