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.