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arXiv:1010.4244 · American Journal of Physics

Less than perfect quantum wavefunctions in momentum-space: How ϕ(p) senses disturbances in the force

M. Belloni, R. W. Robinett

Published Dec 13, 2010• 5 citations
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Abstract

We develop a systematic approach to determine the large |p| behavior of the momentum-space wave function, ϕ(p), of a one-dimensional quantum system for which the position-space wave function, ψ(x), has a discontinuous derivative at any order. We find that if the kth derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in ψ(k+2)(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function proportional to 1/pk+3. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.

DOI
10.1119/1.3492723

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We develop a systematic approach to determine the large |p| behavior of the momentum-space wave function, ϕ(p), of a one-dimensional quantum system for which the position-space wave function, ψ(x), has a discontinuous derivative at any order. We find that if the kth derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in ψ(k+2)(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function proportional to 1/pk+3. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.
The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.
In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.
Abstract: Quantum theory does not require the existence of discontinuities: neither in time (quantum jumps), nor in space (particles), nor in spacetime (quantum events). These apparent discontinuities are readily described objectively by the continuous process of decoherence occurring locally on a very short time scale according to the Schrdinger equation for interacting systems, while the observerÕs Ôincrease of informationÕ is appropriately represented by the resulting dynamical decoupling of the corresponding components of the global wave function. Physics Letters A172, 189 (1993) 2As far as is known, all properties of closed quantum systems are perfectly described by means of wave functions in configuration space (in general, wave functionals of certain fields) dynamically evolving smoothly according to the time-dependent Schrdinger equation. However, the condition of being closed (or shielded against interactions with the environment) can easily be estimated to be quite exceptional. It characterizes very special (usually atomic) systems from which the laws of quantum mechanics were derived. When the shielding ceases, most notably during measurements, discontinuous
The momentum representation of the Morse potential is presented analytically by hypergeometric function. The properties with respect to the momentum p and potential parameter β are studied. Note that |Ψ(p)| is a nodeless function and the mutual orthogonality of functions is ensured by the phase functions arg[Ψ(p)]. It is interesting to see that the |Ψ(p)| is symmetric with respect to the axis p = 0 and the number of wave crest of |Ψ(p)| is equal to n + 1. We also study the variation of |Ψ(p)| with respect to β. The amplitude of |Ψ(p)| first increases with the quantum number n and then deceases. Finally, we notice that the discontinuity in phase occurs at some points of the momentum p and the position and momentum probability densities are symmetric with respect to their arguments.
In classical physics one needs to know the velocity or momentum of a particle in order to determine its evolution. In quantum mechanics the wave function ψ already contains information of possible momenta. What one has to do is to apply harmonic analysis to the function ψ. Sometimes it is helpful to describe quantum states in terms of wave function of momentum. This requires decomposition of ψ in terms of various momentum states and constructing a new function C(k). The value of C(k) of each k gives the strength of the contribution to the k−momentum state of ψ. The space of k is called momentum space.