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 $m = \omega = \hbar = 1$)

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)