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Electron Spin and Probability Current Density in Quantum Mechanics W Electron spin and probability current density in quantum mechanics W. B. Hodgea) and S. V. Migirditch Department of Physics, Davidson College, Davidson, North Carolina 28035 W. C. Kerrb) Olin Physical Laboratory, Wake Forest University, Winston-Salem, North Carolina 27109 (Received 13 August 2013; accepted 26 February 2014) This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time- dependent Schrodinger€ equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrodinger€ equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4868094] I. INTRODUCTION occurs early in the course (and the textbook) when only the simplest form of the time-dependent Schrodinger€ equation has The probability interpretation of the wave function w of a been introduced. That form is for a scalar wave function with particle is a fundamental building block of our current under- a scalar potential energy function. The idea that the equation standing of matter. In the historical development of quantum 1 2 for the probability current density depends on the specific theory Born introduced the idea that jwj (suitably normal- form of the Hamiltonian is usually not developed. ized) is the probability density function. This expression in The purpose of this paper is to present the modified form conjunction with the concept of superposition of states for the probability current density when the description of allows a description of the interference property that is an the particle is expanded to include its spin and it experiences ubiquitous feature of the microscopic world. the spin-orbit interaction. In addition to the spin-orbit contri- One consequence of the probability interpretation is that bution, there is another spin-dependent contribution that, the total probability is constant, i.e., theÐ probability that the 2 curiously, seems to be independent of the Hamiltonian. particle is somewhere must be unity: jwj dV ¼ 1 (the inte- There has been some discussion of this contribution in the gral being over the whole configuration space). Furthermore, literature.3,4 It is not obtainable by the usual procedure (illus- this constraint must be satisfied at all times, even when the trated below) starting from the time-dependent Schrodinger€ state is evolving in time. The time evolution of the state for a equation with a non-relativistic Hamiltonian. We will discuss non-relativistic particle is governed by the time-dependent the origin of this term, show that the spin-orbit interaction Schrodinger€ equation ih@w=@t ¼ Hw, where H is the contribution can also be obtained the same way, and illus- Hamiltonian. The constraint on the integral of the absolute- trate the magnitudes of both of these spin-dependent terms in value-squared of the wave function imposes the requirement a specific example. that H must be a self-adjoint operator. The outline of this paper is as follows. Section II reviews The property described in the previous paragraph is the derivation of the probability current density for the case of global, i.e., it is on the integral of the wave function over the a spinless particle. Section III gives the derivation of the prob- whole spatial domain. However, there is also a local prop- ability current density for an electron with spin and spin-orbit erty: if the probability decreases in one region of space and interaction. An additional “magnetization” contribution is dis- increases in another, it must have flowed from one region to cussed in Sec. IV.SectionV applies these general results to the other. This idea implies that there must exist a probability the specific situation of one of the ground states of the hydro- current density function describing this flow. This quantum- gen atom. Finally, Sec. VI summarizes our results. mechanical function describing flow of probability and its global conservation is entirely analogous to matter flow in fluid dynamics and charge flow in classical electrodynamics. II. REVIEW OF DERIVATION FOR A SPINLESS In those fields, we are familiar with the so-called continuity PARTICLE equation relating the rate of change of the mass or charge density at each point in space to the divergence of the associ- To set the stage for subsequent derivations, we briefly ated current density at the same point. There must be an review the derivation of the probability current in introduc- essentially identical equation in quantum mechanics. tory non-relativistic quantum mechanics. It begins from the Because it is so important to understanding the probability time-dependent Schrodinger€ equation with a scalar potential interpretation of the wave function, courses and textbooks energy function VðxÞ for a scalar wave function wðx; tÞ carefully develop the continuity equation with its associated (where x denotes a point in three-dimensional position probability current density. However, this discussion typically space): 681 Am. J. Phys. 82 (7), July 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 681 @ h2 operator that acts on the Hilbert space of ket vectors) no lon- ih wðx; tÞ¼H wðx; tÞ¼À r2wðx; tÞþVðxÞwðx; tÞ: @t 0 2m ger provide a complete set of commuting operators for (1) describing an electron. The electron also has an intrinsic spin, described by an operator S^, whose components obey Here H denotes the Hamiltonian operator, written explicitly angular momentum commutation relations. It is customary 0 to choose as a complete set of commuting operators the set in the right-most part of the equation. The procedure is to ^ multiply this equation by the complex conjugate of the wave fx^; Szg. The wave function generalizes to wðx; r; tÞ, which is function wÃðx; tÞ, then conjugate Eq. (1) and multiply it by the quantum-mechanical amplitude to find the particle at w ; , and finally subtract these two equations. Because point x, with spin orientation r either up or down (the values ðx tÞ ^ VðxÞ is real-valued, terms including it cancel, and after some of r are 61=2, the quantum numbers of Sz ), at time t. The manipulations one obtains two possible values for r are often more pictorially denoted as r ¼"; #. These two functions are usually packaged into a @q h two-component spinor given by þ ðwÃr2w À wr2wÃÞ¼0(2) @t 2mi 2 3 1 "# 6 w x; þ ; t 7 (with the arguments x; t omitted for brevity), where 6 7 wðÞx; "; t 6 2 7 WðÞx; t ¼ 4 5 ¼ : (7) 2 1 wðÞx; #; t qðx; tÞ¼jwðx; tÞj (3) w x; À ; t 2 is the probability density function. Next, to the second term of Eq. (2) one applies the vector identity rðf VÞ The probability density function for finding the electron at ¼ f rV þrf Á V twice, once with the assignments f ! wà point x at time t is and V !rw and a second time with the complex conjugate X assignments. The result is qðÞx; t ¼ W†ðÞx; t WðÞx; t ¼ jwðÞx; r; t j2: (8) r¼";# @q h Âà þrÁ wÃrw rðÞw Ãw ¼ 0; (4) @t 2im Electrons also have a magnetic moment related to their spin angular momentum, given by which has the form of the continuity equation e l l ¼g S ¼g B S; (9) @q 2mc h þrÁj ¼ 0(5) @t P here g is the electron g-factor (approximately equal to 2) and familiar from electricity and magnetism, where it describes lB ¼ eh=ð2mcÞ is the Bohr magneton. the conservation of electric charge. It is customary to take The existence of electron spin raises the possibility of the probability current density to be the vector that appears additional terms in the Hamiltonian and in the time- in braces in Eq. (4), dependent Schrodinger€ equation that describe the interaction of the spin with its environment. An “intrinsic” (not involv- h ing externally applied fields) such term is the spin-orbit inter- j ðx; tÞ¼ ½wÃðx; tÞrwðx; tÞwðx; tÞrwÃðx; tÞ P 2mi action, which is the interaction of the intrinsic magnetic h moment of the electron with the effective magnetic field aris- ¼ =½wÃrw; (6) m ing from the motion of the electron. An heuristic derivation of this interaction is given in many textbooks.5–8 The result- where the subscript P stands for “paramagnetic” and = ing Hamiltonian is denotes the imaginary part. 1 1 We point out here that there is a degree of non-uniqueness H^ ¼ p^2 þ Vðx^Þþ S^ rVðx^Þp^: (10) in this identification of the probability current: additional 2m 2ðmcÞ2 terms with vanishing divergence could be added to the right- hand-side of Eq. (6) and Eq. (4) would still be satisfied. In the atomic physics applications that are typically pre- However, no proposal to modify Eq. (6) has been adopted sented in quantum mechanics textbooks, the potential energy since the formal structure of quantum mechanics was is a central potential: VðxÞ¼VðjxjÞ ¼ VðrÞ.
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