Abstract
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 Schr\"odinger 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.
Highlights
Let us consider a one-dimensional quantum particle in the external finite step electric potential, i.e., φ(x) = V0 Θ(x), (1)where x ∈ R, V0 = const, and Θ(x) is the Heaviside step function (Θ(x > 0) = 1 and Θ(x < 0) = 0)
Let us suppose that the wave function associated with the particle, Ψ = Ψ(x, t), can be normalizable, that is, Ψ(x → ±∞, t) = 0
Only in the Klein-FockGordon theory this quantity cannot be calculated from the corresponding formula of the mean value
Summary
Let us write the one-dimensional Schrodinger, Klein-Fock-Gordon, and Dirac wave equations, in Hamiltonian form: i. The operator hKFG acts on two-component column vectors of the form Ψ = [φ χ]T (the symbol T represents the transpose of a matrix) In this case, the scalar product must be defined as Ψ, Φ KFG = R dx Ψ†τ3Φ (the symbol † denotes the Hermitian conjugate, or the adjoint, of a matrix and an operator) [1,2,3,4]. Which is obtained immediately from the formula to calculate the average value of the operator fin the Schrodinger case, namely, f S =. From the formula to calculate the average value of the operator fin the Dirac case, the result in Eq (10) can be obtained immediately, namely,.
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