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Journal of Mathematical Physics

Quantum dynamics of Hamiltonians perturbed by pulses

Mark J. Christensen, John Piepenbrink

Published Nov 1, 1979• 5 citations
Publisher

Abstract

There has been a recent controversy between Gzyl [J. Math Phys. 18, 1327 (1977)] and Blume [J. Math. Phys. 19, 2004 (1978)] concerning the correct way to integrate the equation i∂ψ/∂t=H0ψ+δ (t) Vψ where δ is the Dirac delta function. In this paper we: (1) Suggest some physical reasons for rejecting Gzyl’s procedure and (2) Extend the limiting procedure suggested the Blume to unbounded Hamiltonians.

DOI
10.1063/1.523996

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An error in a recent paper on this subject by Gzyl is corrected. His results are then identical to those obtained earlier by Clauser and Blume.

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On Blume’s integration of Schrödinger’s equation for a quantum system subject to random pulses

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According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation: i ∂ψ ∂t = Hψ , (6.1) with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy E = p2 2m , (6.2) using the substitution E → i ∂ ∂t , p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation: E2 = p2 +m2 . (6.4) (+m2)ψ = 0 , (6.5) Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained ∂ρ ∂t = ∇j , (6.6) with ρ = ψ∗ ∂ψ ∂t − ψ∂ψ ∂t (6.7) and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8) However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result, ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs E = ± √ p2 +m2 , Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.

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Derivation of energy expressions in the Rayleigh–Schrödinger perturbation theory

Kalbe Razi Naqvi

It is shown that repeated differentiation, with respect to the perturbation parameter λ, of the equation 〈ψm (λ)‖H0+λV‖ψn(λ)〉 =En(λ)δmn, and methodical application of the closure relation ∑‖ψn〉〈ψn‖=1, provide a painless technique for deriving the energy corrections in the Rayleigh–Schrödinger perturbation theory. The procedure is compared with one based on solving the secular equation D(E)=0, where D is the determinant whose elements are 〈ψi‖H0+V‖ψj〉−Eδij.

American Journal of Physics•Mar 1, 1987•1 citation
There has been a recent controversy between Gzyl [J. Math Phys. 18, 1327 (1977)] and Blume [J. Math. Phys. 19, 2004 (1978)] concerning the correct way to integrate the equation i∂ψ/∂t=H0ψ+δ (t) Vψ where δ is the Dirac delta function. In this paper we: (1) Suggest some physical reasons for rejecting Gzyl’s procedure and (2) Extend the limiting procedure suggested the Blume to unbounded Hamiltonians.
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.
A symmetrical treatment of the incoming and outgoing channels can be done in a basically very simple way by formulating the Schrödinger equation (1.1) in the form of a projection equation (see, e.g., ref. [WI 72]) (2.1) ⟨δψ∣H+ℏi∂∂t∣ψ⟩=0 \langle \delta \psi \left| {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right|\psi \rangle = 0 ⟨δψ​H+iℏ​∂t∂​​ψ⟩=0 As usual, the Dirac bracket denotes the integration over spatial coordinates and the summation over spin and isospin coordinates. If δΨ represents a completely arbitrary variation of Ψ at a given instant t in the complete Hilbert space, then eq. (2.1) implies that the vector (H+ℏi∂∂t)∣ψ⟩ \left( {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right)\left| {\psi \rangle } \right. (H+iℏ​∂t∂​)∣ψ⟩ must be orthogonal to any arbitrary vector in this space. This evidently will be the case only if Ψ obeys eq. (1.1). Therefore, eq. (2.1) is merely another formulation of the time-dependent Schrödinger equation. However, as we shall see later, eq. (2.1) does allow us to treat the incoming and outgoing channels in a symmetrical way.
According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation: i ∂ψ ∂t = Hψ , (6.1) with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy E = p2 2m , (6.2) using the substitution E → i ∂ ∂t , p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation: E2 = p2 +m2 . (6.4) (+m2)ψ = 0 , (6.5) Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained ∂ρ ∂t = ∇j , (6.6) with ρ = ψ∗ ∂ψ ∂t − ψ∂ψ ∂t (6.7) and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8) However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result, ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs E = ± √ p2 +m2 , Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.
It is shown that repeated differentiation, with respect to the perturbation parameter λ, of the equation 〈ψm (λ)‖H0+λV‖ψn(λ)〉 =En(λ)δmn, and methodical application of the closure relation ∑‖ψn〉〈ψn‖=1, provide a painless technique for deriving the energy corrections in the Rayleigh–Schrödinger perturbation theory. The procedure is compared with one based on solving the secular equation D(E)=0, where D is the determinant whose elements are 〈ψi‖H0+V‖ψj〉−Eδij.