Geometry-induced wave-function collapse

When a quantum particle moves in a curved space, a geometric potential can arise. In spite of a long history of extensive theoretical studies, to experimentally observe the geometric potential remains a challenge. What are the physically observable consequences of such a geometric potential? Solving the Schrödinger equation on a truncated conic surface, we uncover a class of quantum scattering states that bear a strong resemblance to the quasiresonant states associated with atomic collapse about a Coulomb impurity, a remarkable quantum phenomenon in which an infinite number of quasiresonant states emerge. A characteristic defining feature of such collapse states is the infinite oscillations of the local density of states (LDOS) about the zero energy point separating the scattering from the bound states. The emergence of such states in the curved (Riemannian) space requires neither a relativistic quantum mechanism nor any Coulomb impurity: they have zero angular momentum and their origin is purely geometrical, hence the term “geometry-induced wave-function collapse.” We establish the collapsing nature of these states through a detailed comparative analysis of the behavior of the LDOS for both the zero and finite angular momentum states as well as the corresponding classical picture. Potential experimental schemes to realize the geometry-induced collapse states are articulated. Not only does our paper uncover an intrinsic connection between the geometric potential and atomic collapse, it also provides a method to experimentally observe and characterize geometric potentials arising from different subfields of physics. For example, in nanoscience and nanotechnology, curved geometry has become increasingly common. Our finding suggests that wave-function collapse should be an important factor of consideration in designing and developing nanodevices.

Figure 1

A truncated conic surface with angular deficit $2\pi (1-\alpha )$ for $0<\alpha <1$. The truncation is physically infinitesimal in the sense that the truncated distance away from the apex of the cone ${\rho }_0$ is chosen to have the size of only one or two atoms: ${\rho }_0\approx 2\mathring{\mathrm{A}}$.

Figure 2

Bound states for $\alpha =1/6$. (a) The bound states determined by the zeros of the function $K_{i\tilde{\alpha }}$ (the middle curve), the asymptotic behaviors of which in the small and large energy regimes are given by $sin[\tilde{\alpha }ln(x/2)-{\varphi }_{\tilde{\alpha },0}]$ (the oscillatory curve for a small energy) and $e^{-x}/\sqrt{x}$ (the upper curve for a large energy), respectively, where $x=\sqrt{-\epsilon }$. (b) The ground-state wave function (dashed curve) with the energy ${\epsilon }_{n_0}=-1$ (the solid horizontal line) for the potential of the form $-{\hslash }^2{\tilde{\alpha }}^2/\left(2M{\rho }_0^2{r}^2\right)$ (solid trace) for $r\ge 1$. There is a hard wall at $r=1$. The depth of the potential well, the minimal potential for the whole region, is about $U\approx -8$. The vertical dotted line denotes the “center of mass” $\langle r\rangle$ of the wave function, which is to the left of $r^{*}$, the classical forbidden region. All quantities plotted are dimensionless.

Figure 3

Normalization of the quantum states at large distances from the conical apex: (a) a collapse state at zero angular momentum ($l=0$ and $E>0$) and (b) a conventional scattering state at a finite angular momentum ($\left|l\right|>0$ and $E>0$). The dimensionless distance is defined as $r\equiv \rho /{\rho }_0$ (Sec. 2).

Figure 4

Behavior of the total LDOS for different conic surfaces at $r=10$. (a) LDOS for the two extreme cases: $\alpha =0.99$ and 0.01 (the solid curves and the right inset for small energy). For $\alpha =0.99$, the LDOS oscillates about the value 1, a characteristic of the normal scattering states for $\alpha =1$ in the 2D plane with a hole of radius ${\rho }_0$. For $\alpha =0.01$, no quantum states exist to contribute to the LDOS. In a higher-energy regime, the asymptotic behavior of the LDOS for $\alpha =1$ is shown in the left inset. [(b)–(d)] LDOS plots for $\alpha =5/6, 4/6$, and $3/6$, respectively, where the top curve in each panel is for $\alpha =1$. In these cases, the values about which the LDOS oscillates are between zero and one. In a small energy interval near zero, the LDOS exhibits infinite oscillations with the energy, as shown in the respective insets of the three cases. As argued in the text, in a near zero energy interval, the main contribution to the LDOS comes from the zero angular momentum states, where the infinite oscillations are indicative of the collapse nature of these states. In each panel, the dimensionless energy $\epsilon$ is defined as $\epsilon \equiv E/E_0$ (Sec. 2).

Figure 5

Contributions to the LDOS from the collapse and conventional scattering states. [(a)–(c)] The contribution from the collapse states for three distance values with the asymptotic behavior (dashed curves) in a large energy interval. (d) The contribution from the conventional scattering states at two distances: $r=3\text{and}5$. The inset shows the decay behavior in the higher-energy regime for $r=3$, where the dashed curve indicates the asymptotic behavior. (e) Oscillations of the LDOS due to the collapse states in a small near zero energy interval for different values of the distance, as represented by a 3D plot of the LDOS in terms of both the energy and distance. The amplitudes of LDOS oscillations for different distances are projected to the 2D plane. All quantities plotted are dimensionless.

Figure 6

Effect of different conic geometry on the LDOS. [(a), (b)] LDOS vs energy contributed to by the collapse and conventional scattering states, respectively, for $r=5$ and four different values of the sector angle $2\pi \alpha$. The dashed curves in (a) show the asymptotic behaviors in the high-energy region. The dashed curve in (b) show the LDOS of the conventional scattering states in the 2D plane with a hard hole. (c) The LDOS contributed to by the collapse states in an infinitesimal energy interval near zero, the frequencies of which decrease while the amplitudes increase as $\alpha$ increases from zero to one. The average LDOS $\overline{N}\left(r\right)$ over the near zero energy point is plotted vs $\alpha$, which is projected to the 2D plane. The solid curve in the back plane is the prediction of Eq. (34) and the data points are the corresponding numerical results. All quantities plotted are dimensionless.

Figure 7

Representative classical particle trajectories on a truncated conic surface from the Hamiltonian (35). [(a)–(c)] Three different potential profiles, and [(d)–(f)] the corresponding classical trajectories. [(a), (d)] For $|L_{\mathrm{eff}}|>|L_z|/\alpha$ and $E<0$, the effective potential is attractive and the particle is confined in the region $(\rho ,{\rho }^{*})$, corresponding to bound states. [(b), (e)] For $|L_{\mathrm{eff}}|>|L_z|/\alpha$ and $E>0$, the classical particle can collapse to ${\rho }_0$ in finite time but would eventually escape to infinity due to the reflective boundary condition at ${\rho }_0$. Geometry-induced wave-function collapse occurs in this case. [(c), (f)] For $|L_{\mathrm{eff}}|<|L_z|/\alpha$ and $E>0$, the effective potential is repulsive and the particle is confined in the region $({\rho }^{*},\infty )$, corresponding to the conventional quantum scattering case.

Figure 8

Feasibility of experimentally observing geometry-induced wave-function collapse through LDOS oscillations. (a) For the sector angle $2\pi \alpha =\pi$ of the truncated cone, LDOS oscillations associated with total, collapse, and normal states with the corresponding behavior near zero energy, where $\epsilon \in [{10}^{-6},{10}^{-3}]$ corresponds to $\mathrm{\Xi }\in 2\times [1,{10}^3]\mu \mathrm{eV}$. Experimental observation of the oscillations is feasible [55, 56] (see text for an analysis). (b) The energy-momentum dispersion relation of a massive Dirac fermion and a Schrödinger particle with the dimensionless energy gap $\tilde{\mathrm{\Delta }}=0.05$, corresponding to $\mathrm{\Delta }=0.1$ eV.