The Schrödinger Equation. Bound States

Ajoy Kumar Ghatak, S. Lokanathan

Maurice A. de Gosson

Stefano Longhi

Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schrödinger equation is retrieved, spatial discretization and lattice effects can deeply modify wave dynamics. Here we discuss implications of breakdown of exact Galilean invariance of the discrete Schrödinger equation on the bound states sustained by a smooth potential well which is uniformly moving on the lattice with a drift velocity v . While in the continuous limit the number of bound states does not depend on the drift velocity v , as one expects from the covariance of ordinary Schrödinger equation for a Galilean boost, lattice effects can lead to a larger number of bound states for the moving potential well as compared to the potential well at rest. Moreover, for a moving potential bound states on a lattice become rather generally quasi-bound (resonance) states.

G. Kunstatter, Saurya Das

Francesco R. Ruggeri

In the literature (e.g. (1)) the Schrodinger equation is sometimes derived using classical fluid equations which follow from Boltzmann’s transport equations. In particular a continuity equation and a conservation of momentum equation involving the potential dV(x)/dx are used. We have argued in Part I that the continuity equation leads to 0+0=0 for a free particle wavefunction exp(-iEt+ipx) because density is 1. Furthermore a bound quantum state also leads to 0+0=0 partly because d/dt partial density(x) =0. Following the analysis of (1) for which a new field S(x,t) is introduced through u(x,t) =velocity field = d/dx partial S(x,t) one may see that the solution S(x,t)= f(t) for any function f yields u(x,t)=0 for a bound state. In fact, for a bound state, the continuity equation contributes nothing to the derivation of the Schrodinger equation except for the result density(x) (no t present) which is known already. The Schrodinger equation for the bound state follows completely from the fluid transport equation containing dV/dx. For this to occur, however, one must use S(x,t)=-Et. It seems, however, that for a bound state with u(x,t) the natural approach is to have S(x,t) =0 Setting S(x,t)= -Et is equivalent to having exp(-iEt) appear in the wavefunction and we argue in this note that this is an independent assumption which does not follow from fluid mechanics. In particular, we link -Et for the bound state to A(x,t)=classical action = -Et+px. For a bound state there is an average energy value, but overall p average is 0. For the free particle quantum system we argued that two flow flux equations d/dx partial A = p ((1a)) and d/dt partial A = -E ((1b)) govern the system. These may be written separately as eigenvalue equations yielding: id/dt partial exp(iA) = E exp(iA) ((2a)) and -id/dx partial exp(iA) = p exp(iA) ((2b)). We show that exp(-iEt) for the bound state is a built-in assumption needed to make the fluid mechanical equation approach to the Schrodinger equation work. As a result, this assumption id/dt exp(iA)=E exp(iA) needs to be presented together with the two fluid equations, otherwise one does not derive the Schrodinger equation. In fact, one may use ((1a)) and ((1b)), the two flux/flow equations we propose for a free particle and create a momentum distribution P(x/p) = a(p) exp(ipx)/W(x) where W(x)=wavefunction = Sum over p a(p) exp(ipx). Using conservation of energy at each x point: [Sum over p a(p)pp/2m exp(ipx)] / W + V(x) = E leads to the Schrodinger equation without the use of fluid equations.

Francisco Chinesta, Emmanuelle Abisset‐Chavanne