The temporal energy spectrum Es,a(t) and the uncertainty product Delta psa(t) Delta x,s,a(t) are derived from an analytical solution psi (x,t) of the initial-boundary-value problem for the Schrodinger equation of a particle contained between moving potential walls at x=+or-s(t), which are set in motion according to an arbitrary (non-relativistic) translation s=s(t) at time t=0. Both symmetrical (s) and antisymmetrical (a) particle states are considered as initial conditions in the region -s(0)<or=x<or=+s(0). The physical implications of the compression and expansion of the probability density by the moving walls on the wave mechanics of the particle are discussed. The results are understandable within the statistical interpretation of quantum theory.
The temporal energy spectrum Es,a(t) and the uncertainty product Delta psa(t) Delta x,s,a(t) are derived from an analytical solution psi (x,t) of the initial-boundary-value problem for the Schrodinger equation of a particle contained between moving potential walls at x=+or-s(t), which are set in motion according to an arbitrary (non-relativistic) translation s=s(t) at time t=0. Both symmetrical (s) and antisymmetrical (a) particle states are considered as initial conditions in the region -s(0)<or=x<or=+s(0). The physical implications of the compression and expansion of the probability density by the moving walls on the wave mechanics of the particle are discussed. The results are understandable within the statistical interpretation of quantum theory.
Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
We prove the uncertainty relation σTσE≥ℏ/2 between the time T of detection of a quantum particle on the surface ∂Ω of a region Ω⊂R3 containing the particle's initial wave function, using the "absorbing boundary rule" for detection time, and the energy E of the initial wave function. Here, σ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since T is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schrödinger. We also prove that if there is nonzero probability that the particle never reaches ∂Ω (in which case we write T=∞), and if σT denotes the standard deviation conditional on the event T<∞, then σTσE≥(ℏ/2)Prob(T<∞).
The origin and the physical interpretation of boundary conditions for a free particle confined to a half-line are analysed. The authors consider potentials V(x,a, . . .) (depending upon parameters a, . . . and x in (- infinity , + infinity )) for which solutions psi (x,a, . . ., t) of the corresponding Schrodinger equation reduce, in a suitable limit a from a 0 , . . ., to solutions psi (x,t) of the Schrodinger equation for a free particle confined to a half-line satisfying the boundary condition ( delta psi / delta x)(x=0, t)=a psi (x=0, t) with real alpha = alpha (a 0 , . . .).
Due to the space and time dependence of the wavefunction in the time-dependent Schrödinger equation, different boundary conditions are possible. The equation is usually solved as an `initial-value problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to `source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for `free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.
The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.
Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.