An Old Thought

Last updated 2015-04-07 08:52:56 SGT

In the pedagogy of the quantum isotropic harmonic oscillator (in 3D) it is usual to effect a change of basis from [\{A_x, A_y, A_z\}] to [\{A_\pm, A_0\}], with [A_\pm = {1 \over \sqrt{2}}\left(A_x \mp i A_y\right),~ A_0 = A_z]. The commutator relations (e.g. [[A_i, A_j^\dagger] = \delta_{ij}]) are used to demonstrate that the change of basis is valid; Dirac's postulate leads us to suspect that this might have some correspondence in classical physics.

In fact, what's really going on is that we are performing a canonical transformation from one (sub)set of phase space coordinates to another; in this case, our definition of the [A_\pm] operators results in a linear canonical transformation in the [x] and [y] coordinates as (for [m = \omega = \hbar = 1])

[\begin{bmatrix} q_+ \\ q_- \\ p_+ \\ p_- \end{bmatrix} = \underbrace{{1 \over \sqrt{2}} \begin{bmatrix} 1 & 0 & 0 & 1\\ 1 & 0 & 0 & -1\\ 0 & -1 & 1 & 0\\ 0 & 1 & 1 & 0\\ \end{bmatrix}}_{=A\in \mathrm{Sp}(4, \mathbb{C})} \begin{bmatrix} x \\ y \\ p_x \\ p_y \end{bmatrix}, \{q_0, p_0\} = \{z, p_z\}]

This transformation [A] preserves the [n=2] symplectic form (and so the Poisson algebra) as [A^T\Omega_2 A = \Omega_2]. The lesson to be learned here is that the validity of a canonical transformation can be checked easily upon quantisation by verifying that the correct commutator relations are satisfied. However, this remains a heuristic procedure at best, since Dirac's postulate does not hold absolutely.

At no point have we sought explicit recourse to a representation of the [\left|n_+, n_-, n_0\right>] eigenstate in any coordinate basis; there is simply no need to. However, since the Hamiltonian takes the same form in the [\left|n_+, n_-, n_0\right>] basis as it does in the [\left|n_x, n_y, n_z\right>] basis (owing to the [\mathrm{U}(3)] symmetry that permitted the canonical transformation in the first place), if we wished to recover the position wavefunctions we could first make the substitution [\{x, y, z\}\to\{q_+, q_-, q_0\}] in the original wavefunctions before applying an analogous integral transform to that here.

More interestingly, since the canonical transformation is linear, the [\left|n_+, n_-, n_0\right>] eigenstates must be linear combinations of the [\left|n_x, n_y, n_z\right>] eigenstates, as is the case with rotations. In principle, one could use the same arguments and procedures later recounted for Clebsch-Gordan coefficients to find unambiguous linear transformations between the [\left|n_+, n_-, n_0\right>] and [\left|n_x, n_y, n_z\right>] eigenkets (up to sign and phase etc).

Like the correspondence between the [\left|m_1, m_2\right>] and [\left|j, m\right>] eigenstates, what's happening here is a decomposition of a symmetry group into irreducible representations. For the angular momentum Clebsch-Gordan coefficients, we have decomposition into representations of the group [\mathrm{SO}(3)] of spatial rotations; in this case, we're dealing with transformations in phase space (and so, the space of linear canonical transformations in [x] and [y]) via the group [\mathrm{SU}(2)\times \mathrm{U}(1) \sim \mathrm{U}(2) \subset \mathrm{U}(3)], under which action the Hamiltonian remains invariant.

This connection with [\mathrm{SU}(2)] permits a nice conceptual analogy between these transformations in 4-space with 3D rotations (for example, it can be shown that the above canonical transformation corresponds to a rotation by [\theta = -{2\pi \over 3}] around the axis Pauli vector [{1 \over \sqrt{3}}\left(\sigma_x+\sigma_y+\sigma_z\right)] together with a phase shift of [\pi \over 4], up to sign). Through Noether's Theorem, this symmetry implies a conservation law, and indeed we can verify that an analogue to the Laplace-Runge-Lenz vector (usually seen with the Coulomb potential) is conserved in this instance (see here)

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