Propagation of Vortex Electron Wave Functions in a Magnetic Field
The physics of coherent beams of photons carrying axial orbital angular momentum (OAM) is well understood and such beams, sometimes known as vortex beams, have found applications in optics and microscopy. Recently electron beams carrying very large values of axial OAM have been generated. In the absence of coupling to an external electromagnetic field the propagation of such vortex electron beams is virtually identical mathematically to that of vortex photon beams propagating in a medium with a homogeneous index of refraction. But when coupled to an external electromagnetic field the propagation of vortex electron beams is distinctly different from photons. Here we use the exact path integral solution to Schrodingers equation to examine the time evolution of an electron wave function carrying axial OAM. Interestingly we find that the nonzero OAM wave function can be obtained from the zero OAM wave function, in the case considered here, simply by multipling it by an appropriate time and position dependent prefactor. Hence adding OAM and propagating can in this case be replaced by first propagating then adding OAM. Also, the results shown provide an explicit illustration of the fact that the gyromagnetic ratio for OAM is unity. We also propose a novel version of the Bohm-Aharonov effect using vortex electron beams.
Coherent beams of photons carrying axial orbital angular momentum (OAM), sometimes referred to as vortex beams, are well understood. and have various uses in optics and microscopy. Recently electron beams carrying very high amounts of axial OAM have been generated and the properties of such beams have been studied. Mathematically the propagation of a vortex photon beam in a medium with a homogeneous index of refraction is virtually identical to that of a freely propagating vortex electron beam. This is obviously not the case when the electrons are propagating in an external electromagnetic field. Here we use the exact path integral solution to examine how an electron wave function carrying axial OAM evolves in time. We find that the propagation of a wave function carrying nonzero axial OAM is equivalent to the the propagation of a zero OAM wave function multiplied by an appropriate position and time dependent prefactor. Also, the results provide an explicit illustration of the fact the the (non-radiatively corrected) gyromagnetic ratio for OAM is unity as it must be. We will see that from a practical point of view this means that the OAM vector rotates at half the rate of that the electron circulates in a magnetic field, i.e., at half the cyclotron or Landau frequency
The paper is organized as follows Section 2 briefly reviews the derivation of the gyromagnetic ratios for orbital and spin angular momentum from the Dirac equation Section 3 discusses the path integral solution for the (non-relativistic) propagation of the electron wave function in a magnetic field. Section 4 uses the path integral solution to study how a vortex electron beam, actually a wave packet, evolves in a magnetic and shows explicitly that the gyromagnetic ratio for OAM is unity.
2 Dirac to Schrodinger
For completeness we provide a brief review of the derivation of the Schrodinger equation from the Dirac equation which shows explicitly that the (non-radiatively corrected) gyromagnetic ratio for orbital angular momentum is unity.
The Dirac equation in SI units is
where is a four-component Dirac spinor and Here is the four-vector potential and is the electron charge. The indices take the values 0,1,2,3 which correspond to the directions, respectively . The Einstein summation convention wherein repeated indices are summed over their appropriate range is used throughout, e.g.,
Multiplying Eq (1) by and using
which follows from where are the gamma matrices, is the Minkowski metric, and is the field strength tensor we get
Consider a constant magnetic field pointing the in the direction. Using gauge invariance we can write or equivalently . Here and are the totally antisymmetric Levi-Civita tensors. is when is an even(odd) permutation of and is zero otherwise and is for and is zero otherwise Note that . We now have Working in the so called ”weak field limit”, i.e. dropping the term, gives
In the Dirac basis
where the are the Pauli matrices. In terms of two-component spinors and and for a slowly moving electron (in the Dirac basis) we can set and so finally
Here is the orbital angular momentum and is the spin angular momentum, both in the direction. More generally we can write
for a constant field. Thus we see that the OAM, couples to the magnetic field as whereas the spin angular momentum, couples as and so the (non-radiatively corrected) gyromagnetic ratio for orbital angular momentum whereas for spin angular momentum This difference has the effect that electron helicity, i.e., the spin projected in the direction of propagation, remains tangent to the trajectory, i.e, it rotates at the same rate that the electron circulates in a magnetic field. We will see below that because this is not the case for electron beams carrying axial OAM. Note that the values of and are a property of the Hamiltonian and not of the wave function. The vortex wave function studied below, which carries nonzero axial OAM, still couples to the magnetic field with a value of unity
3 Path Integral Solution for Propagation in a Magnetic Field
We are interested in OAM and not spin and so we will drop the spin term in (7) and let be a single component wave function. To reduce to the nonrelativistic case substitute
with slowly varying compared to into (7) and dropping the term we get the standard Schrodinger equation
with where is the unit vector in the direction.
Because (9) is linear and first order in the time derivative the solution can be written in the form
Here is the classical Lagrangian corresponding to the quantum Hamiltonian, and the integral is over all paths or trajectories which go from at time to at time The Lagrangian corresponding to (9) has the form
where is the vector potential with the magnetic field Using the form for given above we get, for a constant magnetic field in the direction,
It should be noted that the Lagrangian in (12) and (13) is the full Lagrangian, not the weak field approximation . This can be seen simply by calculating the corresponding classical Hamiltonian which yields .with
In terms of the new variables the Lagrangian corresponds to free propagation in the direction and a harmonic oscillator in the directions with radian frequency The path integral solutions for free propagation and for a harmonic oscillator are well known. Using these results and transforming back to the non-rotating coordinates we get
which is the standard cyclotron frequency and In (15) the combination always occurs divided by 2 and so we should expect various aspects of the wave function to evolve at half the rate at which the electron circulates in the magnetic field.
Note that in the limit as the propagator in (15) reduces to the free propagator
which is explicitly space and time translation invariant as it should be.
4 Evolution of a Gaussian wave function with and without OAM
The propagator given in (15) is Gaussian in form and so if we choose a Gaussian for the wave function at it will remain Gaussian. Also, in this case the integral in (10) can be evaluated analytically.
First consider propagation perpendicular to the magnetic field. In this case let the initial normalized wave function be a Gaussian centered at the origin and propagating in the direction
where we have switched from the notation to the more convenient at this stage notation with . This wave function is roughly in width in the and directions and has length in the direction. If we specify the values of and the radius of the classical orbit of the electron then If we take and to be much larger than the nominal de Broglie wavelength of then we expect mininal ”diffraction” effects to occur during propagation and as shown explicitly below this is exactly the case. This initial wave function has zero OAM about it’s direction of propagation, the direction, since
To generate axial OAM the so called ladder operator approach is used. Consider an operator with eigenstate such that We now want to generate a state such that To do this we only need to find an operator such that since then and so the state up to normalization and phase factors. Noting that
it follows that a state with 1 unit of axial OAM, is given (up to normalization and phase factors) by
Here and increases in the counterclockwise direction when looking in the direction and is measured from the axis. Using the fact that we immediately see that and so carries one unit of axial OAM. The factor of which appears automatically, is necessary since at (= the axis in this case) the phase is not defined and the wave function must vanish there.
To propagate we can write
The integral is still Gaussian and can be evaluated as above by letting and in (22). Taking the derivative with respect to and setting then yields
with the same as in (23).
Even though both these analytic solutions can be manipulated into somewhat more convenient forms, this is not very illuminating and so we will simply plot these solutions for a set of conditions which nicely illlustrate the relevant aspects of their time evolution. On the other hand it is worthwhile to examine the factor to get a better understanding of how it evolves and controls the orientation of the OAM. Substituting from above we find, after some algebra,
We see that at as it should, and that it rotates in time in the plane at a radian frequency of The origin of this factor obvious. In operator notation, ignoring the , (21) becomes
The time evolution is given by
where is the quantum Hamiltonian corresponding to the Lagrangian (13). Note this is the full Hamiltonian, not the weak field approximation.
The position of the node of follows from the solution to At this is the axis as shown above. For arbitrary we have the solution
This solution is illustrated in Figure 1 for several values of . This ”nodal line” rotates only by during one full period, of the electron cyclotron orbit and since this factor is the origin of the OAM carried by this shows explicity that the OAM rotates at half the cyclotron frequency, i.e., This also shows that the OAM is axially oriented only at times with , and its direction switches between being parallel and antiparallel to the direction of propagation at each of these times.
Note that and are not simply propagating Gaussian envelope functions multiplied by a propagating plane wave factor of the form with constant (but rotating at radian frequency and . For both wave functions the de Broglie wavelength varies in time. This is to be expected since the coupling to the vector potential contributes an extra phase to the wave function of the form which varies with position in generally an nonlinear fashion . Figures 2 and 3 show slices of the modulus squared and the real parts of and in the plane at different positions in the electron orbit. The values chosen for and are such that the size of the wave packet at , in the direction and in the direction are both much larger than the wavelength (so that diffraction effects are minimal) and is much larger than . The actual ratios used for the plots are and hence the spatial range of the and plots is about 5 orders of magnitude smaller than for the and plots so that the phase variation is visible. In Figure 2 we see that the long axis of the wave function tracks the nodal line and the spatial extent of the wave function varies with period and thus the length and width return, up to diffraction effects to their initial values at every This periodic variation in the spatial extent of the wave function can be traced back to the fact that in the rotating frame the Lagrangian is that of a harmonic oscillator.The free propagation part of the Langrangian, cause the wave function to expand or diffract as it propagates. The harmonic oscillator part, causes the wave function to contract and unless these two effects are precisely balanced the wave function will oscillate in size This is exactly analogous to the propagation of a paraxial Gaussian optical beam.centered on the axis and propagating in the direction in a medium with an index of refraction of the form , i.e, a harmonic osciallator potential. In the paraxial approximation the propagator for the photon beam has the same Gaussian form as the propagator for the harmonic oscillator. The quadratic variation of the index of refraction will case the beam to focus or shrink in size as it propagates whereas diffraction effects cause the beam to expand as it propagates. If the beam is large, so that the focusing effect dominates, then the beam will shrink in size as it propagates. Eventually it reaches a size where the diffraction effect dominates and it begins to expand. This process repeats itself causing the beam to oscillate in size with a fixed period along its length. These oscillations can be prevented if the size of the beam is fine tuned so that the diffraction and focusing effects exactly cancel out. Figure 3 shows the propagation of the wave function carrying a single unit of OAM. The node in the center of the wave function maintains its alignment on the nodal line during each cycle. The spiral form the phase of is apparent in the plots. Clearly the OAM is rotating at half the cyclotron frequency .
Now consider propagation parallel to the magetic field. In this case we let
where is the same as in (23) but now
Because depends on and only in the combination it follows that the initial Gaussian wave function chosen here does not pick up angular momentum as it propagates along the magnetic field. In fact for propagation parallel to the magnetic field the axial OAM of an eigenstate of is conserved. This follows directly from
where again and Indeed it can be shown that which obviously yields (33).
Using the exact path integral solution for the propagator in a constant magnetic field we have derived the evolution of a Gaussian wave function and shown explicitly that the (non-radiatively corrected) gyromagnetic ratio for OAM is unity. This must be the case since is a property of the Hamiltonian and not of the wave function.
The results presented above a novel version of the Aharonov-Bohm effect. Consider a long thin solenoid aligned along the axis. Outside the solenoid (far from the ends) varies as and so is zero outside. Inside the solenoid varies as and so is constant and nonzero. A Gaussian wave function like those considered above carrying nozero OAM that propagates along the axis has a node on the axis. In fact wave functions carrying large values of OAM have a very large region around the axis where the wave function is effectively zero. As in the standard Aharonov-Bohm experiment this is a case where there is no overlap between the wave function and the magnetic field. The wave function only overlaps with the magnetic vector potential. Hence the presence of the solenoid will cause a change in how the wave function propagates relative to the no solenoid case. This effect will be predominantly a change in the focus position of the wave function. Experimental verification of this would provide yet another example of the fact is the fundamental quantity and not and
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