Quantum beats of the rigid rotor
The dynamics is investigated of a free particle on a sphere (rigid rotor or rotator) that is initially in a coherent state. The instability of coherent states with respect to the free evolution leads to nontrivial time-development of averages of observables representing the position of a particle on a sphere that can be interpreted as quantum beats.
pacs:03.65.-w, 03.65.Sq, 42.50.-p
The rigid rotor model is a very important concept in quantum mechanics that is discussed in most textbooks. We only recall that it is basic for understanding rotational spectroscopy and collisions of molecules. As is well known a mathematical model for rigid rotor is a free particle constrained to a surface of a sphere 1 (). In this work we study the quantum dynamics of a free particle on a sphere in the case when the initial condition is a coherent state. The coherent states for the quantum mechanics on a sphere are unstable with respect to the free evolution, nevertheless, in opposition to the motion in a plane, the corresponding wave packets do not spread. Consequently, the dynamics is nontrivial. In particular, the evolution of the wave packets leads to beats in the expectation values of position observables for a particle on a sphere.
Ii Quantum mechanics on a sphere
We begin with a brief account of the quantum mechanics on a sphere. As shown in 2 () the most natural algebra for the study of the motion on a sphere is the algebra of the form
where is the angular momentum operator and is the position operator for a particle on a sphere. In fact, the algebra (1) has two Casimir operators given in a unitary irreducible representation by
In the following we restrict to the irreducible representations of (1) with i.e. the unit sphere and the “most classical” case of . Then the angular momentum basis spanned by the common eigenvectors of operators , , and is defined by
where is a nonnegative integer and . Besides of the standard action of the angular momentum operators on the basis vectors such that
we have the following relations satisfied by the position operators 2 ()
where . We finally write down the orthogonality relations satisfied by the vectors such that
and the completeness condition which in the case of takes the form
Iii Coherent states for the sphere
We now summarize the basic facts about the coherent states for a particle on a sphere. The coherent states for the quantum mechanics on a sphere were introduced by us very recently 2 (). These states were generalized by Hall and Mitchell 3 () to the case involving -dimensional sphere. For they reduce to the coherent states for a particle on a circle introduced by us in the paper 4 () (see also 5 ()) and utilized for the construction of coherent states on a torus 6 (). An alternative construction of coherent states for the sphere , , was described by Díaz-Ortiz and Villegas-Blas in a very recent work 7 (). The coherent states can be defined as the solution of the eigenvalue equation 2 ()
where is given by
where the cross designates the vector product. The operator and satisfy
The first equation of (10) is evident in view of the relation 8 ()
The coherent states can be generated from the coherent state (fiducial vector), where , via
The projection of the coherent state on the basis is given by
where are the Gegenbauer polynomials and is the sign of . The coherent states are not orthogonal. We have 8 ()
where are the Legendre polynomials.
The natural parametrization of by points of the classical phase space is given by
where the vectors , fulfil and , that is we assume that is the classical angular momentum, is the norm of the vector , and is the radius vector of a particle on a unit sphere. Introducing the spherical coordinates , and parametrizing the tangent vector by its norm and the angle between and the meridian passing through the point with the radius vector , we get from (16) the following natural parametrization of the phase space compatible with the constraints (see 8 ()):
where and are the unit mutually orthogonal vectors. The correctness of the introduced coherent states for a sphere is confirmed by the good behavior of quantum averages. Namely, we have
where , with given by (16), and the approximation is very good. From computer calculations it follows that for , the relative error is of order 1%. Furthermore, the computer simulations indicate that
where the approximation is as good as in (18). We point out that the factor in (19) is related to the fact that is not diagonal in the coherent state basis. Proceeding analogously as in the case of the coherent states for a circle 4 () one can introduce relative expectation value of with respect to averages in the coherent states labelled by unit vectors , so that . Yet another evidence of correctness of the described coherent states is behavior of the corresponding wave functions such that 8 ()
where , and are the common eigenvectors of the position operators spanning the coordinate representation. Namely, the probability density , where the squared norm of the coherent state given by (15) is
where , is peaked at and .
Iv Quantum beats
Consider now a free particle on a sphere. For the sake of simplicity we assume that the particle has a unit mass and it moves in a unit sphere. Clearly the quantum Hamiltonian is given by
As with standard coherent states of harmonic oscillator the discussed coherent states for the quantum mechanics on a sphere are not stable with respect to the free evolution. This can be demonstrated easily for the coherent state given by (13) using first equation of (3a) and the relation 2 ()
Therefore, in view of (12) none of the coherent states is stable. Nevertheless, in opposition to coherent states for a particle on a plane the corresponding wave packet (20) do not spread. Indeed, we have
where . Hence taking into account (20) we find that the time-dependent coherent state in the coordinate representation is
It thus appears that is -periodic function of time. From (25) and (21) it follows immediately that the probability density for the coordinates at time is
The probability density (26) is a periodic function of time with period . Thus it turns out that the wave packets on a sphere referring to coherent states do not spread but rather resemble maintaining their shape solitons. As a result of oscillations of the probability density an interesting phenomenon takes place that can be regarded as quantum beats on a sphere. More precisely, consider the following natural counterparts of the classical spherical coordinates for a free particle on a sphere
where and the use was made of the relation (19), and
where , and , so corresponds to the position and to the angular momentum of a particle. We point out that correctness of the formula corresponding to (28) was demonstrated in the case of the quantum mechanics on a circle 4 (). The explicit formulas for the expectation values of and that can be derived with the help of (7), (3a), (5) and (14) are too complicated to reproduce them herein. From numerical calculations it follows that whenever the condition holds then the dynamics of showing amplitude modulation is similar to well known acoustical beats (see Fig. 1). Clearly, the condition is a counterpart of the requirement of slightly different frequencies of two waves whose superposition is the “beat” wave. The values of oscillate around marking the coherent states via (see (16)). The minimum in the amplitude of oscillations of is reached at , where is integer, so the period of beats is . The dynamics of resembles the classical one in the limit , that is the uniform motion in a circle. Indeed, the plot of is piecewise linear with constant slope for , , and . The only exception is when the amplitude of is minimal. Namely at we have two kinds of behaviour of the angle shown in Fig. 2. First is the simple pulse and the second one the oscillation around . Interestingly, these two types of graphs of the function at occur alternately i.e. for we have oscillation, for we have a pulse, for an oscillation and so on. It is worth mentioning that that the distinguished role of is also present in the behavior of the probability density (26) having at the saddle point (see Fig. 3). As a result of beats the trajectory on a sphere parametrized as is not a grand circle as in the classical case but resembles the family of grand circles possessing a common diameter (see Fig. 4).
It should be noted that the discussed exotic dynamics of the rigid rotor is related to the free dynamics on a sphere that does not preserve the coherence. In fact, the coherent states for the quantum mechanics on a sphere are stable with respect to the evolution generated by the Hamiltonian
where is a constant vector. The stability of the coherent states is an immediate consequence of the identity 2 ()
implying via (8)
where and is given by
In order to compare the quantum dynamics in the case of the free evolution discussed previously and that given by (28) we now restrict to , so . Evidently, with such there exists a classical solution corresponding to the uniform motion in the equator. Using the identities
we find that relations (27) and (28) take the form
It thus appears that and follow the classical uniform circular motion on a sphere. In view of (19) for and utilized in the case of the free evolution, where is a classical position marking the coherent state, the motion takes place in the equator. Bearing in mind the behavior at analyzed in the case of the free evolution we now discuss the probability density for the coordinates corresponding to the Hamiltonian . From (20) and (31) it follows that the probability density is
where is given by (32) with . The probability density (36) is periodic function of time with period . In opposition to (26) it shows maximum and has no saddle points.
In summary, the rigid rotor in the coherent state shows unusual behavior that can be interpreted as quantum beats. It seems plausible to relate these beats to quantum interference. However, the concrete scenario is not clear. An intriguing point is the correlation of behavior of the rigid rotor with saddle points of the probability density. The authors do not know any example of analogous relationship in the literature.
This work was supported by the grant N202 205738 from the National Science Centre.
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