week ending PRL 113, 240404 (2014) PHYSICAL REVIEW LETTERS 12 DECEMBER 2014

Is the Angular Momentum of an Electron Conserved in a Uniform Magnetic Field?

Colin R. Greenshields, Robert L. Stamps, Sonja Franke-Arnold, and Stephen M. Barnett* SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom (Received 25 July 2014; published 9 December 2014) We show that an electron moving in a uniform magnetic field possesses a time-varying “diamagnetic” angular momentum. Surprisingly this means that the kinetic angular momentum of the electron may vary with time, despite the rotational symmetry of the system. This apparent violation of angular momentum conservation is resolved by including the angular momentum of the surrounding fields.

DOI: 10.1103/PhysRevLett.113.240404 PACS numbers: 03.65.-w, 11.30.-j, 41.75.-i, 42.50.Tx

Introduction.—There is an intimate relation between question of whether the angular momentum of the electron angular momentum and rotational symmetry, as encapsu- is in fact conserved. lated by Noether’s theorem [1]. In particular, if a system is In this Letter, we show that for an electron in a nonsta- symmetric under rotation about a given axis, the angular tionary state, the changing radius of the electron’s prob- momentum along that axis will be conserved. This is the ability distribution in fact gives rise to a time-varying case with free electron vortices, which in the last few years kinetic angular momentum. The canonical angular momen- were predicted and observed in electron microscopes [2–5]. tum however remains constant. The apparent violation of Electron vortex beams in field-free space have a cylindrically angular momentum conservation is resolved by considering symmetric wave function and maintain a constant orbital the angular momentum in the fields surrounding the angular momentum in the direction of propagation [2]. electron. We show that the total kinetic angular momentum, Based on the same argument of rotational symmetry it including that of the fields, is conserved, as indeed it would seem that for an electron exposed to a uniform must be. magnetic field its orbital angular momentum in the direc- Results.—We consider an electron moving in a uniform tion of the field must be conserved. This is indeed true of magnetic field, and take the direction of this field to define the angular momentum about the axis of the classical the z axis. The (nonrelativistic) Hamiltonian for this system cyclotron orbit, and, furthermore, the energy eigenstates can be written in the form of an electron in a uniform magnetic field—the Landau 1 — H ¼ ðpkinÞ2; ð Þ states have constant angular momentum [6,7]. 2m 1 On the other hand, it is known that the kinetic angular momentum of an electron, which describes its mechanical where pkin ¼ mv ¼ pcan − eA is the kinetic momentum, motion, is not necessarily constant even when the electron pcan ¼ −iℏ∇ is the canonical momentum, e ¼ −jej is the interacts with external fields which are rotationally sym- charge of the electron, and m its mass. We choose a vector metric [8]. The balance and redistribution of momentum potential which in cylindrical polar coordinates (ρ; ϕ;z) has and angular momentum between matter and fields is a the form – fundamental problem of great general interest [9 12]. Bρ Recent investigations of vortex electron states in uniform A ¼ ϕˆ ⇒ B ¼ ∇ × A ¼ Bzˆ: ð2Þ and quasiuniform magnetic fields have revealed that the 2 angular velocity of the electron depends not only on the We note, for later reference, that this choice of vector poten- field strength but also on the azimuthal quantum number tial corresponds to the Coulomb gauge, in that ∇ · A ¼ 0. and the radial position [7,13,14]. Furthermore, in these The Hamiltonian (1) can then be rewritten as [17] quantum states the average radial position of the electron 1 1 is not in general constant, but rather changes as the wave can 2 2 2 can H ¼ ðp Þ þ mωLρ þ ωLLz ; ð3Þ function diffracts [14–16]. This contrasts sharply with the 2m 2 classical orbit in a uniform magnetic field, leading to the where ωL ¼ −eB=ð2mÞ is the Larmor frequency and ∂ Lcan ¼ðr pcanÞ ¼ −iℏ ð Þ z × z ∂ϕ 4

Published by the American Physical Society under the terms of z the Creative Commons Attribution 3.0 License. Further distri- is the component of canonical orbital angular momentum. kin can bution of this work must maintain attribution to the author(s) and As the z component of linear momentum pz ¼ pz the published article’s title, journal citation, and DOI. commutes with the Hamiltonian, the motion of the electron

0031-9007=14=113(24)=240404(5) 240404-1 Published by the American Physical Society week ending PRL 113, 240404 (2014) PHYSICAL REVIEW LETTERS 12 DECEMBER 2014 in the z direction is unaffected by the magnetic field. This be understood as the radial diffraction of the electron wave means that for an electron beam propagating along the z axis function in a harmonic potential generated by the inter- we need only consider the z component of angular momen- action with the magnetic field [15]. The energy associated tum. Furthermore, as the Hamiltonian, in our chosen gauge, with the motion perpendicular to the magnetic field remains is independent of ϕ it commutes with the canonical angular constant and has the expectation value can can momentum ½Lz ;H¼0 and, hence, Lz is conserved [15]. 1 Consider an electron wave function with cylindrical kin 2 E⊥ ¼ H − ðpz Þ : ð10Þ symmetry so that 2m

Ψ ¼ uðρ;z;tÞeilϕ: ð5Þ We will obtain the time behavior of the radial width, and later also of the kinetic orbital angular momentum, from ’ We make no assumption about the form of the function u, Heisenberg s formalism: so that in general this will not be an energy eigenstate. 2 2 can ∂ hρ iðtÞ 1 This state is, however, an eigenstate of Lz and hence the ¼ − h½½ρ2;H;HiðtÞ 2 2 expectation value of its angular momentum has the time- ∂t ℏ 2 2 ~2 independent value ¼ −ωcðhρ iðtÞ − ρ Þ; ð11Þ

can hLz i¼lℏ: ð6Þ where ωc ¼ 2ωL ¼ −eB=m is the classical cyclotron ~2 2 frequency and ρ ¼ðE⊥ − ωLlℏÞ=ðmωLÞ is the constant This is eminently reasonable as the system is symmetric z steady-state value which depends on the energy, the under rotation about the axis, so that according to canonical angular momentum, and the magnetic field. It ’ z Noether s theorem the component of the angular momen- can be seen from (11) that the mean-square radius oscillates tum should be conserved. ρ~2 In the presence of a magnetic field, the kinetic orbital sinusoidally about the value at the cyclotron frequency. t ¼ 0 angular momentum differs from its canonical counterpart: Setting to correspond to a stationary point of this oscillation, we have Lkin ¼ðr pkinÞ ¼ Lcan þ mω ρ2 ð Þ z × z z L 7 2 ~2 2 ~2 hρ iðtÞ¼ρ þðhρ ið0Þ − ρ Þ cosðωctÞ: ð12Þ [8], where we have used the definition of the kinetic linear According to the relation (8) this is intrinsically linked to an momentum and the specific form of the vector potential (2). oscillation of the kinetic angular momentum: We see that the field-dependent contribution to the kinetic angular momentum is associated with a rotation of the kin ~ kin kin ~ kin hLz iðtÞ¼Lz þðhLz ið0Þ − Lz Þ cosðωctÞ; ð13Þ electron probability distribution at constant angular veloc- ω ’ ity L. This is consistent with Larmor s theorem [18] and where the steady-state value of the kinetic angular can be interpreted as a diamagnetic response of the electron momentum to the external magnetic field [19]. 2 The expectation value of the kinetic orbital angular ~ kin ~2 Lz ¼ lℏ þ mωLρ ¼ E⊥ ð14Þ momentum can be expressed as ωc

kin hLz i¼lℏ þhIziωL; ð8Þ coincides with the classical value of the kinetic angular momentum for an electron with rotational kinetic energy where we have used the fact that the expectation value of E⊥. In general, the angular momentum oscillates sinus- the z component of the electron’s moment of inertia is oidally about the classical value, with the same frequency ωc as the classical cyclotron motion. Only if the kinetic 2 hIzi¼mhρ i: ð9Þ angular momentum is equal to the classical value does its expectation value remain constant. In this sense (14) ~ This means that the kinetic angular momentum of the defines the steady-state value ρ2. electron will be constant only if the radial probability We can obtain an exact solution for (12) in the case when 2 distribution is constant. The squared radius ρ does not at t ¼ 0 the wave function (5) has the Laguerre-Gaussian commute with the Hamiltonian, however, meaning this form quantity is not a constant of motion [6]. This means that, in pﬃﬃﬃ contrast with the classical motion, the mean value hρ2i will ρ 2 jlj ρ2 2ρ2 Ψ ð0Þ¼ ∝ − Ljlj not, in general, be a constant. n;l LGn;l exp 2 n 2 ρ0 ρ0 ρ0 It can be seen from the form of the Hamiltonian (3) that ½iðlϕ þ k zÞ; ð Þ the radial coordinate exhibits a harmonic motion. This can × exp z 15

240404-2 week ending PRL 113, 240404 (2014) PHYSICAL REVIEW LETTERS 12 DECEMBER 2014 jlj where l ∈ Z, n ¼ 0; 1; 2; …, and Ln is an associated Laguerre polynomial. The index n specifies the number of radial nodes in the wave function, while ρ0 is the width of the Gaussian envelope. The Laguerre-Gaussian wave functions can be used to describe electron vortex beams which have intrinsic orbital angular momentum lℏ [2] as well as electron beams with no intrinsic orbital angular momentum in the case when l ¼ 0. The mean-square radius of the Laguerre-Gaussian wave function (15) is equal to 1 hρ2i ð0Þ¼ ð2n þjljþ1Þρ2 ð Þ n;l 2 0 16

[20]. As this depends on the radial index n, it follows that the kinetic angular momentum of the electron also depends on n, which is not the case in the absence of a magnetic field. The steady-state mean-square radius for the same electron energy obtained from (10) and (14) is 4 ~2 1 ρB 2 ρ n;l ¼ ð2n þjljþ1Þ 1 þ ρ0; ð17Þ 4 ρ0

2 where ρB ¼ 2ℏ=ðmjωLjÞ ¼ 4ℏ=jeBj. As the phase of the wave function (15) does not depend on ρ,wehave

∂hρ2i ð0Þ 1 n;l ¼ h½ρ2;Hi ð0Þ¼0; ð Þ ∂t ℏ Im n;l 18 FIG. 1 (color online). Time evolution of the expectation meaning that this wave function corresponds to a stationary value of the electron’s kinetic orbital angular momentum for 2 point of the oscillation. The time evolution of hρ in;l Laguerre-Gaussian states with n ¼ 0 and a magnetic field in can therefore be obtained by substituting (16) and (17) the positive z direction (B>0). This is shown in (a) as a function into (12). of the initial width ρ0 for l ¼ 1, and in (b) for different values l ρ ¼ 1 5ρ In the special case when ρ0 ¼ ρB the Laguerre-Gaussian of assuming 0 . B. wave functions in (15) become the Landau energy eigen- kin states and we obtain the constant values of hLz in;l The combination of the externally imposed magnetic expected in this case [7]. In general, however, the angular field and the electric field of the electron itself gives rise to a momentum will oscillate around the classical value, as nonzero angular momentum density ε0r × ðE × BÞ [11]. illustrated in Fig. 1(a) for different ratios ρ0=ρB and in The z component of the field angular momentum is l Fig. 1(b) for different quantum numbers corresponding to Z canonical angular momenta lℏ. If the canonical angular Lkin field ¼ dVε ½r ðE BÞ momentum is in the opposite direction to the magnetic field z 0 × × z then the kinetic angular momentum may even change Z ⊥ direction, as clearly seen for l ¼ −4 in Fig. 1(b). We note ¼ dVε0ð∇ · EÞðr × A Þz ð19Þ that the moment of inertia increases with the radial quantum number n, resulting in larger amplitude oscillations, and A⊥ that a reversal of the direction of the magnetic field [21], where is the (manifestly gauge invariant) trans- ∇ A⊥ ¼ 0 corresponds to a shift of the phase of the oscillations verse part of the vector potential, defined by · . by 180°. In rewriting the second line we have used the fact that As we have seen, the kinetic angular momentum may the electric field of the electron is longitudinal, that is, ∇ E ¼ 0 change with time despite the fact that the system is entirely × . To complete our resolution we need only note rotationally symmetric—seemingly contradicting Noether’s that the first Maxwell equation is theorem. In order to restore angular momentum conserva- ϱ e tion we have to include the angular momentum contained ∇ · E ¼ ¼ jΨj2; ð20Þ in the field. ε0 ε0

240404-3 week ending PRL 113, 240404 (2014) PHYSICAL REVIEW LETTERS 12 DECEMBER 2014 where ϱ is the charge density, so that dependent on the length scale. As can be seen from (8), it is characterized by the constant −e=2 ¼ 7.60 × eB −4 −1 −2 kin field 2 10 ℏ T nm . In an atomic bound state with rms radius Lz ¼ hρ i: ð21Þ 2 1 Å, even in a field of strength 1 T the diamagnetic angular momentum is negligible compared to a single unit of Using (8) and (21) the total kinetic angular momentum of canonical angular momentum. However, for unbound elec- the electron plus the field is, therefore, trons, which can be distributed over a much larger area, the diamagnetic angular momentum can become significant, and kin total can Lz ¼ lℏ ¼ Lz ; ð22Þ may be the dominant contribution, both to the electron’s kinetic angular momentum and to E⊥. This can certainly be which is conserved, as it should be. Adding the kinetic the case in transmission electron microscopes, where the angular momentum of the field to that of the electron electron beam may have a radius hρ2i1=2 ∼ 1 nm–100 μm has restored the unique and conserved total angular and a field ∼1 T is provided by the objective lens. Note that in momentum. an electron microscope the radial dynamics occur in a Effect of gauge transformations.—We have worked reference frame moving with the electron along the z axis throughout in the Coulomb gauge and uncovered the result [2], and so can be observed as a function of the propagation that the canonical angular momentum of the electron is distance [14,16]. equal to the total angular momentum (with the value lℏ). While the creation of electron vortices has aroused a We can and should ask what we would have found had we considerable interest in the orbital angular momentum of worked in a different gauge. The kinetic angular momen- electron beams [22–24], little attention has been given tum of the field is, as we have seen, gauge invariant. To previously to the angular momentum which arises in a underline this point, we note that it depends only on the magnetic field. Our diamagnetic angular momentum will 2 magnetic field B and the mean-square radius hρ i, both of occur with any electron beam, even those with no canonical which are gauge invariant. Making a gauge transformation orbital angular momentum. The canonical angular momen- does change, however, both the vector potential and the tum is a manifestation of the cylindrical symmetry and phase of the wave function. The natural way to introduce a is restricted to integer multiples of ℏ. In contrast, the gauge transformation in quantum theory is through a local diamagnetic contribution, and hence the kinetic angular change in the phase of the wave function: momentum of the electron, may take any value. It is interesting to ask why our electron carries two Ψ → eiχðr;tÞΨ: ð23Þ distinct angular momenta and where each of these might be expected to appear in experiment. The total angular kin In this case, the action of our canonical momentum operator momentum contains a part Lz that may be ascribed kin field on the state changes to to the electron, and a second part Lz that depends on both the externally imposed magnetic field and iχðr;tÞ iχðr;tÞ pcanΨ → −iℏ∇e Ψ ¼ e ð−iℏ∇ þ ℏ∇χÞΨ: ð24Þ the electric field due to the electron. This latter part may be assigned either to the field, which gives the kinetic This change is counterbalanced by the corresponding momentum, or to the electron, giving the canonical momen- transformation of the vector potential tum. This situation is reminiscent of the linear momentum of a photon in a dielectric medium, where two rival ℏ A → A þ ∇χ ð25Þ momenta, due to Abraham and Minkowski, are the kinetic e and canonical momenta [25]. As with the photon, we can associate the canonical and kinetic momenta of the electron pkin z so that the kinetic momentum is unchanged. The with wavelike and particlelike properties, respectively. component of the canonical angular momentum is similarly Hence, an electron interference pattern should reveal the changed by a gauge transformation to eilϕ dependence associated with the canonical angular ∂ ∂χ momentum [26]. Absorption of an electron by an initially Lcan → −iℏ þ ℏ : ð Þ neutral target, however, should transfer the kinetic angular z ∂ϕ ∂ϕ 26 momentum of the electron to the rotational motion of the target, with the remaining angular momentum retained by Interestingly, the expectation value of this quantity for the electric field of the now charged target. our cylindrically symmetric state will still be lℏ, as jeiχðr;tÞΨj2 ¼jΨj2 is independent of ϕ, and, in particular, C. G. is supported by a SUPA Prize Studentship. We also hℏ∂χ=∂ϕi¼0. acknowledge support from the UK EPSRC under Program Discussion.—The magnitude of the diamagnetic contri- Grants No. EP/I012451/1 and No. EP/L002922/1, and bution to the kinetic angular momentum is strongly Bridging the Gap.

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