06 Jun 2026
III: A Fixed-Confinement Angular-Momentum Quench
The quench studied here changes one term of the Hamiltonian, not the complete magnetic field. An exceptional-Hermite product state is first prepared exactly in two rotated coordinates. At $t=0$, a signed angular coupling is activated while the coordinate confinement—including its diamagnetic dressing—is held fixed.
Conventions and preparation coordinates
We use
\[\hbar=2M=1,\]and retain the charge sign in the half-cyclotron frequency
\[\omega_c=\frac{q_{\mathrm{el}}B}{2Mc}.\]Thus reversing the field or the charge sends $\omega_c\mapsto-\omega_c$. The two preparation coordinates are assigned once and kept fixed:
\[A\equiv +,\qquad B\equiv -, \qquad \mathcal H_A\otimes\mathcal H_B =\mathcal H_+\otimes\mathcal H_-.\]They arise from an orthogonal rotation of the laboratory coordinates. The corresponding coordinate scales are
\[\Omega_\pm^2 =\frac{\omega_x^2+\omega_y^2+2\omega_c^2}{2} \pm\frac12\sqrt{(\omega_x^2-\omega_y^2)^2+4g^2}.\]The $\omega_c^2$ contribution is already included in these scales during preparation. The labels $+$ and $-$ describe coordinate directions, not the full phase-space normal modes of the magnetic oscillator.
The rotation preserves the angular generator,
\[L_z=q_+p_- -q_-p_+.\]This invariance lets us state the quench directly in the preparation coordinates.
Exact exceptional-Hermite factors
For an even nonnegative codimension $m$, define
\[z=\sqrt{\frac{\Omega}{2}}q, \qquad \mathcal H_m(z)=(-i)^mH_m(iz),\]where $\mathcal H_m$ is the pseudo-Hermite seed. Its absence of real zeros for even $m$ makes the rational extension regular on the full line. The one-dimensional potential is
\[V_m^-(q;\Omega) =\frac{\Omega^2q^2}{4} -\Omega\left[ 1+\frac{\mathcal H_m''}{\mathcal H_m} -\left(\frac{\mathcal H_m'}{\mathcal H_m}\right)^2 \right],\]with factorization energy
\[\epsilon_m(\Omega)=-\left(m+\frac12\right)\Omega.\]It is convenient to write
\[U_\alpha(q_\alpha) =V^-_{m_\alpha}(q_\alpha;\Omega_\alpha) -\epsilon_{m_\alpha}(\Omega_\alpha), \qquad h_{m_\alpha}=p_\alpha^2+U_\alpha(q_\alpha).\]The preparatory Hamiltonian is exactly additive:
\[H_{\mathrm{sep}}^{\mathrm{RE}} =h_{m_+}(q_+;\Omega_+)+h_{m_-}(q_-;\Omega_-).\]Its real, definite-parity product eigenstates are
\[\lvert\Psi_0\rangle =\lvert m_+,\ell_+\rangle_A \otimes\lvert m_-,\ell_-\rangle_B,\]with one-factor parity $(-1)^{\ell_\alpha}$. This is an exact preparation statement: no Gaussian approximation and no finite basis cutoff is used to define $\lvert\Psi_0\rangle$.
What is quenched
At $t=0$, the post-quench Hamiltonian becomes
\[H_f=H_{\mathrm{RE},B} =H_{\mathrm{sep}}^{\mathrm{RE}}-\omega_cL_z,\]and the state evolves as
\[\lvert\Psi(t)\rangle=e^{-iH_ft}\lvert\Psi_0\rangle.\]Only the angular generator $-\omega_cL_z$ is switched on. The scales $\Omega_\pm$, and therefore the diamagnetically dressed coordinate confinement, are unchanged across the quench.
This distinction is physical. Switching a real magnetic field would generally change both the angular term and the $B^2$-dependent diamagnetic term. That is a different protocol. The present construction is naturally interpreted as a synthetic rotation or engineered angular coupling added to a fixed trap.
For the underlying magnetic oscillator, lower boundedness requires the bare-trap conditions
\[\omega_x^2>0, \qquad \omega_y^2>0, \qquad \omega_x^2\omega_y^2-g^2>0.\]With these conditions and regular even seeds, the rational terms are bounded relative to the confining oscillator and $H_f$ has the standard self-adjoint, lower-bounded realization on the oscillator domain.
The fixed-coordinate commutator
The decisive algebraic test is
\[\boxed{ [H_{\mathrm{sep}}^{\mathrm{RE}},L_z] =i\left[q_+U_-'(q_-)-q_-U_+'(q_+)\right]. }\]For a generic anisotropic or rationally deformed preparation, the right-hand side does not vanish. Consequently,
- $H_f$ is not additive as an operator on the fixed preparation tensor product $\mathcal H_+\otimes\mathcal H_-$; and
- the product eigenbasis of $H_{\mathrm{sep}}^{\mathrm{RE}}$ is not stationary under $H_f$.
These are coordinate-specific conclusions. The commutator does not prove generic nonseparability under every possible canonical transformation, and it does not rule out all diagonalizations. In the purely quadratic limit, a symplectic transformation can diagonalize the magnetic oscillator. With rational local potentials, however, such a transformation generally sacrifices locality in the original $q_+:q_-$ preparation split.
Exact preparation is not exact post-quench propagation
Three levels of description must remain separate:
- The exceptional-Hermite factors and their product preparation are exact.
- Nested commutators give exact derivatives at $t=0$, but truncating the resulting Taylor series is only a short-time approximation.
- Expanding the evolving state in an exceptional-product basis provides a numerical finite-time method. Any finite cutoff is an approximation that must be tested for convergence; there is no claim of an exact finite truncation.
This separation prevents the known spectrum of $H_{\mathrm{sep}}^{\mathrm{RE}}$ from being mistaken for the spectrum or dynamics of $H_f$.
Discussion