Research Letter | Open Access
M. Apostol, "Quantum Transitions by Change of Parameters and in Noninertial Motion", Physics Research International, vol. 2008, Article ID 570167, 4 pages, 2008. https://doi.org/10.1155/2008/570167
Quantum Transitions by Change of Parameters and in Noninertial Motion
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.
As it is well known , 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 , its (orthogonal) eigenfunctions , and energy eigenvalues depend on a parameter denoted generically by . This dependence is written explicitly in the eigenvalue equationA time dependence is assumed for the parameter , and Schrodinger's equation is written asIn the adiabatic limit , the original eigenstate is preserved during the temporal evolution, and the solution of (2) readswhere is given byThe bracket in (4) indicates a scalar (inner) product. For a circuit described by the parameter , this is Berry's geometric phase .
3. Transitions by Change of Parameters
This result implies that, in general, for nonvanishing , the quantum system may exhibit transitions between its various states. Indeed, the general solution of (2) can be written aswhere the coefficients obey the equation
This is a generalization of the Berry phase; the latter corresponds towhere the integration is performed along the path described by the parameter in its motion from to . The are the matrix elements of the operator , , where may be viewed formally as the momentum associated with the parameter . Then (6) gives the transition amplitudes caused by a perturbation , where is the velocity of the parameter .
Equation (6) is solved in the first order of the perturbation theory, with the initial conditions , for . The transition amplitudesare obtained, where an additional label has been given to the coefficient in order to indicate the transition from state to state . At the same time,From (9) and (10), one can see that in the adiabatic limit the Berry phase is recovered in for a circuit , where is the period during which the parameter describes the circuit .
In the first order of the perturbation theory, the -dependence of the matrix elements and energy eigenvalues in the exponential factor in (9) may be neglected. The transition amplitudes can then be written aswhere .
For a uniform change of parameters, that is, for = const, the transition amplitudes are vanishing (, ). The diagonal amplitude given by (10) contains the correction to the energy of the state in the first order of the perturbation theory. The gauge transformation leaves Schrodinger's equation unchanged.
Let velocity have a sudden variation from for to = const for , such that . The transition amplitudes given by (11) becomeThe first term in the rhs of this equation corresponds to the change in the wave function under the action of the constant perturbation for . The transition amplitude is given by the second term in the rhs of (12), so the transition probability is .
If the velocity is periodic in time with frequency , , the transition probability per unit time is given by 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 in the variation spectrum of the parameter 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 , the quantum transitions described above may induce an increase in temperature. For a periodic change of parameters, the frequency is proportional to the ratio of the average acceleration to the average velocity , so the increase in temperature is . It is similar with the Unruh temperature .
4. Some Simple Applications
Let a particle of mass move in an infinite square potential well in one dimension. The eigenfunctions are and the energy eigenvalues are given by , where . The width of this potential well is taken as parameter . The wall of the potential well, placed at distance from the origin, is subjected to an oscillatory motion of frequency as described by , where . Making use of (11), we get the transition probabilities per unit time in the limit of the infinite time. The diagonal matrix element is vanishing in this case, .
Following Berry , we consider a spin placed in a magnetic field . The hamiltonian reads , where is the gyromagnetic factor and is the Bohr magneton. The energy eigenvalues are given by , where . In order to calculate the matrix elements entering (11), it is convenient to use the identity for . We write then and take the angles and as parameters . First, we set and let describe a circuit according to , where . Making use of (9), we get transition probabilities in the limit of the infinite time. Since , these transition probabilities are vanishing, in fact, as we get by using (11). We may also set = const and let describe a conical circuit of semiangle . The results are similar, the amplitudes containing now the factor .
Another example is provided by the electronic terms of the molecules, which depend parametrically on the nuclear coordinates . The interaction can easily be estimated as , where is a characteristic electronic term of the molecule and is the ratio of the electron mass to the nuclear mass . 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 , , be a translation, where , denote the position of the system and is the displacement vector. The hamiltonian, its eigenfunctions, and energy eigenvalues do not depend on the displacement , so it can be taken as the general parameter in the previous sections. Schrodinger's (2) becomeswhere . The last term in the rhs of (13) can be viewed as an interaction , where is the momentum associated to the coordinate . The transition amplitudes are given by (11), where is replaced by .
For a free particle, the transition amplitudes are vanishing since for . Similarly, for an ensemble of (in general interacting) particles, the momentum 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 corresponds to a gauge transformation of the -state, which, in general, has no determined energy (it is not a stationary state in general). For constant velocity = const, the phase of this gauge transformation is the first-order correction to the energy of the -state. It is easy to check that the gauge transformation , where is the mass of the ensemble, preserves Schrodinger's equation in accordance with Galileo's principle of relativity. The unitary transformation takes the Schrodinger equation into . Making use of 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 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 , , be a change of coordinates (), where is a rotation matrix of angle and angular velocity about some axis, such that , . Making use of , where is the totally antisymmetric unit tensor, we get easily that an interaction appears in hamiltonian, similar with the interaction given by (13), where is the total (orbital) angular momentum. The parameter introduced in the previous sections is the rotation angle 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 may generate a gauge transformation, which reflects, in general, the nonstationarity of the rotating state. For uniform rotations, that is, for = const, the gauge transformation , where 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, denotes the total momentum of inertia and the first term in the phase is the kinetic energy .
For particles in an external field, the angular momentum may have nonvanishing offdiagonal matrix elements, so nonuniform (accelerated) rotations may induce quantum transitions.
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.
- M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society of London. Series A, vol. 392, no. 1802, pp. 45–57, 1984.
- W. G. Unruh, “Notes on black-hole evaporation,” Physical Review D, vol. 14, no. 4, pp. 870–892, 1976.
- Y. Q. Cai and G. Papini, “Neutrino helicity flip from gravity-spin coupling,” Physical Review Letters, vol. 66, no. 10, pp. 1259–1262, 1991.
- H. Casini and R. Montemayor, “Chirality transitions in gravitational fields,” Physical Review D, vol. 50, no. 12, pp. 7425–7429, 1994.
- D. Singh, “Neutrino helicity and chirality transitions in Schwarzschild space-time,” Physical Review D, vol. 71, no. 10, Article ID 105003, p. 12 pages, 2005.
- D. Singh and N. Mobed, “The implications of noninertial motion on covariant quantum spin,” Classical and Quantum Gravity, vol. 24, no. 10, pp. 2453–2463, 2006.
- D. M. Greenberger and A. W. Overhauser, “Coherence effects in neutron diffraction and gravity experiments,” Reviews of Modern Physics, vol. 51, no. 1, pp. 43–78, 1979.
- D. M. Greenberger, “Inadequacy of the usual Galilean transformation in quantum mechanics,” Physical Review Letters, vol. 87, no. 10, Article ID 100405, p. 4 pages, 2001.
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.