Quantum Transitions by Change of Parameters and in Noninertial Motion

Abstract

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 𝜑_𝑘, and energy eigenvalues 𝐸_𝑘 depend on a parameter denoted generically by 𝐑. This dependence is written explicitly in the eigenvalue equation𝐻(𝐑)𝜑_𝑘(𝐑)=𝐸_𝑘(𝐑)𝜑_𝑘(𝐑).(1)A time dependence 𝐑(𝑡) is assumed for the parameter 𝐑, and Schrodinger's equation is written as𝑖ℏ𝜕𝜓(𝑡)𝜕𝑡=𝐻(𝐑)𝜓(𝑡).(2)In the adiabatic limit 𝐑̇→0, the original eigenstate 𝜑_𝑛(𝐑) is preserved during the temporal evolution, and the solution of (2) reads𝜓_𝑛(𝑡)=exp−𝑖ℏ^𝑡0𝐸_𝑛𝐑(𝑡^)𝑑𝑡^⋅𝑒^{𝑖𝛾𝑛(𝑡)}𝜑_𝑛𝐑(𝑡),(3)where 𝛾_𝑛(𝑡) is given by𝛾̇_𝑛(𝑡)=𝑖𝜑_𝑛,𝜕𝜑^𝑛𝜕𝐑𝐑̇.(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 𝛾_𝑛 [1].

3. Transitions by Change of Parameters

This result implies that, in general, for nonvanishing 𝐑̇, the quantum system may exhibit transitions between its various states. Indeed, the general solution of (2) can be written as𝜓(𝑡)=_𝑘𝑎_𝑘(𝑡)exp−𝑖ℏ^𝑡0𝐸_𝑘𝐑(𝑡^)𝑑𝑡^𝜑^𝑘𝐑(𝑡),(5)where the coefficients 𝑎_𝑘(𝑡) obey the equation𝑎̇_𝑛=⋅𝑖ex_𝑘p𝑎_𝑘𝛾𝑖_𝑛𝑘(𝑡)𝐑̇ℏ^𝑡0𝐸^𝑛𝐑(𝑡^𝛾)−𝐸_𝑘𝐑(𝑡^)𝑑𝑡^,(6)_𝑛𝑘(𝑡)=𝑖𝜑_𝑛,𝜕𝜑^𝑘𝜕𝐑.(7)

This 𝛾_𝑛𝑘(𝑡) is a generalization of the Berry phase; the latter corresponds to𝛾_𝑛(𝑡)=^𝑡0𝑑𝐑(𝑡^)𝛾_𝑛𝑛(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 𝛾_𝑛𝑘(𝑡) are the matrix elements of the operator −𝐏/ℏ, 𝛾_𝑛𝑘=−𝐏_𝑛𝑘/ℏ, 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 𝐕=𝐑̇ is the velocity of the parameter 𝐑.

Equation (6) is solved in the first order of the perturbation theory, with the initial conditions 𝑎_𝑛(0)=1, 𝑎_𝑘(0)=0 for 𝑘≠𝑛. The transition amplitudes𝑎_𝑘𝑛(𝑡)=⋅𝑖ex^𝑡0p𝑑𝐑𝑖(𝑡^)𝛾_𝑘𝑛(𝑡^)ℏ^𝑡′₀𝐸^𝑘𝐑(𝑡^)−𝐸_𝑛𝐑(𝑡^)𝑑𝑡^(9)are obtained, where an additional label 𝑘 has been given to the coefficient 𝑎_𝑛 in order to indicate the transition from state 𝑛 to state 𝑘. At the same time,𝑎_𝑛𝑛(𝑡)=1+𝑖^𝑡0𝑑𝐑(𝑡^)𝛾_𝑛𝑛(𝑡^)=1+𝑖𝛾_𝑛(𝑡).(10)From (9) and (10), one can see that in the adiabatic limit 𝐑̇→0 the Berry phase 𝛾_𝑛=𝛾_𝑛(𝑇) is recovered in 𝑎_𝑛𝑛(𝑇)=𝑒^{𝑖𝛾𝑛(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 𝛾_𝑘𝑛 and energy eigenvalues in the exponential factor in (9) may be neglected. The transition amplitudes can then be written as𝑎_𝑘𝑛(𝑡)=−𝑖ℏ^𝑡0𝑑𝑡^⋅𝐕(𝑡^)𝐏_𝑘𝑛exp𝑖𝜔_𝑘𝑛𝑡^,(11)where 𝜔_𝑘𝑛(𝑡)=(𝐸_𝑘−𝐸_𝑛)/ℏ.

For a uniform change of parameters, that is, for 𝐕 = const, the transition amplitudes are vanishing (𝑎_𝑘𝑛(𝑡)=0, 𝑘≠𝑛). The diagonal amplitude 𝑎_𝑛𝑛(𝑡)=1−(𝑖/ℏ)𝐕𝐏_𝑛𝑛𝑡≃exp(−𝑖𝐕𝐏_𝑛𝑛𝑡/ℏ) given by (10) contains the correction 𝐕𝐏_𝑛𝑛 to the energy of the state 𝜑_𝑛 in the first order of the perturbation theory. The gauge transformation 𝜓^𝑛=exp(−𝑖𝐕𝐏_𝑛𝑛𝑡/ℏ)𝜓_𝑛 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(𝑡−𝑡₀). The transition amplitudes given by (11) become𝑎_𝑘𝑛(𝑡)=−𝐕𝐏^𝑘𝑛𝑒^{𝑖𝜔𝑘𝑛𝑡}𝐸_𝑘−𝐸_𝑛+𝐕𝐏^𝑘𝑛𝐸_𝑘−𝐸_𝑛𝑒^{𝑖(𝐸𝑘−𝐸𝑛)𝑡0/ℏ}.(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 𝑤_𝑘𝑛=[𝐕𝐏_𝑘𝑛/(𝐸_𝑘−𝐸_𝑛)]².

If the velocity is periodic in time with frequency 𝜔, 𝐕(𝑡)=𝐕𝑒^𝑖𝜔𝑡+c.c., the transition probability per unit time is given by 𝑤_𝑘𝑛=(2𝜋/ℏ)(𝐕𝐏_𝑘𝑛)²𝛿(𝐸_𝑘−𝐸_𝑛±ℏ𝜔) 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 ℏ𝜔×integer, the quantum transitions described above may induce an increase 𝛿𝑇∼ℏ𝜔 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 𝛿𝑇∼ℏ𝑣/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 𝜑_𝑛(𝑥)=√2/𝑎sin(𝜋𝑛𝑥/𝑎) and the energy eigenvalues are given by 𝐸_𝑛=𝜋²ℏ²𝑛²/2𝑚𝑎², 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/𝑎₀≪1. Making use of (11), we get the transition probabilities 𝑤_𝑘𝑛=2𝜋ℏ[𝜀𝜔𝑘𝑛/𝑎₀(𝑘²−𝑛²)]²𝛿(𝐸_𝑘−𝐸_𝑛±ℏ𝜔) per unit time in the limit of the infinite time. The diagonal matrix element 𝛾_𝑛𝑛 is vanishing in this case, 𝛾_𝑛𝑛=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 𝐸_𝑛=−𝑔𝜇𝐵𝑛, where 𝑛=−𝑆,…,𝑆. In order to calculate the matrix elements entering (11), it is convenient to use the identity (𝐸_𝑛−𝐸_𝑘)(𝜑_𝑘,𝜕𝜑_𝑛/𝜕𝐑)=(𝜑_𝑘,(𝜕𝐻/𝜕𝐑)𝜑_𝑛) for 𝑘≠𝑛. We write then 𝐁𝐒=𝐵(𝑆_𝑥sin𝜃cos𝜙+𝑆_𝑦sin𝜃sin𝜙+𝑆_𝑧cos𝜃) 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 𝜔≪𝑔𝜇𝐵/ℏ. Making use of (9), we get transition probabilities 𝑤_𝑘𝑛=(𝜋ℏ𝜔²/8) in the limit of the infinite time. Since [𝑆(𝑆+1)−𝑛(𝑛±1)]𝛿_𝑘,𝑛±1𝛿(𝐸_𝑛−𝐸_𝑘±ℏ𝜔), these transition probabilities are vanishing, in fact, as we get by using (11). We may also set 𝜔≪𝑔𝜇𝐵/ℏ = 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)𝐸_el is a characteristic electronic term of the molecule and 𝐸_el 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 𝑖ℏ𝜕𝜓(𝑡^,𝐫^)𝜕𝑡^=𝐻(𝐫^)𝜓(𝑡^,𝐫^)+𝑖ℏ𝐕𝜕𝜓(𝑡^,𝑟^)𝜕𝐫^,(13). The last term in the rhs of (13) can be viewed as an interaction 𝐕=𝐑̇, where 𝐻₁=−𝐕𝐩 is the momentum associated to the coordinate 𝑝=−𝑖ℏ𝜕/𝜕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 𝐩_𝑘𝑛=0. Similarly, for an ensemble of (in general interacting) particles, the momentum 𝑘≠𝑛 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 𝑎_𝑛𝑛(𝑡) of the exp[𝑖∫^𝑡0𝑑𝑡₁𝐕(𝑡₁)𝐩_𝑛𝑛]-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/ℏ)(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/ℏ)𝜓^ into 𝑖ℏ𝜕𝜓/𝜕𝑡=𝐻𝜓. Making use of 𝑖ℏ𝜕𝜓^/𝜕𝑡=𝐻𝜓−𝐕𝐩𝜓^+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 (𝜑_𝑘,(𝜕𝐻/𝜕𝐫)𝜑_𝑛)=(𝐸_𝑛−𝐸_𝑘)(𝜑_𝑘,𝜕𝜑_𝑛/𝜕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 𝐩_𝑘𝑛, 𝑟_𝑖=𝛼_𝑖𝑗(𝑡^)𝑟_𝑗, be a change of coordinates (𝑡=𝑡^), where 𝑖,𝑗=1,2,3 is a rotation matrix of angle 𝛼_𝑖𝑗 and angular velocity 𝜙 about some axis, such that 𝜙̇=𝛀, 𝑟^𝑖=𝛼_𝑗𝑖(𝑡)𝑟_𝑗. Making use of 𝛼_𝑗𝑖𝛼_𝑗𝑘=𝛿_𝑖𝑘, where 𝛼_𝑙𝑖𝛼̇_𝑙𝑗=𝜀_𝑖𝑗𝑘Ω_𝑘 is the totally antisymmetric unit tensor, we get easily that an interaction 𝜀_𝑖𝑗𝑘 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 𝑎_𝑛𝑛 = const, the gauge transformation 𝛀, where 𝜓^(𝑡,𝐫^)=exp[−(𝑖/ℏ)(𝑚𝜌²Ω²/2−𝑚𝜌²𝛀𝜙/ℏ)]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 [3–8].