Classical and quantum mechanics of a charged particle in oscillating electric and magnetic fields

 

V.L.B. de Jesus, A.P. Guimarães, and I.S. Oliveira
Centro Brasileiro de Pesquisas Físicas
Rua Dr. Xavier Sigaud 150, Rio de Janeiro - 22290-180, Brazil

 

Received 27 May, 1999

 

 

The motion of a particle with charge q and mass m in a magnetic field given by B = kB₀ + B₁ [icos(wt) + jsin(wt)] and an electric field which obeys Ñ ´ E = -¶B/¶t is analysed classically and quantum-mechanically. The use of a rotating coordinate system allows the analytical derivation of the particle classical trajectory and its laboratory wavefunction. The motion exhibits two resonances, one at w = w_c = -qB_o/m, the cyclotron frequency, and the other at w = w_L = -qB_o/2m, the Larmor frequency. For w at the first resonance frequency, the particle acquires a simple closed trajectory, and the effective hamiltonian can be interpreted as that of a particle in a static magnetic field. In the second case a term corresponding to an effective static electric field remains, and the particle orbit is an open line. The particle wave function and eigenenergies are calculated.

PACS: 41.75.A, 52.50.G, 03.50.D, 12.20
Keywords: magnetic resonance, ion trapping, isotope separation, magnetic confinement

 

 

I   Introduction

The dynamics of charged particles in electric and magnetic fields is of both academic and practical interest. The areas where this problem finds applications include the development of cyclotron accelerators [1], free electron lasers [2], plasma physics [4] and so on.

In this paper one considers the classical and quantum dynamics of a particle with charge q and mass m acted by a magnetic field given by

B = kB₀ + B₁[icos(wt)+jsin(wt) ]   (1)

and an electric field which relates to B through the Faraday law:

Both fields can be derived from the vector potential:

When B₁ is given by two pairs of crossed Helmholtz coils, the approximation of homogeneity implicit in equation (1) is valid in a region of about 20% of the volume enclosed by the coils. Similar assumption have been taken by other authors [2, 3].

The classical equations of motion can be obtained from the lagrangian:

whereas to study the quantum dynamics we need the hamiltonian:

 

II  The classical motion

The classical equations of motion can be easily obtained from (4). In order to eliminate the time dependence of the lagrangian, we perform the following transformation of coordinates:

i = i^¢cos(wt) - j^¢sin(wt)

j = i^¢sin(wt) + j^¢cos(wt)        (6)

k = k^¢

In the nuclear magnetic resonance (NMR) literature [5] these transformations are interpreted as leading to a system of reference which rotates with angular frequency w in respect to the laboratory coordinate system.

From (6), using lower case to indicate the variables in the rotating system, the effective lagrangian becomes:

This lagrangian can be written in the usual compact form:

where T is the particle kinetic energy, A_eff the effective vector potential given by:

with B_eff being the effective magnetic field (defined below), and the effective scalar potential:

where g = q/m is the particle charge-to-mass ratio. Thus, one has for the particle in the rotating frame the following equations of motion:

It is useful to define an effective electric field E_eff. The expressions for the effective fields are:

Therefore, for a fixed value of w, each particle with a given charge-to-mass ratio, g, will feel different effective fields. Note that B_eff differs from that in the NMR case by a factor '2' in the ''apparent field'' w/ g [5]. With these definitions, the form of the Lorentz force is preserved in the rotating system:

F_eff = qE_eff + qv ×B_eff          (14)

One can clearly see from (11) that there are two resonance frequencies in the motion: one at w = w_c = -qB₀/m, the cyclotron frequency, and another at w_L = -qB₀/2m, the Larmor frequency. For a frequency equal to the first one (w_c) the particle feels the following effective fields:

and for the second frequency (w = w_L), 

Now we consider a particle incident in region of fields B₀ and B₁, at the origin of the coordinate system, with initial velocity parallel to B₀. That is, x(0) = y(0) = z(0) = 0, v_x(0) = v_y(0) = 0, and v_z(0) = v₀. As in usual NMR, one makes the approximation B₀ >> B₁. In this limit, it is easy to verify by direct substitution the following solutions of (11), for w = w_c:

where w₁ º gB₁. For w = w_L:

The detailed calculation for the obtention of these solutions is given in ref. [8].

Then, we see that whereas for w = w_c the solutions are purely trigonometric functions, for w = w_L there is a mixture of trigonometric and hyperbolic functions. This means that the trajectory of the particle will be a closed path in the first case, and an open line in the second. For a general value of w, which can be very close to w_c, there will also be an exponential drift. As an example, Figure 1 shows the trajectories of particles in a beam containing triply ionized isotopes of uranium. The respective charge-to-mass ratios in MHz/T are as follow: g(²³³U) = 1.242; g(²³⁴U) = 1.237; g(²³⁵U) = 1.231; g(²³⁶U) = 1.226 and g(²³⁸U) = 1.216. For this simulation we set v₀ = 10⁴ m/s, B₀ = 1 T, B₁ = 0.01 T. The oscillating field frequency is tuned to the cyclotron frequency of the isotope ²³⁵U, that is, w = -1.231 MHz. The drawing is in the rotating system. One can have a picture of the trajectories in the laboratory system by rotating the figure about the z-axis. Note that each isotope, according to Eqs. (12) and (13), feels different effective fields. This causes the lighter isotopes to deviate in opposite directions in respect to the heavier ones.

 

 

III  Quantum description

In this section we approach the problem from the quantum-mechanical point of view. The transformation to the rotating frame in this case, made directly on the Schrödinger equation, allows the derivation of the laboratory wave function and the particle eigenenergies.

Using the following straightforward relations:

where, L_x, L_y and L_z are the components of the canonical angular momentum of the particle, the hamiltonian in Eq. (5) becomes¹:

This hamiltonian represents a charged particle moving in a static magnetic field B = B₁i + B₀k. The above operation can be interpreted as the quantum-mechanical transformation of the hamiltonian to the rotating coordinate system.

Defining the wavefunction y¢ through the relation:

and replacing into the Schrödinger equation one obtains:

Since _eff is time-independent, the solution of (22) will be:

and consequently the wavefunction in the laboratory system will be

Note that since [L_z, ] ¹ 0, the two exponential operators in Eq. (24) cannot be gathered into one.

Now we shall analyze the properties of _eff:

where DB º B₀ + 2w/g. By adding and subtracting the quantity

the effective hamiltonian can be re-written as:

which, in turn, has the general form:

where the effective scalar potential is again given by

A_eff being the effective vector potential. The components of A_eff can be obtained by commuting X, Y and Z with _eff, and comparing the result with the definition of the canonical momentum P = m + qA. For instance:

Consequently,

Repeating the procedure for the other components, one obtains:

These results are the same as those obtained in ref. [6], the only difference being a factor '2' in the definition of DB.

Written in the form of Eq. (26), the hamiltonian exhibits the effects of the electric field. Contrary to what happens when this is neglected [6], it shows two resonance frequencies. At the Larmor frequency, DB = 0, and the hamiltonian becomes:

which represents a particle in a static magnetic field along the x direction, plus an electric field potential. The eigenstates of the particle in this case cannot be easily found. On the other hand, at the cyclotron frequency, the second term of f_eff in Eq. (28) vanishes, and the hamiltonian becomes:

which represents a particle moving in a static magnetic field B_eff = B₀k - B₁i. This hamiltonian can easily written in a diagonal form by defining the angle

between the z-axis and B_eff, and writing the operators of _eff in (30) in terms of the new coordinates, X¢, Y¢ and Z¢, where the effective field is axial.

The particle's eigenenergies are given in this case by:

where w_c¢ = gB_eff = is the cyclotron frequency about the effective field in the rotating system. From this one sees that the quantization axis can be rotated continuously by changing the angle q through the change in the ratio B₁/B₀.

 

IV  Conclusions

In this paper we have studied the classical and the quantum dynamics of a charged particle in oscillating magnetic and electric fields which are related through the Faraday law. The equations of motion show two resonance frequencies, one at the Larmor frequency (w_L) and another at the cyclotron frequency (w_c). When the field frequency equals w_c, the particle is confined to a simple closed trajectory, but when w = w_L, it drifts away, the same happening to off resonance particles whose frequencies are very close to w_c. The use of a ''rotating coordinate system'' such as used in conventional nuclear magnetic resonance allows the derivation of analytical solutions for the equations of motion.

By using the corresponding quantum-mechanical transformation, one finds the exact wavefunction of the particle in the laboratory system. When w = w_c, the effective hamiltonian corresponds to that of a charged particle in an static magnetic field. In this case the particle eigenenergies are derived in the rotating coordinate system, and it is shown that the direction of the axis of quantization can be continuously rotated by changing the ratio between the intensities of the fields. On the other hand, when w = w_L, the hamiltonian is a mixture of effective magnetic and electric fields, and the eigenenergies cannot be easily derived.

The Hamiltonian (30) predicts the existence of ''current echoes'', and therefore is in accordance with the results of references [6] and [7]. There is, however, one important difference which appears in the present case where the electric field is considered. Contrary to what happens in [6], the static field term does not vanish at resonance. Thus, in order to describe properly the formation of a current echo in the present situation, one must consider B₁ >> B₀ when the pulse is ''on'', and obviously B₁ = 0 when it is off. Having this in mind, the calculation for the current echo amplitude can be carried out in the same way as described in [6].

 

Acknowledgements

The authors are in debt to Prof. W. Baltensperger, Prof. N. V. de Castro Faria and to Dr. S. A. Dias, for their useful suggestions. V. L. B. J. acknowledges the support from CNPq, Brazil.

 

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¹ For the quantum treatment we keep capitals throughout the section.