Table of Contents
Research Letters in Physics
Volume 2008 (2008), Article ID 570167, 4 pages
Research Letter

Quantum Transitions by Change of Parameters and in Noninertial Motion

Department of Theoretical Physics, Institute of Atomic Physics, Magurele, 077125 Bucharest, Romania

Received 1 October 2007; Accepted 13 February 2008

Academic Editor: Jerry Draayer

Copyright © 2008 M. Apostol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


As it is well known, a quantum system depending on parameters exhibits the (geometric) Berry phase when parameters are varying in the adiabatic limit. A generalization of the Berry phase is given in the present paper for a nonadiabatic change of parameters, which leads to quantum transitions in the system. This generalization is applied to noninertial motions and it is shown that such motions may induce quantum transitions for a system in an external field governed by Schrodinger's equation.

1. Introduction

As it is well known [1], a quantum system subjected to a change of parameters exhibits the (geometric) Berry phase, providing the change proceeds in the adiabatic limit. A careful examination of the derivation of the Berry phase suggests that a nonadiabatic change in parameters may induce quantum transitions in the system. It is shown in the present paper that the Berry phase can be generalized in such a way as to describe quantum transitions for such nonadiabatic changes in parameters. Moreover, it is shown that the displacement vector in a (nonuniform) translation or the rotation angle in a (nonuniform) rotation may play the role of such nonadiabatic parameters, such that a noninertial motion may cause quantum transitions for a system placed in an external field.

2. Berry Phase

Let the hamiltonian 𝐻, its (orthogonal) eigenfunctions 𝜑𝑘, and energy eigenvalues 𝐸𝑘 depend on a parameter denoted generically by 𝐑. This dependence is written explicitly in the eigenvalue equation𝐻(𝐑)𝜑𝑘(𝐑)=𝐸𝑘(𝐑)𝜑𝑘(𝐑).(1)A time dependence 𝐑(𝑡) is assumed for the parameter 𝐑, and Schrodinger's equation is written as𝑖𝜕𝜓(𝑡)𝜕𝑡=𝐻(𝐑)𝜓(𝑡).(2)In the adiabatic limit ̇𝐑0, the original eigenstate 𝜑𝑛(𝐑) is preserved during the temporal evolution, and the solution of (2) reads𝜓𝑛𝑖(𝑡)=exp𝑡0𝐸𝑛𝐑(𝑡)𝑑𝑡𝑒𝑖𝛾𝑛(𝑡)𝜑𝑛,𝐑(𝑡)(3)where 𝛾𝑛(𝑡) is given bẏ𝛾𝑛𝜑(𝑡)=𝑖𝑛,𝜕𝜑𝑛̇𝜕𝐑𝐑.(4)The bracket in (4) indicates a scalar (inner) product. For a circuit 𝐶 described by the parameter 𝐑, this is Berry's geometric phase 𝛾𝑛 [1].

3. Transitions by Change of Parameters

This result implies that, in general, for nonvanishing ̇𝐑, the quantum system may exhibit transitions between its various states. Indeed, the general solution of (2) can be written as𝜓(𝑡)=𝑘𝑎𝑘𝑖(𝑡)exp𝑡0𝐸𝑘𝐑(𝑡)𝑑𝑡𝜑𝑘𝐑(𝑡),(5)where the coefficients 𝑎𝑘(𝑡) obey the equatioṅ𝑎𝑛=𝑖𝑘𝑎𝑘𝛾𝑛𝑘̇𝐑𝑖(𝑡)exp𝑡0𝐸𝑛𝐑(𝑡)𝐸𝑘𝐑(𝑡)𝑑𝑡,𝛾(6)𝑛𝑘𝜑(𝑡)=𝑖𝑛,𝜕𝜑𝑘.𝜕𝐑(7)

This 𝛾𝑛𝑘(𝑡) is a generalization of the Berry phase; the latter corresponds to𝛾𝑛(𝑡)=𝑡0𝑑𝐑(𝑡)𝛾𝑛𝑛(𝑡),(8)where the integration is performed along the path described by the parameter 𝐑 in its motion from 𝐑(𝑡=0) to 𝐑(𝑡). The 𝛾𝑛𝑘(𝑡) are the matrix elements of the operator 𝐏/, 𝛾𝑛𝑘=𝐏𝑛𝑘/, where 𝐏 may be viewed formally as the momentum associated with the parameter 𝐑. Then (6) gives the transition amplitudes caused by a perturbation 𝐻1=𝐕𝐏, where ̇𝐑𝐕= is the velocity of the parameter 𝐑.

Equation (6) is solved in the first order of the perturbation theory, with the initial conditions 𝑎𝑛(0)=1, 𝑎𝑘(0)=0 for 𝑘𝑛. The transition amplitudes𝑎𝑘𝑛(𝑡)=𝑖𝑡0𝑑𝐑(𝑡)𝛾𝑘𝑛(𝑡)𝑖exp𝑡0𝐸𝑘𝐑(𝑡)𝐸𝑛𝐑(𝑡)𝑑𝑡(9)are obtained, where an additional label 𝑘 has been given to the coefficient 𝑎𝑛 in order to indicate the transition from state 𝑛 to state 𝑘. At the same time,𝑎𝑛𝑛(𝑡)=1+𝑖𝑡0𝑑𝐑(𝑡)𝛾𝑛𝑛(𝑡)=1+𝑖𝛾𝑛(𝑡).(10)From (9) and (10), one can see that in the adiabatic limit ̇𝐑0 the Berry phase 𝛾𝑛=𝛾𝑛(𝑇) is recovered in 𝑎𝑛𝑛(𝑇)=𝑒𝑖𝛾𝑛(𝑇) for a circuit 𝐶, where 𝑇 is the period during which the parameter 𝐑 describes the circuit 𝐶.

In the first order of the perturbation theory, the 𝐑-dependence of the matrix elements 𝛾𝑘𝑛 and energy eigenvalues in the exponential factor in (9) may be neglected. The transition amplitudes can then be written as𝑎𝑘𝑛𝑖(𝑡)=𝑡0𝑑𝑡𝐕(𝑡)𝐏𝑘𝑛exp𝑖𝜔𝑘𝑛𝑡,(11)where 𝜔𝑘𝑛(𝑡)=(𝐸𝑘𝐸𝑛)/.

For a uniform change of parameters, that is, for 𝐕 = const, the transition amplitudes are vanishing (𝑎𝑘𝑛(𝑡)=0, 𝑘𝑛). The diagonal amplitude 𝑎𝑛𝑛(𝑡)=1(𝑖/)𝐕𝐏𝑛𝑛𝑡exp(𝑖𝐕𝐏𝑛𝑛𝑡/) given by (10) contains the correction 𝐕𝐏𝑛𝑛 to the energy of the state 𝜑𝑛 in the first order of the perturbation theory. The gauge transformation 𝜓𝑛=exp(𝑖𝐕𝐏𝑛𝑛𝑡/)𝜓𝑛 leaves Schrodinger's equation unchanged.

Let velocity 𝐕 have a sudden variation from 𝐕=0 for 0<𝑡<𝑡0 to 𝐕 = const for 𝑡0<𝑡, such that 𝜕𝐕/𝜕𝑡=𝐕𝛿(𝑡𝑡0). The transition amplitudes given by (11) become𝑎𝑘𝑛(𝑡)=𝐕𝐏𝑘𝑛𝑒𝑖𝜔𝑘𝑛𝑡𝐸𝑘𝐸𝑛+𝐕𝐏𝑘𝑛𝐸𝑘𝐸𝑛𝑒𝑖(𝐸𝑘𝐸𝑛)𝑡0/.(12)The first term in the rhs of this equation corresponds to the change in the wave function under the action of the constant perturbation 𝐕𝐏 for 𝑡>𝑡0. The transition amplitude is given by the second term in the rhs of (12), so the transition probability is 𝑤𝑘𝑛=[𝐕𝐏𝑘𝑛/(𝐸𝑘𝐸𝑛)]2.

If the velocity is periodic in time with frequency 𝜔, 𝐕(𝑡)=𝐕𝑒𝑖𝜔𝑡+c.c., the transition probability per unit time is given by 𝑤𝑘𝑛=(2𝜋/)(𝐕𝐏𝑘𝑛)2𝛿(𝐸𝑘𝐸𝑛±𝜔) in the limit of the infinite time. The calculations are not restricted to the discrete spectrum, so there may appear transitions in the continuum. It is worth noting that frequencies 𝜔 in the variation spectrum of the parameter 𝐑 must be comparatively high of the order of the frequencies of the quantum system in order to have such quantum transitions. For a quantum-statistical system with a characteristic spectrum 𝜔×integer, the quantum transitions described above may induce an increase 𝛿𝑇𝜔 in temperature. For a periodic change of parameters, the frequency 𝜔 is proportional to the ratio of the average acceleration 𝑎 to the average velocity 𝑣, so the increase in temperature is 𝛿𝑇𝑣/𝑎. It is similar with the Unruh temperature [2].

4. Some Simple Applications

Let a particle of mass 𝑚 move in an infinite square potential well in one dimension. The eigenfunctions are 𝜑𝑛(𝑥)=2/𝑎sin(𝜋𝑛𝑥/𝑎) and the energy eigenvalues are given by 𝐸𝑛=𝜋22𝑛2/2𝑚𝑎2, where 𝑛=1,2,. The width 𝑎 of this potential well is taken as parameter 𝑅. The wall of the potential well, placed at distance 𝑎 from the origin, is subjected to an oscillatory motion of frequency 𝜔 as described by 𝑎=𝑎0+𝜀cos𝜔𝑡, where 𝜀/𝑎01. Making use of (11), we get the transition probabilities 𝑤𝑘𝑛=2𝜋[𝜀𝜔𝑘𝑛/𝑎0(𝑘2𝑛2)]2𝛿(𝐸𝑘𝐸𝑛±𝜔) per unit time in the limit of the infinite time. The diagonal matrix element 𝛾𝑛𝑛 is vanishing in this case, 𝛾𝑛𝑛=0.

Following Berry [1], we consider a spin 𝐒 placed in a magnetic field 𝐁. The hamiltonian reads 𝐻=𝑔𝜇𝐁𝐒, where 𝑔 is the gyromagnetic factor and 𝜇 is the Bohr magneton. The energy eigenvalues are given by 𝐸𝑛=𝑔𝜇𝐵𝑛, where 𝑛=𝑆,,𝑆. In order to calculate the matrix elements entering (11), it is convenient to use the identity (𝐸𝑛𝐸𝑘)(𝜑𝑘,𝜕𝜑𝑛/𝜕𝐑)=(𝜑𝑘,(𝜕𝐻/𝜕𝐑)𝜑𝑛) for 𝑘𝑛. We write then 𝐁𝐒=𝐵(𝑆𝑥sin𝜃cos𝜙+𝑆𝑦sin𝜃sin𝜙+𝑆𝑧cos𝜃) and take the angles 𝜃 and 𝜙 as parameters 𝑅. First, we set 𝜙=0 and let 𝜃 describe a circuit according to 𝜃=𝜔𝑡, where 𝜔𝑔𝜇𝐵/. Making use of (9), we get transition probabilities 𝑤𝑘𝑛=(𝜋𝜔2/8) in the limit of the infinite time. Since [𝑆(𝑆+1)𝑛(𝑛±1)]𝛿𝑘,𝑛±1𝛿(𝐸𝑛𝐸𝑘±𝜔), these transition probabilities are vanishing, in fact, as we get by using (11). We may also set 𝜔𝑔𝜇𝐵/ = const and let 𝜃 describe a conical circuit of semiangle 𝜙=𝜔𝑡. The results are similar, the amplitudes containing now the factor 𝜃.

Another example is provided by the electronic terms of the molecules, which depend parametrically on the nuclear coordinates sin𝜃. The interaction 𝐑 can easily be estimated as 𝐻1=𝐕𝐏, where 𝐻1(𝑚/𝑀)𝐸el is a characteristic electronic term of the molecule and 𝐸el is the ratio of the electron mass 𝑚/𝑀 to the nuclear mass 𝑚. It is of the same order of magnitude as the accuracy of the adiabatic decoupling of the electronic motion from the nuclear motion, so it gives a natural width of the electronic terms in molecules.

5. Noninertial Motion Translations

A similar analysis can be carried out for noninertial motion. Let 𝑀, 𝐫=𝐫+𝐑(𝑡), be a translation, where 𝑡=𝑡, 𝐫 denote the position of the system and 𝐫 is the displacement vector. The hamiltonian, its eigenfunctions, and energy eigenvalues do not depend on the displacement 𝐑, so it can be taken as the general parameter 𝐑 in the previous sections. Schrodinger's (2) becomes𝐑where 𝑖𝜕𝜓(𝑡,𝐫)𝜕𝑡=𝐻(𝐫)𝜓(𝑡,𝐫)+𝑖𝐕𝜕𝜓(𝑡,𝑟)𝜕𝐫,(13). The last term in the rhs of (13) can be viewed as an interaction ̇𝐑𝐕=, where 𝐻1=𝐕𝐩 is the momentum associated to the coordinate 𝑝=𝑖𝜕/𝜕𝐫. The transition amplitudes are given by (11), where 𝐫 is replaced by 𝐏.

For a free particle, the transition amplitudes are vanishing since 𝐩 for 𝐩𝑘𝑛=0. Similarly, for an ensemble of (in general interacting) particles, the momentum 𝑘𝑛 is the total momentum, that is, the momentum of the center of mass of the ensemble, so there are no transitions as expected. The coefficient 𝐩 corresponds to a gauge transformation 𝑎𝑛𝑛(𝑡) of the exp[𝑖𝑡0𝑑𝑡1𝐕(𝑡1)𝐩𝑛𝑛]-state, which, in general, has no determined energy (it is not a stationary state in general). For constant velocity 𝑛 = const, the phase of this gauge transformation is the first-order correction to the energy of the 𝐕-state. It is easy to check that the gauge transformation 𝑛, where 𝜓(𝑡,𝐫)=exp[(𝑖/)(𝑀𝑉2𝑡/2+𝑀𝐕𝐫)]𝜓(𝑡,𝐫) is the mass of the ensemble, preserves Schrodinger's equation in accordance with Galileo's principle of relativity. The unitary transformation 𝑀 takes the Schrodinger equation 𝜓=exp(𝑖𝐑𝐩/)𝜓 into 𝑖𝜕𝜓/𝜕𝑡=𝐻𝜓. Making use of 𝑖𝜕𝜓/𝜕𝑡=𝐻𝜓𝐕𝐩𝜓+𝐑(𝜕𝐻/𝜕𝐫)𝜓+ one can show by direct calculation that the additional interacting term in the hamiltonian has no relevance. Such a unitary transformation is different from the coordinate change.

The situation is different for particles in an external field. There, in general, the offdiagonal matrix elements (𝜑𝑘,(𝜕𝐻/𝜕𝐫)𝜑𝑛)=(𝐸𝑛𝐸𝑘)(𝜑𝑘,𝜕𝜑𝑛/𝜕𝐫), of the momentum of the particles are nonvanishing, and they may cause transitions. For instance, if one or more particles in an ensemble of interacting particles acquire a large mass, then they may be viewed as being at rest during the motion of the rest of particles. Their interaction with the rest of particles becomes now an external field for the latter, whose motion depends parametrically on the positions of the former. The coordinates of the heavy particles do not appear anymore in the momentum, so there may exist nonvanishing matrix elements of this momentum between states of the moving particles. It follows that noninertial motion may give rise to quantum transitions for particles in an external field.

6. Noninertial Motion Rotations

A similar result holds also for rotations. Let 𝐩𝑘𝑛, 𝑟𝑖=𝛼𝑖𝑗(𝑡)𝑟𝑗, be a change of coordinates (𝑡=𝑡), where 𝑖,𝑗=1,2,3 is a rotation matrix of angle 𝛼𝑖𝑗 and angular velocity 𝜙 about some axis, such that ̇𝜙=𝛀, 𝑟𝑖=𝛼𝑗𝑖(𝑡)𝑟𝑗. Making use of 𝛼𝑗𝑖𝛼𝑗𝑘=𝛿𝑖𝑘, where 𝛼𝑙𝑖̇𝛼𝑙𝑗=𝜀𝑖𝑗𝑘Ω𝑘 is the totally antisymmetric unit tensor, we get easily that an interaction 𝜀𝑖𝑗𝑘 appears in hamiltonian, similar with the interaction given by (13), where 𝐻1=𝛀𝐥 is the total (orbital) angular momentum. The parameter 𝐥 introduced in the previous sections is the rotation angle 𝐑 in this case. The discussion is similar with the one given above for translations. For a free particle, or an ensemble of interacting particles, the total angular momentum has no offdiagonal matrix elements. The coefficient 𝜙 may generate a gauge transformation, which reflects, in general, the nonstationarity of the rotating state. For uniform rotations, that is, for 𝑎𝑛𝑛 = const, the gauge transformation 𝛀, where 𝜓(𝑡,𝐫)=exp[(𝑖/)(𝑚𝜌2Ω2/2𝑚𝜌2𝛀𝜙/)]𝜓(𝑡,𝐫) is the distance of particles to the axis of rotation, leaves Schrodinger's equation unchanged, in accordance with its invariance under uniform rotations. In this gauge transformation, 𝜌 denotes the total momentum of inertia 𝑚𝜌2 and the first term in the phase is the kinetic energy 𝐼.

For particles in an external field, the angular momentum may have nonvanishing offdiagonal matrix elements, so nonuniform (accelerated) rotations may induce quantum transitions.

7. Conclusion

The main conclusion of the results described herein is that noninertial motion may cause quantum transitions for systems in external fields governed by Schrodinger's equation. It follows that an observer who is set in noninertial motion may record such quantum transitions. Similar transitions may be caused by changes of parameters associated with Berry's phase. The acceleration of the change of coordinates or of parameters must be fast enough in order to match the excitation spectrum of the quantum system and have such transitions.

Similar problems appear also in the field theory. For similarities with quantization in gravitational fields, we refer to [38].


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