Quantum Mechanics of Bending of a Nonrelativistic Charged Particle

arXiv:1911.02472v1 [quant-ph] 6 Nov 2019 atceba pia eie.We atceo charge of particle a man When of devices. components optical important beam are particle systems deflection, or Bending, Introduction 1 corrections Quantum optics, beam particle t charged Foldy-Wouthuysen Quantum tion, representation, Sch Feshbach-Villars Nonrelativistic Mechanics, equation, Quantum map, Transfer magnet, ing Keywords smvn ihspeed with moving is 2 1 unu ehnc fbnigo nonrelativistic a of bending of mechanics Quantum eie Faculty, Retired Email olg fAt n ple cecs(AS,Doa Univers Dhofar (CAAS), Sciences Applied and Arts of College t rs tet eta ntttso ehooy(I)Ca (CIT) Technology of Institutes Central Street, Cross 4th fcasclmcaisi hre atceba optics. beam particle charged terms. remarkab in correction the mechanics quantum classical explain tiny of corrections transf are quantum classical there small and the negligibly part der contains main is map the system transfer as the quantum of Schrödinger equation. the axis nonrelativistic that p optic the transverse curved two with the moment between starting on and beam points position the par different of the the at particle in for a studied map of is transfer components magnet The dipole a approximation. by beam particle charged [email protected] : hre atceba yadpl magnet dipole a by beam particle charged unu ehnc fbnigo orltvsi monoener nonrelativistic a of bending of mechanics Quantum aenAmdKhan Ahmed Sameen lsia hre atceba pis antcdpl,Bend- dipole, Magnetic optics, beam particle charged Classical : otBxN.20,Psa oe 1,sllh Oman salalah, 211, Code: Postal 2509, No. Box Post a eateto ahmtc n Sciences and Mathematics of Department E-mail b h nttt fMteaia Sciences Mathematical of Institute The hrmn,Ceni601,India 600113, Chennai Tharamani, v ln tagtln nahrzna ln,t edits bend to plane, horizontal a in line straight a along [email protected] : a 1 Abstract n aawm Jagannathan Ramaswamy and q n etmass rest and ti found is It esuccess le rmap er b ransforma- mpus 2 lanes axial getic charged y ived The . um rödinger ity m path in a circular arc of radius ρ in the horizontal plane the constant magnetic field B to be applied in the vertical direction is given by the relation γmv2 = qvB, (1) ρ so that the magnetic Lorentz force provides the required centripetal force, where γ =1/ 1 − (v2/c2) with c as the speed of light. In other words, q γmv p Bρ = = , (2) q q where p is the momentum of the particle. Thus, if the path of a particle of charge q and momentum p is to be bent along a circular arc of radius ρ by a dipole magnet the constant magnetic field in the vertical direction obtained in the gap between the two poles of the magnet should be such that

qB = κp, (3) where κ =1/ρ is the curvature of the circular orbit. Note that this formula is applicable irrespective of whether the beam is nonrelativistic (γ ≈ 1) or relativistic (γ ≫ 1). Let us now consider a dipole magnet bending system with a circular arc of curvature κ, as the optic axis, or the design trajectory, and a magnetic ﬁeld κp B = 0 , (4) 0 q matched to the design momentum p0. The reference particle moving along the design trajectory will have p0 as the longitudinal component of momentum at each point of the optic axis and zero transverse momentum components. The arclength s measured along the optic axis from some reference point is the natural choice for the independent coordinate for studying the motion of the charged particle. Let the reference particle carry an orthonormal (x, y)- coordinate frame with it. The x-axis is taken to be perpendicular to the tangent to the design orbit and in the same horizontal plane as the trajectory. The y-axis is taken to be in the vertical direction perpendicular to both the x-axis and the trajectory. The curved s-axis is along the design trajectory and perpendicular to both the x and y axes at any point on the design trajectory. The instantaneous position of the reference particle in the design trajectory at an arc length s from the reference point corresponds to x = 0

2 and y = 0. Let any particle of the beam have coordinates (X,Y,Z) with respect to a fixed right handed Cartesian coordinate frame with its origin at the reference point on the design trajectory from which the arclength s is measured. Then, the two sets of coordinates of any particle of the beam, (X,Y,Z) and (x,y,s), are related as: X = ρ(1 + κx) cos(κs) − ρ, Y = y, Z = ρ(1 + κx) sin(κs). (5) In classical charged particle beam optics, Newtonian mechanics, or the equivalent Hamiltonian mechanics, is used to study the trajectory of the charged particle moving under the Lorentz force. Theoretical, practical, and historical aspects of classical mechanics of electron optics have been pre- sented extensively, with detailed references to the literature, in the first two volumes of the encyclopedic text book of Hawkes and Kasper [1, 2] and the third volume [3] presents the electron wave optics, or essentially, the quantum mechanics of electron optical imaging (for classical electron optics see also, e.g., [4]-[8]). In the context of accelerator physics, detailed accounts of classical charged particle beam optics are available in several books (see, e.g., [9]-[17]). In electron wave optics used for studying imaging in electron microscopy nonrelativistic Schrödinger equation is used, and in relativistic situations either a Schrödinger equation with relativistic correction for mass or an approximate scalar wave equation derived from the relativistic Dirac equation is used (see [3] for details). Wondering how classical mechanics is so successful in the design and operation of electron beam optical devices like electron microscopes when the microscopic electron should be obeying quantum mechanics, a systematic study of quantum mechanics of electron beam optics based on the Dirac equation, the proper equation for the electron, was initiated in [18] and the first quantum mechanical derivation of the classical Busch formula for the focal length of a round magnetic electron lens was obtained. Subsequent studies have led to the formalism of quantum charged particle beam optics applicable to devices from low energy electron microscopes to high energy particle accelerators (see [19]-[31]). In [31], which presents a consolidated account of the formalism of quantum mechanics of charged particle beam optics, we have treated the dipole magnet, besides other charged particle beam optical elements like round magnetic lenses and magnetic quadrupoles, using the relativistic wave equations, namely, the Klein-Gordan equation for spin-0 particle and the Dirac equation 1 for spin- 2 particle. The Klein-Gordan equation can also be used for a particle when its spin is ignored. We have shown, in general, how the relativistic

3 quantum theory of any charged particle beam optical system can be approx- imated leading to the nonrelativistic quantum theory of the system and we have not given the details for any speciﬁc system. This article shows explic- itly how the bending of a nonrelativistic paraxial charged particle beam by a dipole magnet can be understood based on the nonrelativistic Schr¨odinger equation.

2 Nonrelativistic classical mechanics of the optics of dipole magnet

Let us consider a nonrelativistic paraxial monoenergetic beam propagating through the dipole magnet. Let si (for sin) and so (for sout) be, respectively, the entry and exit points, the points on the curved optic axis of the dipole magnet at which the beam particle enters from the free space and exits into the free space. Classical charged particle optical Hamiltonian for the system is given by 1 2 1 2 2 Ho = −p0 + p⊥ + p0κ x , (6) 2p0 2 2 2 2 where p⊥ = px + py, p0 is the design momentum, and the design energy is 2 E0 = p0/2m. We shall derive this Hamiltonian in the next section as the classical approximation of the corresponding quantum Hamiltonian. Hamilton’s equations of motion along the s-axis are

dx ∂Ho px dpx ∂Ho 2 = = , = − = −p0κ x, ds ∂px p0 ds ∂x dy ∂H p dp ∂H = o = y , y = − o =0, (7) ds ∂py p0 ds ∂y leading to the paraxial equations of motion x′′ + κ2x =0, y′′ =0, (8) where x′′ = d2x/ds2 and y′′ = d2y/ds2. The general solutions for x, x′ = ′ dx/ds = px/p0, y and y = dy/ds = py/p0 at any point s on the optic axis, with si ≤ s ≤ so, are given by 1 x(s) cos(κ (s − si)) κ sin(κ (s − si))0 0 x′(s) −κ sin(κ (s − s )) cos(κ (s − s )) 0 0   =  i i  y(s) 0 01 s − si      ′     y (s)   0 0 01      4 x (si) x′ (s ) ×  i  . (9) y (si)    ′   y (si)    Note that the position coordinates of the particle (x(s),y(s)), and the components of the tangent to the trajectory (x′(s),y′(s)), are the coordinates of the charged particle ray at the point s. Thus the transfer map for the ray coordinates from si to any s on the optic axis is given by Eq.(9). The transfer map in Eq.(9) shows that if a particle of the paraxial beam enters the ′ dipole from free space along the optic axis i.e., with x (si) = 0, x (si) = 0, ′ y (si) = 0, and y (si) = 0, then at any point s on the optic axis it will have x(s) = 0, x′(s) = 0, y(s) = 0, and y′(s) = 0, or in other words, it will continue to move along the optic axis, or the design trajectory, until it exits the dipole into the free space. It is instructive to arrive at the transfer map in Eq.(9) using the Poisson bracket formalism of Hamilton’s equations. Let us deﬁne the Poisson bracket between any two functions A (r⊥, p⊥,s) and B (r⊥, p⊥,s) as ∂A ∂B ∂A ∂B ∂A ∂B ∂A ∂B : A : B = {A, B} = − + − , (10) ∂x ∂px ∂px ∂x ! ∂y ∂py ∂py ∂y ! Hamilton’s equation for s-evolution of any observable of the particle, without explicit s-dependence, say O (r⊥, p⊥), is dO =: −H : O. (11) ds o

Since Ho is independent of s it is possible to integrate this equation and write O(s) = e(s−si):−Ho:O s=si

2 (s − si) 2 = 1+(s − si): −Ho :+ : −Ho : 2! 3 (s − si) 3 + : −Ho : + ··· O 3! ! s=si

2 (s − si) = O +(s − si) {−Ho,O} + {−Ho, {−Ho,O}} 2! 3 (s − si) + {−Ho, {−Ho, {−Ho,O}}} + ··· (12). 3! ! s=si

5 With px px 2 {−Ho, x} = , −Ho, = −κ x, (13) p0 ( p0 ) we get

2 2 κ (s − si) x(s) = 1 − + ··· x (si) 2! ! 3 3 1 κ (s − si) px (si) + κ (s − si) − + ··· κ 3! ! p0

1 px (si) = cos(κ (s − si))x (si)+ sin(κ (s − si)) , κ p0 3 3 px(s) κ (s − si) = −κ κ (s − si) − + ··· x (si) p0 3! ! κ2 (s − s )2 p (s ) + 1 − i + ··· x i 2! ! p0

px (si) = −κ sin(κ (s − si))x (si) + cos(κ (s − si)) . (14) p0

With py py {−Ho,y} = , −Ho, =0, (15) p0 ( p0 ) we get py (si) py(s) py (si) y(s)= y (si)+(s − si) , = . (16) p0 p0 p0 Thus, we have the transfer map

1 x(s) cos(κ (s − si)) sin(κ (s − si))0 0 p (s) κ x −κ sin(κ (s − s )) cos(κ (s − s )) 0 0  p0  =  i i   y(s)  0 0 1(s − si)  py(s)       0 001   p0        x (si) px(si) ×  p0  , (17)  y (si)   py(si)     p0    6 ′ ′ same as in Eq.(9) with x (s) = px(s)/p0 and y (s) = py(s)/p0. For any function A (r⊥, p⊥,s) the operator ∂A ∂ ∂A ∂ ∂A ∂ ∂A ∂ : A := − + − , (18) ∂x ∂px ∂px ∂x ! ∂y ∂py ∂py ∂y ! such that : A : acting on another function B (r⊥, p⊥,s) gives the Poisson bracket {A, B}, is known as a Lie operator. Equation (12) is a simple example of powerful Lie algebraic techniques developed to study photon optical and charged particle optical systems (see [32]-[35], and also e.g., [9, 15, 31, 36]).

3 Nonrelativistic quantum mechanics of the optics of dipole magnet

To understand the quantum mechanics of bending of a nonrelativistic charged particle beam by a dipole magnet we have to change the corresponding Schrödinger equation to the curved (x,y,s)-coordinate system and rewrite it as an s-evolution equation instead of time-evolution equation. This will lead to the quantum beam optical Hamiltonian governing the s-evolution of the beam variables. To this end, we proceed as follows. The free particle Schrödinger equation in the Cartesian (X,Y,Z) coordinate system is ∂Ψ(R, t) ih¯ = HΨ(R, t) ∂t 1 1 H = c P 2 = P 2 + P 2 + P 2 2m 2m X Y Z c 1 b ∂ b2 b ∂b 2 ∂ 2 = −ih¯ + −ih¯ + −ih¯ 2m  ∂X ! ∂Y ! ∂Z !    1 2 2 = −h¯ ∇R , (19) 2m where H is the nonrelativistic free particle Hamiltonian. The line element dS, distance between two infinitesimally close points (X,Y,Z) and (X + dX,Y +cdY,Z + dZ), corresponding to (x,y,s) and (x + dx,y + dy,s + ds), respectively, is given by

dS2 = dX2 + dY 2 + dZ2 = dx2 + dy2 + ζ2ds2, (20)

7 where ζ =1+ κx as follows from Eq.(5). Then, we know (see, e.g., [37]) that 2 the Laplacian ∇R in the (x,y,s)-coordinate system becomes 2 2 2 2 2 2 1 ∂ ∂ ∂ 1 ∂ ∂ ∂ 1 ∂ κ ∂ ∇R = ζ + + = + + + . (21) ζ ∂x ∂x ! ∂y2 ζ2 ∂s2 ∂x2 ∂y2 ζ2 ∂s2 ζ ∂x 2 If we substitute this expression for ∇R in Eq.(19), and change Ψ(R, t) to Ψ(r⊥,s,t), with r⊥ =(x, y), we get 2 2 2 2 ∂Ψ(r⊥,s,t) h¯ ∂ ∂ 1 ∂ κ ∂ ih¯ = − + + + Ψ(r⊥,s,t) . (22) ∂t 2m "∂x2 ∂y2 ζ2 ∂s2 ζ ∂x # Note that the transformation of Eq.(19) from Cartesian coordinates to curved coordinates has resulted in Eq.(22) with a non-Hermitian Hamiltonian operator on the right hand side because of the presence of the last term,(κ/ζ)∂/∂x, 2 in the expression for ∇R in Eq.(21). So, let us replace this term by the Her- mitian term † 1 κ ∂ κ ∂ 1 κ ∂ ∂ κ 1 κ ∂ κ2 + = − = , = . (23) 2  ζ ∂x ! ζ ∂x !  2 ζ ∂x ∂x ζ ! 2 " ζ ∂x # 2ζ2   Note, that ζ is a function of x. Consequently, the free particle Schr¨odinger equation in the curved (x,y,s) coordinate system becomes ∂Ψ(r ,s,t) ih¯ ⊥ = HΨ(r ,s,t) ∂t ⊥ 2 cf h¯ ∂2 ∂2 1 ∂2 κ2 H = − + + + , (24) 2m " ∂x2 ∂y2 ζ2 ∂s2 2ζ2 # c Let us now rewrite Eq.(24)f as

∂ 1 ∂ 2 ∂ 2 ih¯ Ψ(r⊥,s,t) = −ih¯ + −ih¯ ∂t ! 2m  ∂x ! ∂y !  1 ∂ 2 h¯2κ2 + −ih¯ − Ψ(r⊥,s,t) . (25) ζ2 ∂s ! 2ζ2   To proceed from the free particle Schr¨odinger equation to the equation in the presence of a magnetic ﬁeld, we can make use of the principle of electro- magnetic minimal coupling ∂ ∂ ∂ ∂ −ih¯ −→ −ih¯ − qA , −ih¯ −→ −ih¯ − qA , ∂x ∂x x ∂y ∂y y

8 1 ∂ 1 ∂ 1 ∂ −ih¯ −→ −ih¯ − qAs = −ih¯ − qζAs . (26) ζ ∂s ! ζ ∂s ! ζ ∂s !

Here (Ax, Ay, As) are the components of the vector potential such that the magnetic ﬁeld is given by

1 ∂ (ζAs) ∂Ay Bx = (∇ × A)x = − , ζ ∂y ∂s !

1 ∂Ax ∂ (ζAs) By = (∇ × A)y = − , ζ ∂s ∂x !

∂Ay ∂Ax Bs = (∇ × A)s = − , (27) ∂x ∂y ! using the expression for curl of a vector in a curved coordinate system (see, e.g., [37]). Then, the Schrödinger equation in presence of a magnetic field, in the curved (x,y,s) coordinate system, becomes ∂Ψ(r ,s,t) ih¯ ⊥ = HΨ(r ,s,t) , ∂t ⊥ 2 2 c1 2 2 2 h¯ κ H = πx + πy + πs − , (28) 2m " 2ζ2 # c with b b b ∂ πx = px − qAx = −ih¯ − qAx , ∂x ! b b ∂ πy = py − qAy = −ih¯ − qAy , ∂y ! b b −ih¯ ∂ πs = ps − qAs = − qAs . (29) ζ ∂s ! Our aim is to studyb the evolutionb of the beam characteristics along the optic axis of the system. For this we have to rewrite the Schrödinger equation as an s-evolution equation. To this end we proceed as follows. Let us consider a nonrelativistic monoenergetic paraxial charged particle beam propagating through the dipole magnet. We can associate the beam particle with the wave function p2 Ψ(r ,s,t)= e−iE0t/¯hψ (r ,s) , with E = 0 , (30) ⊥ ⊥ 0 2m 9 corresponding to a scattering state, where p0 is the design momentum with which the particle enters the bending magnet from the free space outside. Then, ψ (r⊥,s) satisfies the time-independent Schrödinger equation

2 2 1 2 2 2 h¯ κ Hψ (r⊥,s)= πx + πy + πs − ψ (r⊥,s)= E0ψ (r⊥,s) . (31) 2m " 2ζ2 # c b b b Rearranging the terms in Eq.(31), we can rewrite it as

2 2 2 2 2 h¯ κ πs ψ (r⊥,s) = 2mE0 − πx − πy + ψ (r⊥,s) 2ζ2 ! b 2 2 b 2 b 2 = p0 − πx − πy + p ψ (r⊥,s) , (32) where b b h¯2κ2 p2 =2mE , p2 = . (33) 0 0 2ζ2 For the dipole magnet we can take

κx2 Ax =0, Ay =0, As = −B0 x − , (34) 2ζ ! so that the magnetic ﬁeld is given by

Bx =0, By = B0, Bs =0, (35) according to Eq.(27). Then, we have

2 2 2 2 πs ψ (r⊥,s)= p0 − p⊥ + p ψ (r⊥,s) , (36) 2 2 2b b where p⊥ = px + py. Following the idea of the Feshbach-Villars representation of the Klein- Gordonb equationb b (see, e.g., [38, 39], and also [31, 42, 43]), let us now rewrite Eq.(36) as an equation for a two-component wave function. First, we write it as πs ψ1 0 1 ψ1 = 1 2 2 2 , (37) 2 (p − p + p ) 0 p0 ψ2 ! p0 0 ⊥ ! ψ2 ! b with b ψ1 ψ = πs . (38) ψ2 ! p0 ψ ! b 10 Now, let

ψ 1 ψ + ψ 1 1 1 ψ ψ + = 1 2 = 1 = M 1 . (39) ψ− ! 2 ψ1 − ψ2 ! 2 1 −1 ! ψ2 ! ψ2 ! Then, it follows that

πs ψ+ 1 −ih¯ ∂ ψ+ = − qAs p0 ψ− ! p0 ζ ∂s ! ψ− ! b 0 1 −1 ψ+ = M 1 2 2 2 M 2 (p − p + p ) 0 p0 0 ⊥ ! ψ− ! 1 2 2 1 2 2 1 − 2 (p − p ) − 2 (p − p ) 2p0 ⊥ b 2p0 ⊥ ψ+ = 1 2 2 1 2 2 .  2 (p⊥ − p ) −1+ 2 (p⊥ − p )  ψ− ! 2p0 b 2p0b   b b (40) Rearranging this equation we get

∂ ψ ψ ih¯ + = H + , ∂s ψ− ! ψ− ! b H = −p0σz + E + O,

ζ 2 2 Eb = −qζAsI +b b p⊥ − p − p0κx σz, 2p ! 0 b iζ 2 2 b O = p⊥ − p σy. (41) 2p0 b Let us consider a particle of the paraxialb beam in the free space outside the dipole just before entering the dipole. For this particle s-axis becomes z-axis and its longitudinal momentum pz will be < p0 and ≈ p0. Its transverse momentum components will be ≪ p0 and ≈ 0. In other words, for this particle pz/p0 ≈ 1. For this free particle entering the dipole we can take ipzz/¯h the wave function to be of the form ψ = φ (r⊥) e and correspondingly 1 1 we have ψ+ = 2 [ψ +(pz/p0) ψ] ≈ ψ and ψ− = 2 [ψ − (pz/p0) ψ] ≈ 0. This shows that for a particle of a forward moving paraxial beam ψ+ will be large compared to ψ−. b b In Eq.(41), I is the 2 × 2 identity matrix and

1 0 0 −i σz = , σy = , (42) 0 −1 ! i 0 !

11 are two of the well known triplet of Pauli matrices. The 2×2 matrix operator O which has nonzero entries only along the oﬀ-diagonal and couples the upper and lower components of any two-component wave function on which it acts isb known as an odd term of H. The 2 × 2 matrix operator E which has nonzero entries only along the diagonal and does not couple the upper and lower components of any two-componentb wave function on whichb it acts is known as an even term. Note that σz commutes with E and anticommutes with O: b σzE = Eσz, σzO = −Oσz. (43) b The structure of Eq.(41) isb similarb to theb structureb of the Dirac equation in which also the Hamiltonian can be split in to a constant term mc2β (like −p0σz) and an even and an odd term which, respectively, commute and anticommute with β. The Dirac wave function has four components and for a positive energy Dirac particle the upper pair of components are large compared to the lower pair of components in the nonrelativistic situation. Based purely on the commutation relations between β and the even and odd terms, same as Eq.(43) with β instead of σz, a series of Foldy-Wouthuysen transformations (see, e.g., [38, 39], and also [31, 42, 43]) leads to an expan- sion of the Dirac Hamiltonian into nonrelativistic approximation and a series of relativistic correction terms. Following a similar Foldy-Wouthuysen-like transformation technique we can expand H into paraxial approximation and a series of nonparaxial, or aberration, correction terms. We shall be interested only in the paraxial approximation. b Let us now apply the ﬁrst Foldy-Wouthuysen-like transformation:

(1) ψ+ iS1 ψ+ i (1) = e , with S1 = σzO. (44) ψ− ! ψ− ! 2p0 b b b The result is

(1) (1) ∂ ψ+ H(1) ψ+ ih¯ (1) = (1) , ∂s ψ− ! ψ− ! b ∂ H(1) = eiS1 He−iS1 − ihe¯ iS1 e−iS1 = eiS1 He−iS1 , (45) ∂s b b b b b b b b (1) b since S1 is s-independent. Calculating H , using the operator identity

b 1 1 eABe−A = B + A, B + A, A, Bb + A, A, A, B + ··· , (46) 2! 3! h i h h ii h h h iii b b b b b b b b b b b b b 12 with the commutator bracket A, B = AB − BA , we have h i b b b b b b H(1) 1 2 2 ≈ −qζAsI − ζp0 − p⊥ − ζp σz, (47) " 2p0 # b up to the paraxial approximation in which onlyb terms upto quadratic in r⊥ and p⊥ are retained. From Eq.(41) and Eq.(44) it can be seen that for a (1) (1) forward-moving paraxial beam ψ+ is large compared to ψ− , exactly as we b (1) found in the case of ψ+ and ψ−. In view of this, we can approximate H further by replacing σz by I. Thus, we write b 1 H(1) ≈ −qζA − ζp + p2 − ζp2 . (48) s 0 2p ⊥ 0 2 b 2 2 1 2 2 b We can take ζp =h ¯ κ /2ζ ≈ 2 h¯ κ (1 − κx) since κx ≪ 1 for a paraxial beam. We are interested in the s-evolution of ψ(r⊥,s). To this end, we retrace the above transformations: (1) ∂ ψ1 ∂ −1 −iS1 ψ+ ih¯ = ih¯ M e (1) ∂s ψ2 ! ∂s " ψ− !# b (1) −1 −iS1 ∂ ψ+ = M e ih¯ (1) " ∂s ψ− !# b ψ = M −1e−iS1 H(1)eiS1 M 1 . (49) ψ2 ! b b One may wonder whether such a retracing ofb the transformations would lead back to the original equation (41). It is not so because of the truncation of the series with the paraxial term and the approximation of replacing σz by I in Eq.(48). The result of this step will be the desired paraxial approximation of the original equation (41). Thus, from the equation obtained from (49) we can identify the s-evolution equation for ψ(r⊥,s) since ψ(r⊥,s) = ψ1. The result is:

∂ 1 2 1 2 ih¯ ψ (r⊥,s) = −p0 + p⊥ + qB0κx ∂s " 2p0 2 b h¯2κ2 +(qB0 − κp0) x − (1 − κx) ψ (r⊥,s) . (50) 4p0 # With the curvature of the design trajectory matched to the dipole magnetic ﬁeld as qB0 = κp0, (51)

13 we have ∂ ih¯ ψ (r ,s) = H ψ (r ,s) , ∂s ⊥ o ⊥ c 1 2 1 2 2 Ho = −p0 + p⊥ + p0κ x " 2p0 2 c b h¯2κ2 − (1 − κx) ψ (r⊥,s) . (52) 4p0 #

This Ho is the quantum charged particle beam optical Hamiltonian of the dipole magnet for a paraxial beam. c The wave function ψ (r⊥,s) represents the probability amplitude, and 2 |ψ (r⊥,s)| represents the probability, for the particle to be found at the position (x, y) in the transverse plane at the point s on the optic axis. Since Ho is Hermitian, s-evolution of ψ (r⊥,s) is unitary and the normalization ∞ 2 dxdy |ψ (r⊥,s)| = hψ(s)| ψ(s)i =1, (53) Z−∞ will be preserved. For any observable O (r⊥, p⊥) of the particle, represented by the corresponding Hermitian operator O (r⊥, p⊥), we have ∞ ∗ b O (s)= dxdy ψ (r⊥,s) Oψ (r⊥,s)=b hψ(s)| O |ψ(s)i (54) −∞ D E Z as the averageb value in the transverse planeb at s. We canb take the average value O (s) as corresponding to the value of the classical variable O in the transverseD E plane at s, following the Ehrenfest theorem (see, e.g., [40, 41], and b also [31]). We shall be interested in the s-evolution of hxi (s), hpxi (s)/p0, hyi (s), and hpyi (s)/p0. If we drop the lasth ¯-dependent term from Ho and 2 b b replace the quantum operators x and p⊥, respectively, by the corresponding b b 2 c classical variables x and p⊥ we get the classical charged particle beam optical Hamiltonian of the dipole magnet for ab paraxial beam,

1 2 1 2 2 Ho = −p0 + p⊥ + p0κ x , (55) 2p0 2 same as taken in Eq.(6). Since the quantum beam optical Hamiltonian Ho is independent of s we can integrate Eq.(52) and write c i − ¯ (s−si)Ho ψ (r⊥,s)= e h ψ (r⊥,si) , (56) b 14 or i − ¯ (s−si)Ho |ψ(s)i = e h |ψ (si)i . (57) Then, we can write the average value O b(s) for any O as D E i i ¯ (s−si)Hbo − ¯ (s−si)Ho b O (s)= hψ (si)| e h Oe h |ψ (si)i . (58) D E Using the operatorb identity in Eq.(46),b andb deﬁningb

: A : B = A, B , (59) h i we can write b b b b 2 i i i 1 i ¯ (s−si)Ho − ¯ (s−si)Ho 2 e h Oe h = I + (s − si): Ho :+ (s − si) : Ho : " h¯ 2! h¯ b b b c 3 c 1 i 3 + (s − si) : Ho : + ··· O 3! h¯ # i (s−s ):H : c b = e h¯ i o O. (60)

Thus, we write b b

i i ¯ (s−si):Ho: ¯ (s−si):Ho: O (s)= hψ (si)| e h O |ψ (si)i = e h O (si) . (61) D E It mayb be noted that, consideringb b the quantum averageb b of an observable as the value of the corresponding classical observable following the Ehren- fest theorem, the transition from the quantum transfer map in Eq.(61) to the classical transfer map in Eq.(12) is in accordance with the classical −→ quantum correspondence rule of Dirac: 1 {A, B} −→ A, B . (62) ih¯ h i b b Thus, in the transition from Eq.(61) to Eq.(12), (i/h¯) Ho , · = h i (1/ih¯) −Ho , · gets replaced by {−Ho , ·}. c Fromh Eq.(61)i it follows that c i ¯ (s−si):Ho: hxi(s) e h x (si) 1 1 i (s−s ):H :  D e h¯ i o pE (s )   p0 hpxi (s)  p0 b x i = i . (63) hyi(s)  D h¯ (s−si):Ho: E     e b y (si)  1 b b  hp i s   i   p0 y ( )   1 D ¯ (s−si):Ho: E     e h b py (si)     p0   D E  b b 15 b 2 2 Omitting from Ho the constant terms −p0 and −h¯ κ /4p0 which have zero commutator with the quantum operator corresponding to any observable, we have c 2 3 i i 1 2 1 2 2 h¯ κ px : Ho : x = p⊥ + p0κ x + x , x = , h¯ h¯ "2p0 2 4p0 # p0 b c 2 3 2 3 i px i 1 b2 1 2 2 h¯ κ px 2 h¯ κ : Ho : = p⊥ + p0κ x + x , = −κ x − 2 , h¯ p0 h¯ "2p0 2 4p0 p0 # 4p0 b b c 2 3 i i 1 b2 1 2 2 h¯ κ py : Ho : y = p⊥ + p0κ x + x, y = , h¯ h¯ "2p0 2 4p0 # p0 b c 2 3 i py i 1 b2 1 2 2 h¯ κ py : Ho : = p⊥ + p0κ x + x , =0, (64) h¯ p0 h¯ "2p0 2 4p0 p0 # b b leadingc to the quantum transferb map for the dipole hxi(s) 1 cos(κ (s − si)) κ sin(κ (s − si))0 0 1 hpxi (s) −κ sin(κ (s − s )) cos(κ (s − s )) 0 0  p0  =  i i  hyi(s) 0 01 s − si      1 hpb i (s)   0 0 01   p0 y          hxi (s ) b i 1 hp i (s ) ×  p0 x i   hyi (si)   1 b   p0 hpyi (si)   2   ¯h κ  2 (cos(κ (s − si)) − 1) 4p0 b 2 2  ¯h κ  2 sin(κ (s − si)) + 4p0 . (65)    0     0    This shows that a particle entering the dipole magnet along the design trajectory, i.e., with hxi (si)=0, hpxi (si)=0, hyi (si)=0, hpyi (si) = 0, will follow the curved design trajectory, except for some tiny quantum kicks in 2 b 2 2 b the x coordinate (∼ λ0/ρ) and the x-gradient (∼ λ0/ρ ), where λ0 is the de Broglie wavelength! The quantum transfer map in Eq.(65) is seen to con- tain the classical transfer map as the main part and the quantum correction part is negligibly small. This explains the remarkable eﬀectiveness of classical charged particle beam optics in the design and operation of bending magnets. Analysis of the quantum mechanics of some other charged particle optical elements has also led to the same conclusion (see [31]).

16 4 Concluding Remarks

To summarize, we have studied the bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet in the paraxial approximation using quantum mechanics. This involves rewriting the nonrelativistic Schrödinger equation as an equation of evolution along the curved optic axis of the system and calculating the transfer map for the quantum averages of transverse position and momentum components. Classical mechanics of the optics of dipole magnet is seen to emerge as an approximation of the quantum theory. It is found that a particle entering the dipole magnet along the design trajectory, will follow the curved design trajectory, except for some tiny 2 2 2 quantum kicks in the position (∼ λ0/ρ) and the gradient (∼ λ0/ρ ), where λ0 is the de Broglie wavelength! In the relativistic case the study of quantum mechanics of the optics of the dipole magnet using the Klein-Gordon equation leads to the same transfer map (65) with p0 replaced by its rela- 2 2 2 tivistic expression (E0/c) − m c (see [31] for details), and consequently the quantum correctionsq become still smaller due to shorter de Broglie wave- lengths. Such quantum corrections could have significant effects in the case of ultracold electron beams. The negligibility of quantum corrections explains the remarkable success of classical mechanics in the design and operation of charged particle beam optical devices from electron microscopes to particle accelerators.

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