# Quantum Transitions by Change of Parameters

Authors: M. Apostol
Journal: Research Letters in Physics
 (2008-02-19)

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## 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.


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## Body

## 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 hamiltonianH, its (orthogonal) eigenfunctions φk, and energy eigenvalues Ek depend on a
parameter denoted generically by R. This dependence is written explicitly in the
eigenvalue equation(1)H(R)φk(R)=Ek(R)φk(R).A time dependence R(t) is assumed for
the parameter R, and Schrodinger's equation is written
as(2)iℏ∂ψ(t)∂t=H(R)ψ(t).In the adiabatic limit R˙→0, the original eigenstate φn(R) is preserved
during the temporal evolution, and the solution of (2) reads(3)ψn(t)=exp⁡[−iℏ∫0tEn(R(t′))dt′]⋅eiγn(t)φn(R(t)),where γn(t) is given
by(4)γ˙n(t)=i(φn,∂φn∂R)R˙⁢.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 nonvanishingR˙, the quantum system may exhibit transitions between
its various states. Indeed, the general solution of (2) can be written
as(5)ψ(t)=∑kak(t)exp⁡[−iℏ∫0tEk(R(t′))dt′]φk(R(t)),where the coefficients ak(t) obey the
equation(6)a˙n=i∑kakγnk(t)R˙⋅exp⁡[iℏ∫0t[En(R(t′))−Ek(R(t′))]dt′],(7)γnk(t)=i(φn,∂φk∂R).Thisγnk(t) is a
generalization of the Berry phase; the latter corresponds to(8)γn(t)=∫0tdR(t′)γnn(t′),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/ℏ, γnk=−Pnk/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=R˙ 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 k≠n. The transition amplitudes(9)akn(t)=i∫0tdR(t′)γkn(t′)⋅exp⁡[iℏ∫0t′[Ek(R(t′′))−En(R(t′′))]dt′′]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,(10)ann(t)=1+i∫0tdR(t′)γnn(t′)=1+iγn(t).From (9) and (10), one can see
that in the adiabatic limit R˙→0 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, theR-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 as(11)akn(t)=−iℏ∫0tdt′⋅V(t′)Pknexp⁡(iωknt′),where ωkn(t)=(Ek−En)/ℏ.For a uniform change of parameters, that is, forV = const, the transition amplitudes are vanishing (akn(t)=0, k≠n). The diagonal amplitude ann(t)=1−(i/ℏ)VPnnt≃exp⁡(−iVPnnt/ℏ) 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/ℏ)ψn leaves
Schrodinger's equation unchanged.Let velocityV 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)
become(12)akn(t)=−VPkneiωkntEk−En+[VPknEk−En]ei(Ek−En)t0/ℏ.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/(Ek−En)]2.If the velocity is periodic in time with frequencyω, V(t)=Veiωt+c.c., the transition probability per unit time is given by wkn=(2π/ℏ)(VPkn)2δ(Ek−En±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 massm 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=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+ε⁢cos⁡ωt, where ε/a0≪1. Making use of (11), we get the transition
probabilities wkn=2πℏ[εωkn/a0(k2−n2)]2δ(Ek−En±ℏω) 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 (En−Ek)(φk,∂φn/∂R)=(φk,(∂H/∂R)φn) for k≠n. We write then BS=B(Sxsin⁡θcos⁡ϕ+Sysin⁡θsin⁡ϕ+Sz⁢cos⁡θ) 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 wkn=(πℏω2/8)[S(S+1)−n(n±1)]δk,n±1δ(En−Ek±ℏω) in the limit of
the infinite time. Since ω≪gμB/ℏ, these transition probabilities are vanishing, in
fact, as we get by using (11). We may also set θ = const and let ϕ=ωt describe a
conical circuit of semiangle θ. The results are similar, the amplitudes containing
now the factor sin⁡θ.Another example is provided by the electronic terms
of the molecules, which depend parametrically on the nuclear coordinatesR. The interaction H1=VP can easily be
estimated as H1⁢∼(m/M)Eel, where Eel is a
characteristic electronic term of the molecule and m/M is the ratio of
the electron mass 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. Letr=r′+R(t′), t=t′, be a
translation, where r, 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) becomes(13)iℏ∂ψ(t′,r′)∂t′=H(r′)ψ(t′,r′)+iℏV∂ψ(t′,r′)∂r′,where V=R˙. The last term in the rhs of (13) can be
viewed as an interaction H1=−Vp, where p=−iℏ∂/∂r′ is the momentum
associated to the coordinate r′. The transition amplitudes are given by (11), where P is replaced by p.For a free particle, the transition amplitudes are
vanishing sincepkn=0 for k≠n. Similarly, for an ensemble of (in general
interacting) particles, the momentum p 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 ann(t) corresponds to
a gauge transformation exp⁡[i∫0tdt1V(t1)pnn] of the n-state, which,
in general, has no determined energy (it is not a
stationary state in general). For constant velocity V = const, the phase of this gauge transformation is the
first-order correction to the energy of the n-state. It is
easy to check that the gauge transformation ψ′(t,r′)=exp⁡[−(i/ℏ)(MV2t/2+MVr)]ψ(t,r), where M is the mass of
the ensemble, preserves Schrodinger's equation in accordance with Galileo's
principle of relativity. The unitary transformation ψ=exp⁡(−iRp/ℏ)ψ′ takes the
Schrodinger equation iℏ∂ψ/∂t=Hψ into iℏ∂ψ′/∂t=Hψ−Vpψ′+R(∂H/∂r)ψ′+⋯. Making use of (φk,(∂H/∂r)φn)=(En−Ek)(φk,∂φn/∂r), 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 elementspkn 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. Letri=αij(t′)rj, t=t′, be a change of
coordinates (i,j=1,2,3), where αij is a rotation
matrix of angle ϕ and angular
velocity ϕ˙=Ω about some
axis, such that ri′=αji(t)rj, αjiαjk=δik. Making use of αliα˙lj=εijkΩk, where εijk is the totally
antisymmetric unit tensor, we get easily that an interaction H1=Ωl appears in
hamiltonian, similar with the interaction given by (13), where l is the total
(orbital) angular momentum. The parameter R introduced in
the previous sections is the rotation angle ϕ 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 ann may generate a
gauge transformation, which reflects, in general, the nonstationarity of the
rotating state. For uniform rotations, that is, for Ω = const, the gauge transformation ψ′(t,r′)=exp⁡[−(i/ℏ)(mρ2Ω2/2−mρ2Ωϕ/ℏ)]ψ(t,r), where ρ 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, mρ2 denotes the
total momentum of inertia I and the first
term in the phase is the kinetic energy l2/2I.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].



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