Chapter 5 Motion of a charged particle in a magnetic field Hendrik Antoon Lorentz 1853- Hitherto, we have focussed on applications of quantum mechanics to free parti- 1928 A Dutch physi- cles or particles confined by scalar potentials. In the following, we will address cist who shared the influence of a magnetic field on a charged particle. Classically, the force on the 1902 Nobel Prize in Physics a charged particle in an electric and magnetic field is specified by the Lorentz with Pieter Zee- force law: man for the dis- covery and the- F = q (E + v B) , oretical explana- × tion of the Zee- man effect. He where q denotes the charge and v the velocity. (Here we will adopt a convention also derived the transformation equa- in which q denotes the charge (which may be positive or negative) and e e tions subsequently used by Albert ≡| | Einstein to describe space and time. denotes the modulus of the electron charge, i.e. for an electron, the charge 19 Joseph-Louis Lagrange, born q = e = 1.602176487 10− C.) The velocity-dependent force associated − − × Giuseppe Lodovico Lagrangia with the magnetic field is quite different from the conservative forces associated 1736-1813 with scalar potentials, and the programme for transferring from classical to An Italian-born quantum mechanics - replacing momenta with the appropriate operators - has mathematician and astronomer, to be carried out with more care. As preparation, it is helpful to revise how who lived most the Lorentz force arises in the Lagrangian formulation of classical mechanics. of his life in Prussia and France, mak- ing significant 5.1 Classical mechanics of a particle in a field contributions to all fields of analysis, to number For a system with m degrees of freedom specified by coordinates q , q , the theory, and to classical and celestial 1 ··· m mechanics. On the recommendation classical action is determined from the Lagrangian L(qi, q˙i) by of Euler and D’Alembert, in 1766 Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in S[qi]= dt L(qi, q˙i) . Berlin, where he stayed for over ! twenty years, producing a large body of work and winning several The action is said to be a functional of the coordinates qi(t). According prizes of the French Academy of to Hamilton’s extremal principle (also known as the principle of least Sciences. Lagrange’s treatise on analytical mechanics, written in action), the dynamics of a classical system is described by the equations that Berlin and first published in 1788, minimize the action. These equations of motion can be expressed through the offered the most comprehensive treatment of classical mechanics classical Lagrangian in the form of the Euler-Lagrange equations, since Newton and formed a basis for the development of mathematical d physics in the nineteenth century. (∂ L(q , q˙ )) ∂ L(q , q˙ ) = 0 . (5.1) dt q˙i i i − qi i i " Info. Euler-Lagrange equations: According to Hamilton’s extremal princi- ple, for any smooth set of curves wi(t), the variation of the action around the classical 1 solution qi(t) is zero, i.e. lim! 0 (S[qi + #wi] S[qi]) = 0. Applied to the action, → ! − Advanced Quantum Physics 5.1. CLASSICAL MECHANICS OF A PARTICLE IN A FIELD 45 the variation implies that, for any i, dt (wi∂qi L(qi, q˙i) +w ˙ i∂q˙i L(qi, q˙i)) = 0. Then, integrating the second term by parts, and droping the boundary term, one obtains " d dt w ∂ L(q , q˙ ) ∂ L(q , q˙ ) =0. i qi i i − dt q˙i i i ! # $ Since this equality must follow for any function wi(t), the term in parentheses in the integrand must vanish leading to the Euler-Lagrange equation (5.1). The canonical momentum is specified by the equation pi = ∂q˙i L, and the classical Hamiltonian is defined by the Legendre transform, H(q ,p )= p q L(q , q˙ ) . (5.2) i i i i − i i %i It is straightforward to check that the equations of motion can be written in the form of Hamilton’s equations of motion, q˙i = ∂pi H, p˙i = ∂qi H. Sim´eon Denis Poisson 1781- − 1840 A French From these equations it follows that, if the Hamiltonian is independent of a mathematician, particular coordinate qi, the corresponding momentum pi remains constant. geometer, and 1 physicist whose For conservative forces, the classical Lagrangian and Hamiltonian can be mathematical written as L = T V , H = T + V , with T the kinetic energy and V the skills enabled − potential energy. him to compute the distribution of electrical " Info. Poisson brackets: Any dynamical variable f in the system is some charges on the surface of conduc- function of the phase space coordinates, the q s and p s, and (assuming it does not tors. He extended the work of his i i mentors, Pierre Simon Laplace and depend explicitly on time) its time-development is given by: Joseph Louis Lagrange, in celestial mechanics by taking their results to a d higher order of accuracy. He was also f(qi,pi)=∂q f q˙i + ∂p f p˙i = ∂q f∂p H ∂p f∂q H f, H . dt i i i i − i i ≡{ } known for his work in probability. The curly brackets are known as Poisson brackets, and are defined for any dynamical variables as A, B = ∂ A∂ B ∂ A∂ B. From Hamilton’s equations, we have { } qi pi − pi qi shown that for any variable, f˙ = f, H . It is easy to check that, for the coordinates { } and canonical momenta, q ,q =0= p ,p , q ,p = δ . This was the { i j} { i j} { i j} ij classical mathematical structure that led Dirac to link up classical and quantum mechanics: He realized that the Poisson brackets were the classical version of the commutators, so a classical canonical momentum must correspond to the quantum differential operator in the corresponding coordinate.2 With these foundations revised, we now return to the problem at hand; the infleunce of an electromagnetic field on the dynamics of the charged particle. As the Lorentz force is velocity dependent, it can not be expressed simply as the gradient of some potential. Nevertheless, the classical path traversed by a charged particle is still specifed by the principle of least action. The electric and magnetic fields can be written in terms of a scalar and a vector potential as B = A, E = φ A˙ . The corresponding Lagrangian takes the form:3 ∇× −∇ − 1 L = mv2 qφ + qv A. 2 − · 1i.e. forces that conserve mechanical energy. 2For a detailed discussion, we refer to Paul A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series Number 2, 1964. 3In a relativistic formulation, the interaction term here looks less arbitrary: the relativistic µ version would have the relativistically invariant q A dxµ added to the action integral, where the four-potential A =(φ, A) and dx =(ct, dx , dx , dx ). This is the simplest possible µ µ R 1 2 3 invariant interaction between the electromagnetic field and the particle’s four-velocity. Then, µ in the non-relativistic limit, q A dxµ just becomes q (v A φ)dt. · − R R Advanced Quantum Physics 5.2. QUANTUM MECHANICS OF A PARTICLE IN A FIELD 46 In this case, the general coordinates q x =(x ,x ,x ) are just the Carte- i ≡ i 1 2 3 sian coordinates specifying the position of the particle, and theq ˙i are the three componentsx ˙ i = (x ˙ 1, x˙ 2, x˙ 3) of the particle velocities. The important point is that the canonical momentum pi = ∂x˙ i L = mvi + qAi , is no longer simply given by the mass velocity – there is an extra term! × Making use of the definition (5.2), the corresponding Hamiltonian is given by 1 1 H(q ,p )= (mv + qA ) v mv2 + qφ qv A = mv2 + qφ. i i i i i − 2 − · 2 %i Reassuringly, the Hamiltonian just has the familiar form of the sum of the kinetic and potential energy. However, to get Hamilton’s equations of motion, the Hamiltonian has to be expressed solely in terms of the coordinates and canonical momenta, i.e. 1 H = (p qA(r,t))2 + qφ(r,t) . 2m − Let us now consider Hamilton’s equations of motion,x ˙ i = ∂pi H and p˙ = ∂ H. The first equation recovers the expression for the canonical i − xi momentum while second equation yields the Lorentz force law. To under- stand how, we must first keep in mind that dp/dt is not the acceleration: The A-dependent term also varies in time, and in a quite complicated way, since it is the field at a point moving with the particle. More precisely, ˙ p˙i = mx¨i + qAi = mx¨i + q ∂tAi + vj∂xj Ai , where we have assumed a summation over& repeated indicies.' The right-hand ∂H side of the second of Hamilton’s equation,p ˙i = , is given by − ∂xi 1 ∂ H = (p qA(r,t))q∂ A q∂ φ(r,t)=qv ∂ A q∂ φ. − xi m − xi − xi j xi j − xi Together, we obtain the equation of motion, mx¨ = q ∂ A + v ∂ A + i − t i j xj i qv ∂ A q∂ φ. Using the identity, v ( A)= (v A) (v )A, and j xi j − xi × ∇× ∇ ·& − ·∇ ' the expressions for the electric and magnetic fields in terms of the potentials, one recovers the Lorentz equations mx¨ = F = q (E + v B) . × With these preliminary discussions of the classical system in place, we are now in a position to turn to the quantum mechanics.