Constrained quantum systems as an adiabatic problem

We derive the effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) in the asymptotic limit, where the restoring forces tend to infinity. In contrast to earlier works, we consider, at the same time, the effects of variations in the constraining potential and the effects of interior and exterior geometry, which appear at different energy scales and, thus, provide a complete picture, which ranges over all interesting energy scales. We show that the leading order contribution to the effective Hamiltonian is the adiabatic potential given by an eigenvalue of the confining potential well known in the context of adiabatic quantum waveguides. At next to leading order, we see effects from the variation of the normal eigenfunctions in the form of a Berry connection. We apply our results to quantum waveguides and provide an example for the occurrence of a topological phase due to the geometry of a quantum wave circuit (i.e., a closed quantum waveguide).

Figure 1

In (a), we plotted a potential for a waveguide, which widens near $x=0$. The widening lowers the energy of normal modes and, thus, produces an attractive effective potential for the motion in the $x$ direction. In (b), the modulus of the ground state wave function is sketched. Its variation in the $x$ direction is slower than in the $y$ direction, but its tangential derivatives already grow in $\varepsilon$. In (c), the modulus of an excited state with energy of order 1 above the ground state is sketched. Its variation in the $x$ direction is on the same scale as the confinement (i.e., it oscillates on a scale of order $\varepsilon$). Thus, any analysis that assumes bounded tangential derivatives of the solutions will be restricted to confining potentials with a constant profile.

Figure 2

The curves $E_0(x)$ and $E_1(x)$ are sketches of the lowest normal eigenvalues for a waveguide potential as depicted in Fig. 1a. On the vertical axis, the spectrum of $H^{\varepsilon }$ is drawn: One expects spectral bands, which start at the minima $e_0$ and $e_1$ of the effective potentials with level spacing of order ${\varepsilon }^2$. The continuum edge $\Sigma$ is determined by the threshold of $E_0$. Eigenstates in the lower shaded region vary on a $\sqrt{\varepsilon }$ scale in the $x$ direction as indicated in Fig. 1b. Eigenstates in the upper shaded region with energies of order 1 above $e_0$ have $\varepsilon$ oscillations in the $x$ direction as indicated in Fig. 1c.

Figure 3

Level set of a potential such as $V_{\mathrm{c}}^{x/2}$, which describes a Möbius wave circuit.

Figure 4

First excited transverse eigenfunction in a Möbius wave circuit: $n_1$ and $n_2$ are the normal elements of a global Tang frame; and, with respect to this frame, the confining potential $V_{\mathrm{c}}^{x/2}$ twists by an angle of $\pi$ when going around the circuit once. Due to its symmetry, $V_{\mathrm{c}}^{x/2}$ is still globally smooth; see Fig. 3. However, since the first excited state ${\Phi }_1$ does not have the full symmetry of the potential, it changes sign after a twist by $\pi$.