Relativistic Electron Theory The Dirac Equation Mathematical Physics Project

Transcription

1 Relativistic Electron Theory The Dirac Equation Mathematical Physics Project Karolos POTAMIANOS Université Libre de Bruxelles Abstract This document is about relativistic quantum mechanics and more precisely about the relativistic electron theory. It presents the Dirac equation, a wave equation for massive spin- 1 2 particles. The important advances in the theory arising from that equation, such as the natural way it accounts the spin of the electron and its magnetic moment as well as the existence of the positron, are also discussed. Contents 1 Introduction 3 2 Postulates of the theory 4 3 The relativistic notation 6 4 Lorentz transformations 6 5 The Dirac wave equation 7 6 The Dirac matrices 9 7 Covariant form of the Dirac wave equation 10 8 Dirac γ matrices 11 9 Dirac wave functions The particle current density Invariance under Lorentz transformations Magnetic moment of the electron 15 1

2 CONTENTS 2 13 Solutions of the Dirac equation Exactly solvable problems The Dirac equation in an electric field The sea of negative energy Foldy-Wouthuysen representation Conclusion 24

3 1 INTRODUCTION 3 Acknowledgements I would like to thank Prof. D. Baye for his remarks on the intermediate version of this document as well for his courses which allowed me to confirm my interest in quantum mechanics and provided the incentive for my willingness to learn more about relativistic quantum mechanics. 1 Introduction In 1928, Paul Adrien Maurice Dirac ( ) discovered the relativistic equation which now bares his name while trying to overcome the difficulties of negative probability densities of the Klein-Gordon equation 1. For a long time, it was believed that the Dirac equation was the only valid equation for massive particles. It was only after Pauli reinterpreted the K-G equation as a field theory in 1934 that this belief was shaken. Even now the Dirac equation has special importance because it describes particles of spin- 1 2, which is the case of the electron as well as many of the elementary particles. In fact, it is a theoretical conjecture that all the elementary particles found in Nature obeying Fermi statistics have spin- 1 2 ([8]). It is therefore useful to study the Dirac equation, not only from a theoretical point of view but also from a practical one as some phenomena like the β decay (positron emission) can be explained by the Dirac equation, as well as some of the phenomena were the non-relativistic quantum theory is unable to explain experimental facts such as the anomalous Zeeman effect. At first, this document shall set the postulational basis of the theory, enouncing the frame in which the relativistic electron theory has been built in. This will be followed by a brief review of relativistic notations and of the Lorentz transformations. Then, we will follow Dirac s revolutionary way of thinking which will lead us to the Dirac equation for the free particle, in absence and presence of an electromagnetic field. The role of the spin as an internal degree of freedom and the existence of negative energy particles will be discussed. The next step is to apply the theory to some simple system and to see what the solutions of the equation are and how to interpret them. Finally, a trial for interpretation of the theory will require us to introduce a new representation, the Foldy-Wouthuysen representation, and to highlight its main advantages. 1 This equation is derived by inserting the operator substitutions E i t, p i into E 2 = c 2 p 2 + µ 2 c 4, the relativistic relation between energy and momentum for a free particle of mass µ.

4 2 POSTULATES OF THE THEORY 4 2 Postulates of the theory The relativistic electron theory being a quantum mechanical theory, certain of its postulates are common to general quantum mechanical theories. However, the relativistic theory is consistent with the special principle of relativity. The postulates of the theory are listed here and will be followed by a brief discussion (see [7] for more details). For a more complete discussion, the reader is referred to the work of Dirac [3]. I. The theory shall be formulated in terms of a field, quantitatively represented by an amplitude function ψ, in such way that the statistical interpretation of quantum phenomena will be valid. II. The description of physical phenomena in the theory will be based on an equation of motion describing the development in time of the system or of the field amplitude ψ. III. The superposition principle shall hold, requiring the equation of motion to be linear in ψ. IV. The equations of motion must be consistent with the special principle of relativity 2. This requires that they may be written in covariant form. V. From postulate I, it must be possible to define a probability density ρ such that it is positive definite : ρ 0 and that its space integral satisfies : ρ d 3 x = ρ d 3 x (1a) d dt ρ d 3 x = 0 (1b) Condition (1a) expresses that ρ is a relativistic invariant and both conditions permit a Lorentz-invariant meaning to a normalization condition such as ρ d 3 x = 1 2 Since general relativity is required only when dealing with gravitational forces, which are quite unimportant in atomic phenomena, there is no need to make the theory conform to general relativity.

5 2 POSTULATES OF THE THEORY 5 VI. The theory should be consistent with the correspondence principle and in its non-relativistic limit should reduce to the standard form of quantum mechanics applicable at low velocities. Furthermore, in its non-quantum limit, the theory should yield the mechanics of special relativity. Postulates I and III appear to be necessary in view of such experimental facts as scattering and the attendant diffraction effects observed in such phenomena. The ψ-function referred to will be called a wave function. It will, in general depend on the four space-time coordinates x µ and may be a multi-component wave function (as if the theory is to account spin properties). Postulate II implies the existence of an operator equation of the form : Hψ = i ψ t, (2a) or, switching to a natural unit system by setting = c = 1 (as it will be the case in this document) : Hψ = i ψ t. (2b) In connection with postulate IV, the occurrence of the first time derivative in the equation of motion implies the space derivatives should also occur to first order. The obvious requirement of symmetry in all four space-time variables is clearly not fulfilled by the non-relativistic form of the quantum mechanics. Although the required symmetrical appearance of the four x µ in the equations of motion is not a sufficient condition for relativistic covariance and therefore this covariance must, and will be, demonstrated. About postulate V, the fact that ρ is positive definite implies we speak of a particle and not a charge density. It is not clear whether the goal of ρ being positive definite is attainable in a given theory. We also have to notice that (1b) is assured if a continuity equation exists and if ψ vanishes sufficiently strongly at the boundaries of the system. That is, a particle current density j must exist such that j + ρ t = 0. (3) This equation has the usual interpretation that a particle cannot disappear from a volume of space unless it crosses the surface bounding that volume. In fact, electrons can actually do this by means of pair annihilation. Thus creation or destruction of particles and antiparticles contradict the conservation of particles but not the conservation of charge. There is clearly a contradiction here. However, this difficulty, which disappears in a quantized field theory, raises no real problem in the questions discussed in this document.

6 3 THE RELATIVISTIC NOTATION 6 3 The relativistic notation Before starting on the path toward developing the relativistic wave equation, a few words on relativistic notation are in order. In relativity all 4-vectors and their transformations are the most important quantities. The most fundamental 4-vector is the one that describes space and time, x µ = (t, x, y, z). Its transformation properties are defined in terms of the invariant quantity s 2 = x µ x µ = t 2 x 2 y 2 z 2. This introduces a second quantity x µ = (t, x, y, z). The 4-vector with the upper index is a contravariant vector, while that with the lower index is a covariant vector. To transform between the two types of vectors, we introduce the metric tensor: g µν = xµ = g µν x ν (4) The momentum p, whose components will be written p 1, p 2, p 3, is equal to the operator p r = i x r (r = 1, 2, 3). (5) To bring (5) into a relativistic theory, we must first write it with balanced suffixes, p r = i / x r (r = 1, 2, 3), and extend it to the complete 4-vector notation, p µ = i x µ. (6) We thus have to introduce p 0, equal to the operator i / x 0. Since the last forms a 4-vector when combined with the momenta p r, it must have the physical meaning of the energy of the particle divided by c. We now have to develop the theory treating the four p s on the same footing, just like the four x s. 4 Lorentz transformations The Lorentz group is the group of transformations that preserves the length s 2 = x µ x µ. Some of the continuous transformations that do this are the ordinary (space) rotations and the boosts or imaginary rotations, which correspond to passing from one inertial frame to another one moving relative to the first. For example, the homogenous Lorentz transformation to a frame with

7 5 THE DIRAC WAVE EQUATION 7 velocity v along the x 1 -axis is given by: γ γβ 0 0 Λ µ ν = γβ γ x µ = Λ µ ν x ν, (7) where γ = 1/ 1 β 2 and β = v/c = v. Those transformations, satisfying det Λ = +1, constitute the subgroup of the proper Lorentz transformations while the transformations satisfying det Λ = 1 constitute the improper transformation subgroup. The latter include: (a) space reflections : Λ ik = δ ik, Λ 00 = 1, Λ j0 = Λ 0j = 0 (b) time reflections : Λ ik = δ ik, Λ 00 = 1, Λ j0 = Λ 0j = 0 (c) as well as any product of a proper transformation with a space or time reflection. The covariant vector transforms differently from the contravariant vector x µ = Λ ν µx ν where the two different transformations are defined by the invariance of x µ x µ = x ν Λ µ ν Λ ν µx ν. This imposes the condition that Λ µ ν Λ ν µ = 1, that is they are inverse transformations of each other: γ γβ 0 0 Λ ν µ = γβ γ x µ = Λ ν µx ν (8) The Dirac wave equation The reasoning followed here is, at least for its first part, inspired by Dirac s book [3]. Let us consider the case of the motion of an electron in the absence of an electromagnetic field, so that the problem is that of the free particle, with the possible addition of internal degrees of freedom. The relativistic Hamiltonian provided by the classical mechanics (through the relation E = p 2 + m 2 ) leads to the wave equation {p 0 (m 2 + p p p 2 3) 1 2 }Ψ = 0 (9) where the p s are interpreted as in (6). This equation is however very unsatisfactory as it is very unsymmetrical between p 0 and the other p s. We must therefore search for an other wave equation.

8 5 THE DIRAC WAVE EQUATION 8 Multiplying (9) on the left by {p 0 + (m 2 + p p2 2 + p2 3 ) 1 2 }, we obtain the equation {p 2 0 m 2 p 2 1 p 2 2 p 2 3}Ψ = 0, (10) which is of a relativistically invariant form. However equation (10) is not completely equivalent to (9) since every solution of (10) is not solution of (9), although the converse is true. At this point, equation (10) is not of the form required by the laws of the quantum theory on account of its being quadratic in p 0. We need a wave equation linear in p 0 and roughly equivalent to (10). In order to transform in a simple way under Lorentz transformations, we shall try to make that equation rational and linear in p µ, and thus of the form {p 0 α 1 p 1 α 2 p 2 α 3 p 3 βm}ψ = 0, (11) in which the α s and β are independent of the x s and the p s. They therefore describe some new degree of freedom, belonging to some internal motion in the electron. In fact, as we shall see, they bring in the spin of the electron. Multiplying (11) by {p 0 +α 1 p 1 +α 2 p 2 +α 3 p 3 +βm} on the left, we obtain {p 2 0 [ α 2 i p 2 i + (α i α j + α j α i ) p i p j + (α i β + βα i ) p i m + β 2 m 2] }ψ = 0, (12) summation being implied over repeated suffixes, with the imposed condition i > j. This is the same as (10) with the α s and β satisfying α 2 i = β 2 = 1 ; α i α j + α j α i = 2δ ij ; α i β + βα i = 0. (13) Thus by giving suitable properties to the α s and β we can make equation (11) equivalent to (10), in so far as the motion of an electron as a whole is concerned. We may now assume that or in the (2b) form, {p 0 [α p + βm]}ψ = 0, (14) Eψ = i ψ t = [α p + βm] ψ = H Dψ, (15) is the correct relativistic equation for the motion of an electron in the absence of a field. Taken into account that this equation is not exactly equivalent to (9), we shall, at the moment, consider only those solutions corresponding to positive values of p 0, the negative values not corresponding to any actually observable motion of an electron. We shall come back to that point later. To generalize this equation to the case when there is an electromagnetic field present, we follow the classical rule of replacing p 0 and p by p 0 qa 0

9 6 THE DIRAC MATRICES 9 and p qa, A 0 and A being the scalar and vector potentials of the field at the place where the particle is. This gives the equation {p 0 qa 0 [α (p qa)] βm}ψ = 0, (16) the Hamiltonian of the energy being and H F D = α π + βm + qa 0 (17) π = p qa (18) being the standard kinetic momentum operator in the general case of a particle with a charge q. For an electron, π = p + ea. 6 The Dirac matrices It is obvious that relations (13) require the α s and β to be matrices. To determine the form of the matrices, some conditions need to be imposed : - The wave function should be a column vector in order that the probability density be given as ψ ψ 3. This imposes the condition that the matrices must be square. - The Hamiltonian must be hermitian so that its eigenvalues are real. This forces the four matrices to also be hermitian. The α s and β have similar properties to the Pauli σ matrices (21), which are 2 2 matrices. However, so long as we keep working with 2 2 matrices, we can get a representation of no more than three anticommuting quantities. The rank n of those matrices must be even. This can be shown by observing that for each of the four matrices there is another matrix which anti-commutes with it. Therefore, if b µ is any of the four matrices and b ν is a matrix which anti-commutes with b µ, we have Tr[b µ ] = Tr[b µ b 2 ν] = Tr[b ν b µ b ν ] = Tr[b µ b 2 ν] = 0 (19) since each b 2 ν = 1 and Tr[AB] = Tr[BA]. Each matrix has thus zero trace. There exists a representation in which any b µ can be brought to diagonal form, and, since b 2 µ = 1 and Tr[b µ ] = 0 are independent of the representation, we conclude that the eigenvalues of b µ in diagonal form are ±1 and that there are as many +1 as -1 eigenvalues. Thus the number of rows and columns must be even. 3 Notation denotes the conjugate transpose

10 7 COVARIANT FORM OF THE DIRAC WAVE EQUATION 10 The minimum possible number for n is 4, and a 4 4 representation exist. For example, ( ) ( ) 0 σ I2 0 α =, β =, (20) σ 0 0 I 2 where I 2 is a 2 2 unity matrix and σ represents the three Pauli matrices ( ) ( ) ( ) i 1 0 σ 1 =, σ =, σ i 0 3 =. (21) 0 1 If we consider the direct matrix product between two matrices operating in different spaces, we can write all the Dirac matrices previously defined as a direct product of two 2 2 matrices : one operating in the Dirac space referring to the four areas of the 4 4 matrices and the other operating in the Pauli space referring to the four elements within each of these four areas. Thus α j = ρ 1 σ j, β = ρ 3 I 2 (22) where the three matrices operating in Dirac space ( ) ( ) ( ) i 1 0 ρ 1 =, ρ =, ρ i 0 3 = 0 1 (23) form, with I 2, a complete set like I 2, σ 1, σ 2, σ 3. Since it is to be understood that the direct product is always implied for matrices operating in different spaces, the symbol can be omitted. 7 Covariant form of the Dirac wave equation Although being in Hamiltonian form the Dirac equation given above (15) doesn t include time and space coordinates in a symmetric manner. To transform the equation, we first need rewriting it using the usual operator substitutions (i.e. E i / t, p i ) before multiplying its both sides by β on the left: [ iα + βm] ψ = i ψ t [ iβα + m] ψ = iβ ψ t (24) We can now introduce the Dirac γ matrices γ µ = (β, βα) and rewrite the Dirac equation as : [iγ µ µ m] ψ = 0 ( µ = / x µ ), (25) which puts both time and position coordinates on an equal footing.

11 8 DIRAC γ MATRICES 11 8 Dirac γ matrices We now consider the complete set of matrices which can be constructed from the four γ µ matrices defined in the previous section by multiplications. There are 16 different matrices γ A which can be formed this way. Those can be classified into five groups (as done in [7]): - Group S. This consists of a single matrix, the identity matrix. It can be formed by at least four ways : (γ µ ) 2 = 1. - Group V. These are just the four γ µ matrices. - Group T. These are the six matrices formed by the relation iγ µ γ ν (µ ν), the phase factor i being taken to have in all cases (γ A ) 2 = 1 - Group P. This is the single matrix formed by multiplying all four γ µ : γ 5 = γ 0 γ 1 γ 2 γ 3. - Group A. These are the four possible products formed by products of three γ µ : iγ µ γ ν γ ξ (µ ν ξ). These can be written using the γ 5 matrix in the form iγ 5 γ µ. The designation group used above does not mean that these 16 matrices form a group in the technical sense. Nevertheless, this set does form a mathematical entity : a Clifford algebra. Proceeding from the rules in (13), some relations can be derived for the γ matrices : - Multiplying βα i + α i β = 0 by β from the left, we get : β(βα i ) + (βα i )β = γ 0 γ i + γ i γ 0 = 0. (26) - Now taking α i α j + α j α i = 2δ ij and multiplying it from both sides by β, we get, using the αβ anti-commutation relation : (βα i )(α j β) + (βα j )(α i β) = 2δ ij ββ γ i γ j + γ j γ i = 2δ ij. (27) - Putting the previous two equations together yields: γ µ γ ν + γ ν γ µ = {γ µ, γ ν } = 2g µν (28)

12 9 DIRAC WAVE FUNCTIONS 12 - The hermiticity of the γ µ matrices can be derived in a similar manner, as the α s and β are hermitian. This is obviously the case for γ 0 = β. The other components are given by: (γ i ) = (βα i ) = (α i β) = γ i (29) and these components are shown to be anti-hermitian. For more information about the γ matrices, the reader is referred to [8] and [7]. 9 Dirac wave functions Each wave function is a 4-component vector with 4 rows and 1 column ψ 1 (r, t) ( ) ψ (r, t) = ψ 2 (r, t) ψ ψ 3 (r, t) = u ψ l (30) ψ 4 (r, t) where ψ u and ψ l refer to upper and lower and are each two component spinors. The spin of the electron requires the wave function to have two components 4. The fact our theory gives four is due to our wave equation (11) having twice as many solutions as it ought to have, half of them corresponding to states of negative energy (p 0 < 0). Looking at how operators act on the four-component wave functions, we may for example calculate ( ) iψ l ρ 2 ψ = iψ u (31) or ( ) σψ l αψ = ρ 1 σψ = σψ u. (32) So the matrices operating in the Dirac space act on ψ u and ψ l while the matrices operating in Pauli space act on the two components in ψ u (ψ 1, ψ 2 ) and in ψ l (ψ 3, ψ 4 ). The four-component ψ will be called a spinor (or fourspinor). The Dirac matrices from section 6, like ρ 3, with zero elements in the upper right and lower left quadrants are called even in the Dirac sense; those, like ρ 1 and ρ 2, with zeroes in the upper left and lower right quadrants are called odd. Even Dirac matrices couple ψ u with ψ u and ψ l with ψ l while odd ones couple ψ u and ψ l. This will, as we shall see, have some consequences. 4 In fact, the appearance of a multi-component wave function is characteristic of the existence of a non-vanishing spin [7].

13 10 THE PARTICLE CURRENT DENSITY The particle current density With the Dirac equation (15) and a wave function of the form of a 4- component vector (30), we can have a look at the associated probability current. Taking (15) multiplied on the left by ψ and substracting with its adjoint multiplied on the right by ψ : ψ i t ψ = ψ ( iα + βm)ψ, (i t ψ )ψ = (i ψ α + mψ β)ψ, (33a) (33b) resulting in i t (ψ ψ) = i (ψ αψ), (34) which is the continuity equation (3), expressing the conservation of probability density if we define it the usual way, i.e. as and the probability current as ρ = j 0 = ψ ψ, (35) j = ψ αψ. (36) The postulated property of ρ (1b) is automatically valid with H hermitian and with (35), which is obviously positive definite. For, if (35) is assumed, we have ρ [ t d3 x = i ψ Hψ (Hψ) ψ] d 3 x = 0 (37) by virtue of H = H. Thus with the wave equation defined and the form of the wave function known, (35) allows us to specify the current density j implied by postulate V. We now have the constancy of the total probability of finding the electron at any point of space (37). We have now apparently solved the problem of finding a relativistic generalization of the Schrödinger equation. 11 Invariance under Lorentz transformations From the previous section, it seems like we have our relativistic generalization completed. But we must still verify the invariance of the Dirac equation under Lorentz transformations.

14 11 INVARIANCE UNDER LORENTZ TRANSFORMATIONS 14 As in the preceding section we derived the continuity equation using α and β, we will for this section use the γ matrices which appear in the covariant form of the equation. Starting with the covariant form of the Dirac equation (25) from section 7, we will show (as in [8]) that the Dirac equation is form invariant under an inhomogeneous Lorentz transformation if we define x = Λx + a (38) ψ (x ) = S(Λ) ψ(x) = S(Λ) ψ(λ 1 (x a)), (39) where S(Λ) is a 4 4 matrix operating on the components of ψ satisfying As the Dirac equation S 1 (Λ) γ λ S(Λ) = Λ λ µγ µ. (40) x µ = x ν x µ x ν = Λν µ ν, (41) [iγ µ µ m] ψ(x) = 0 (42) can be re-expressed in the form ] [iλ νµγ µ S 1 x ν ms 1 ψ (x ) = 0, (43) provided that the γ matrices remain unaltered under Lorentz transformation. Multiplying (43) by S on the left yields [ is(λ ν µ γ µ )S 1 ν m ] ψ (x ) = 0, (44) which is the same as (42) provided that S satisfies (40). Theorem Fundamental Pauli theorem [7] If two sets of matrices γ µ and γ λ obey the commutation rules (28), then there must exist a non-singular matrix S which connects the two sets according to γ λ S = Sγ µ. (45) The fundamental theorem of Pauli guarantees the existence of a nonsingular S. In fact, the condition (40) uniquely determines S up to a factor([8]). We now know the Dirac wave equation is form invariant under any Lorentz transformation. The equation now fulfills the main requirements for being a relativistic generalization of the Schrödinger equation.

15 12 MAGNETIC MOMENT OF THE ELECTRON Magnetic moment of the electron As we have defined the frame we will be working in, we will use this basis to highlight some of the advances the theory brought in. Following Dirac in [3], we will start with one of the biggest success of Dirac s theory: the theoretical explanation of the electron having a magnetic moment. Suppose we put the electron in a magnetic field B = A. The Hamiltonian (17) determines the equation of motion. From it, we get for the electron (H F D + ea 0 ) 2 = (α π + βm) 2 = (α π) 2 + m 2 = π 2 + m 2 + e Σ B, (46) as, using the relation (σ B)(σ C) = (B C) + i[σ (B C)] (47) and introducing the spin matrix Σ = I 2 σ = (ρ 1 I 2 ) α = ( ) σ 0 0 σ (48) to make a distinction with the 2 2 Pauli matrices, (α π) 2 = π 2 + iσ (π π) = π 2 + e Σ B, (49) with π π = ie A = ie B. In the non-relativistic limit, i.e. for an electron moving slowly, with a small momentum, we may expect an Hamiltonian of the form m+h 1, where H 1 is small compared to m. Putting this Hamiltonian for HD F in (46) and neglecting H1 2 and other terms involving e2, we get, on dividing by 2m, H 1 + ea 0 = 1 2m (π2 + eσ B). (50) The Hamiltonian given by this last equation is the same as the classical Hamiltonian for a slow electron, except for his last term, e Σ B, 2m which may be considered as an additional potential energy which a slow electron has. This extra energy can be interpreted as arising from the electron having a magnetic moment µ = e Σ, (51) 2m

16 13 SOLUTIONS OF THE DIRAC EQUATION 16 which implies that the g-factor of the electron is 2, which is very nearly the case. It is remarkable to notice that the Uhlenbeck-Goudsmit hypothesis, which is that the observed spectral features on the anomalous Zeeman effect are matched by assigning to the electron a magnetic moment given in terms of the operator µ = e/m s, where s = 1 2 σ, emerges from the Dirac theory. This discussion suggests that the relativistic particle has a intrinsic angular momentum 1/2, so that the total angular momentum is J = L + 1 Σ, L = r p. (52) 2 The Hamiltonian from (15) commutes with the total angular momentum [J, H] = [L, H] + 1 [Σ, H] = iα p iα p = 0, (53) 2 which can be obtained using Σ = 1 2i α α. invariance of the Dirac equation. This verifies the rotational 13 Solutions of the Dirac equation The Dirac equation admits of plane wave solutions of the form ψ(x) = e ip r u(p) (54) where u(p) is a four-component spinor which satisfies the equation (γ p m) u(p) = (γ µ p µ m) u(p) = 0. (55) Equation (55) is a system of four linear homogenous equations for the components u µ, for which non trivial solution exist only if det(γ µ p µ m) = (p 2 0 p 2 m 2 ) 2 = 0. (56) Solutions therefore only exist if p 2 0 = p2 + m 2, i.e. only if p 0 = ± p 2 + m 2. Let u + (p) be a solution for p 0 = E(p) = + p 2 + m 2 so that u + (p) satisfies the Dirac equation (α p + βm) u + (p) = E(p) u + (p). (57) Using the decomposition of the wave function in Dirac space, like in the second relation of (30), we may write ( ) uu u + =, u l

17 13 SOLUTIONS OF THE DIRAC EQUATION 17 where u u and u l have two components each, and, adopting the representation (20) for α and β, we find that u u and u l obey the following equations: (σ p) u l + m u u = E(p) u u (58a) (σ p) u u m u l = E(p) u l. (58b) Since E(p) + m 0, u l = and substituting this value back into (58a), we find Using (47), (σ p) 2 = p 2 and σ p E(p) + m u u (59) (σ p)2 ( E(p) + m + m) u u = E(p) u u. (60) p 2 E(p) + m = E2 (p) m 2 E(p) + m = E(p) m, we get equation (60) is identically satisfied. There are therefore two linearly independent positive energy solutions for each momentum p, which correspond, for example, to choosing u u equal to ( ) ( ) 1 0 or, 0 1 which are respectively equal to χ +1/2 and χ 1/2 when using Pauli s spinor notation. This can also be seen using the operators and some of their properties. The Hamiltonian operator H D = α p + βm commutes with the hermitian operator s(p) = Σ p, (61) p where Σ is defined by (48). s(p) is the helicity operator, or helicity of the particle, and physically corresponds to the spin of the particle parallel to the direction of motion. The solutions can therefore be chosen as simultaneous eigenfunctions of H

18 14 EXACTLY SOLVABLE PROBLEMS 18 and s(p). Since s 2 (p) = 1, the eigenvalues of s(p) are ±1. The solutions can therefore be classified according to the eigenvalues +1 or 1. A similar classification can be made for the negative energy solutions for which p 0 = p 2 + m 2 and where, for a given momentum, there are again two linearly independent solutions. So, for a given four-momentum p, there are four linearly independent solutions of the Dirac equation. These are characterized by p 0 = ±E(p) and s(p) = ±1. As an example, we may explicit two linearly independent solutions for positive energy and momentum p : ( ) u (u) + (p) = E(p) + m (62a) 2E(p) u (l) + (p) = ( E(p) + m 2E(p) χ +1/2 σ p E(p)+m χ+1/2 χ 1/2 σ p E(p)+m χ 1/2 ), (62b) where the normalization constant is determined by the requirement that u u = 1. In the non-relativistic limit (v 1 p = mv and E(p) m) 5, the components u l of a positive energy solution are of order v times u u and therefore small. For a negative energy particle, it is the two upper components of the wave function who will be small. 14 Exactly solvable problems There are only few problems for which the Dirac equation can be solved exactly ([8]). Some of them are, in (3+1)-dimensional space-time : - The Coulomb potential. - The case of a homogeneous magnetic field extending over all space. - The field of an electromagnetic plane wave. - The so-called Dirac oscillator, which is a relativistic extension of the oscillator problem. In (2+1)-dimensional space-time, we may cite the Dirac oscillator. 5 As we are in natural units for quantum mechanics, i.e. = c = 1.

19 15 THE DIRAC EQUATION IN AN ELECTRIC FIELD The Dirac equation in an electric field As in [6], we are starting with the Dirac Hamiltonian in the presence of a filed (17), we may, as we did in section 13, express the Dirac equation as two coupled equations with time independent solutions u u, u l : From those, we get by substitution [σ π]u l + [m qa 0 ]u u = Eu u (63a) [σ π]u u [m qa 0 ]u l = Eu l. (63b) 1 (σ π) (σ π)u u + qa 0 u u = (E m)u u. (64) E + m qa 0 Now we will assume A = 0, E = E + m and that We thus have 1 E 1 + 2m qa 0 2m [1 E qa 0 ]. (65) 2m 1 2m [1 E qa 0 2m ](σ )2 u u q 4m 2 (σ A 0)(σ u u )+qa 0 u u = E u u. (66) Using the relation (47) and assuming that A 0 (r) is spherically symmetric, we get, as the orbital momentum operator L = r p and the Pauli spin operator s = 1 2 σ: [ 1 2m 2 +qa qa 0 2m [E 2m ] 2 + q 1 2m 2 r knowing that da 0 dr L s q da 0 4m 2 dr A 0 (r) = 1 da 0 r dr r A 0 u u = da 0 u u dr r iσ [ A 0 u u ] = 2 1 da 0 r dr L s Let us now look at equation (67) more closely: r ]u u = E u u, (67) - The first and second term are in the non-relativistic Hamiltonian for a particle of mass m and charge q in a central potential A 0 (r). - The third term is a relativistic correction to the kinetic energy operator. It can be written as E qa 0 p 2 /2m. 1 qa 0 2m [E 2m ] 2 p4 8m 3, (68)

20 16 THE SEA OF NEGATIVE ENERGY 20 - The fourth term is the spin-orbit interaction. - The fifth in non-hermitian. C.G. Darwin showed [2] it could be written as q 8m 2 2 A 0 (r), (69) which is 4π 8m 2 ( Ze2 )δ(r) 4πɛ 0 (70) for a Coulomb potential. This term only affects the s-states. It comes from the fact that, in quantum mechanics, the electrons wavefunction is spread out. In the nonrelativistic limit, the electron therefore feels the electric field of the proton over a finite volume of approximate radius given by the Compton wavelength of the electron, /mc ([8]). The Dirac equation can be solved exactly for the hydrogen atom. The energy eigenvalues are given by ([8]) : Enj D Zα = m{1 + ( n j 1/2 + (j + 1/2) Z 2 α 2 } 1/2, (71) where α = e2 4π c If we expand, we get ] Enj D = m [1 (Zα)2 2n 2 + (Zα)4 (6j + 3 8n) 8(2j + 1)n 4 + O(Zα) 6, (72) which leads to where E nj = E D nj m = E (0) n [ ] 1 + (Zα)2 (6j + 3 8n) 4(2j + 1)n 2 + O(Zα) 4, (73) E n (0) = m(zα)2 2n 2 (74) are the eigenvalues obtained with the non-relativistic Schrödinger equation. 16 The sea of negative energy We have noted that the Dirac equation admits of negative energy solutions. Their interpretation presented a great deal of difficulty for some time; as for example the fact a negative energy particle would be accelerated in the opposite direction of the external force. In a classical theory, the negative energy states cause no trouble because no transition between positive and negative energy states occur. Therefore, if a particle occupies a positive energy state at any time, it will never appear

21 16 THE SEA OF NEGATIVE ENERGY 21 in a negative energy state. The anomalous negative energy states are then eliminated as a result of initial conditions stipulating that no such state occurred in the past. In a quantum theory, this device is no longer admissible, as spontaneous emission of radiation can occur as long as a state of lower energy is unoccupied and as long as conservation of angular and linear momenta can be fulfilled. These conservation principles can always be fulfilled under appropriate conditions. There is nothing to prevent an electron from radiating energy in making a transition to lower and lower states. ([7]) In 1930, Dirac resolved the difficulties of interpretation by suggesting his so-called hole theory which he formulated as follows ([3]) : Assume that nearly all the negative energy states are occupied, with one electron in each state in accordance with the exclusion principle of Pauli. The exclusion principle makes it impossible for positive energy electrons to make transition to negative energy states unless they are emptied by some means. Such an unoccupied negative energy state will now appear as something with positive energy, since to make it disappear, i.e. to fill it up, we should have to add an electron with negative energy. We assume that these unoccupied negative-energy states are the positrons. The hole would have a charge opposite of that of the positive energy particle. The positron was experimentally discovered in 1932 by Carl D. Anderson [1]. It has the same spin operator as the electron but has opposite energy, momentum and angular momentum operators. Therefore it has also the opposite helicity, which is well known as an experimental result in beta decay [7]. Dirac also suggested there has to be a distribution of electrons of infinite density everywhere in the world and that a perfect vacuum is a region where all the states of positive energy are unoccupied and all those of negative energy are occupied. However, this infinite distribution doesn t contribute to the electric field, as, of course, Maxwell s equation in a perfect vacuum, E = 0, must be valid. Thus only departures from the distribution in a vacuum will contribute to the electric density. There will be a contribution e for each occupied state of positive energy and a contribution +e for each unoccupied state of negative energy. The exclusion principle will operate to prevent a positive-energy electron ordinarily from making transitions to states of negative energy. It will still be possible, however, for such an electron to drop into an unoccupied state of negative energy. In this case we should have an electron and a positron disappearing simultaneously, their energy being emitted in the form of radiation. The converse process would consist in the creation of an electron and a positron from electromagnetic radiation.

22 17 FOLDY-WOUTHUYSEN REPRESENTATION 22 Although the prediction of the positron is a brilliant success of Dirac s theory, some questions still arise. With a completely filled negative energy sea, the theory can no longer be a single-particle theory. The treatment of problems of electrodynamics is complicated by the requisite elaborate structure of the vacuum. However, the effects of the crowded vacuum on the mass and charge of a Dirac particle is to change them to new values, which must be identified with the observed mass and charge. 17 Foldy-Wouthuysen representation The Dirac equation in the form described above does not lend itself easily to a simple interpretation. Let us consider for example the operator ẋ = i [H D, x] = α, (75) which we would like to call the velocity operator. Since αi 2 = 1, the absolute magnitude of the velocity in any given direction is always 1, which is, since we have set c = = 1, the speed of light. This is, of course, not physically reasonable. From this example, we have to conclude that there must exist another representation of the Dirac equation in which the physical interpretation is more transparent. This can also be inferred from the fact that the two independent states associated with each value of the momentum of a positive energy Dirac particle, which correspond to the two possible directions of the spin, have, according to quantum mechanics, to be represented by exactly two vectors in Hilbert space ([8]). There exist therefore a redundancy in the representation of those vectors. This problem was solved in 1950 by Foldy and Wouthuysen ([4]) who noticed that the main reason for this redundancy is the presence of odd operators, i.e., as has been said before, an operator which connects upper and lower components of the wave function. If it were possible to perform a canonical transformation on the Hamiltonian H D and bring it to a form free of odd operators, it would be possible to represent the solutions by two-component spinors. The suggested transformation, ψ e is ψ = ψ (76a) with S of the form H e is H D e is = H D, (76b) S = ( i )βα pω( p ), (77) 2m m

23 17 FOLDY-WOUTHUYSEN REPRESENTATION 23 ω being a real function to be determined such that H is free of odd operators, leads to a new position operator X = e is xe is = x + i βα β(α p) p + i [σ p] p i 2E(p) 2E(p) (E(p) + m) p (78) and to a new spin operator, called the main spin operator, Σ M = Σ iβ(α p) E(p) p (σ p) E(p) (E(p) + m), (79) where Σ = e is Σ M e is = 1 (α α) (80) 2i is the spin operator defined by (48). The complete reasoning can be followed in the original article [4] as well as in [8]. Here are some of the consequences of this canonical transformation: - In this representation, positive and negative energy states are separately represented by two-component wave functions. - Position and spin operators differ from the conventional representation. - The components of the time derivative of the new position operator all commute and have for eigenvalues all values between 1 and +1, i.e. between c and +c in non-reduced units. - The new spin operator is now a constant of the motion, which was not the case before. - It is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The Foldy-Wouthuysen representation is particulary useful for the discussion of the non-relativistic limit of the Dirac equation, since the operators representing physical quantities are in one-to-one correspondence with the operators of the Pauli theory. There exists also another limit which is of a considerable interest, namely the ultrarelativistic, where the mass of the particle can be neglected in comparison with its kinetic energy. Such a form of the Dirac equation is obtained by choosing an appropriate ω (see [8] for more details).

24 18 CONCLUSION Conclusion In this document, we have built up a part of the Dirac theory. We have used it in the case of an electron in a magnetic and a electric field. However, only a small part of the potential of the theory has been discussed. This theory has been the subject of many studies, articles and books and is still today an important field of research in quantum mechanics, as it provides a practical way to study relativistic problems. The interested reader is referred to the references for further readings in the incredible world of relativistic quantum mechanics, which is a gate to Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD).

25 REFERENCES 25 References [1] C. D. Anderson, Phys. Rev., 43 (1933), p [2] C. G. Darwin, Proc. Roy. Soc (London), A118 (1928), p [3] P. Dirac, The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Ely House, London W.1, [4] L. L. Foldy and S. A. Wouthuysen, On the dirac theory of spin 1/2 particles and its non-relativistic limit, Physical Review, 71 (1950), p. 29. [5] W. Greiner, Relativistic Quantum Mechanics - Wave Equations, Springer-Verlag, New York, [6] D. Ritchie, Advanced quantum physics lectures 2004/5. Semiconductor Physics Group, University of Cambridge dar11/pdf/. [7] M. Rose, Relativistic Electron Theory, John Wiley & Sons, Inc., New York, [8] S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson and Company, Elmsford, New York, 1961.