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European Journal of Physics

On quantum mechanics on a half-line

V. Pažma, Peter Prešnajder

Published Jan 1, 1989• 8 citations
Publisher

Abstract

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 , . . .).

DOI
10.1088/0143-0807/10/1/007

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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 , . . .).
One-dimensional particle states are constructed according to orthogonality conditions, without requiring boundary conditions. Free particle states are constructed using Dirac's delta function orthogonality conditions. The states (doublets) depend on two quantum numbers: energy and parity. With the aid of projection operators the particles are confined to a constrained region, in a way similar to the action of an infinite well potential. From the resulting overcomplete basis only the mutually orthogonal states are selected. Four solutions are found, corresponding to different non-commuting Hamiltonians. Their energy eigenstates are labeled with the main quantum number n and parity "+" or "-". The energy eigenvalues are functions of n only. The four cases correspond to different boundary conditions: (I) the wave function vanishes on the boundary, (II) the derivative of the wavefunction vanishes on the boundary,(III) periodic (symmetric) boundary conditions, (IV) periodic (antisymmetric)boundary conditions . Among the four cases, only solution (III) forms a complete basis in the sense that any function in the constrained region, can be expanded with it. By extending the boundaries of the constrained region to infinity, only solution (III) converges uniformly to the free particle states. Orthogonality seems to be a more basic requirement than boundary conditions. By using projection operators, confinement of the particle to a definite region can be achieved in a conceptually simple and unambiguous way, and physical operators can be written so that they act only in the confined region.
The effect of a δδδ distribution potential on a quantum mechanical particle in a box
We study the effect of a δδδ distribution potential placed at x0≥0x_0\geq 0x0​≥0 and multiplied by a parameter ααα on a quantum mechanical particle in an infinite square well over the segment [− L2,L2]\left[-\,\frac{L}{2},\frac{L}{2}\right][−2L​,2L​]. We obtain the limit of the eigenfunctions of the time independent Schrödinger equation as α↗+∞α\nearrow+\inftyα↗+∞ and as α↘−∞α\searrow-\inftyα↘−∞. We see how each solution of the Schrödinger equation corresponding to α=0α=0α=0 changes as ααα runs through the real line. When x0x_0x0​ is a rational multiple of LLL, there exist solutions of the Schrödinger equation which vanish at x0x_0x0​ and are unaffected by the value of ααα. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking ∣α∣→∞|α|\to\infty∣α∣→∞. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is x0x_0x0​.
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.