January 2016 PX408: Relativistic Quantum Mechanics Tim Gershon (T.J.Gershon@warwick.ac.uk) Handout 1: Revision & Notation Relativistic quantum mechanics, as its name implies, can be thought of as the bringing together of relativity (specifically, special relativity) and (non-relativistic) quantum mechanics. Therefore let us start by revising these constituents. One of the purposes of this revision is to define the notations that will be used. Relativity Einstein s principle of special relativity can be formulated thus The laws of physics are the same in all inertial frames of reference. This has several far reaching consequences, including The speed of light c is constant. The world s most famous equation: E = mc 2. (Note that m in this equation is not a constant, rather m = γm 0 where m 0 is the (constant) rest mass and γ is defined below. However, in what follows we will use m to be the rest mass.) The relativistic energy-momentum relation: E 2 = p 2 c 2 + m 2 0 c4 Space and time, rather than being distinct entities, become mixed together as components of the 4-dimensional space-time any relativistically invariant theory has to treat space and time on the same footing To illustrate the last point, consider the Lorentz transformation caused by a boost along the x axis to an inertial frame with relative velocity (in units of the speed of light) β = v/c. x = ( ct x ) = ct x y z ct x y z = γ(ct βx) γ(x βct) y z (1) Here γ = (1 β 2 ) 1/2. Note that we are using a notation in which the time-like component appears as the zeroth element of the 4-vector: x 0 = ct (some texts put the time-like component as the fourth element, which is then labelled x 4 ). The Lorentz transformation can be written in various ways. One useful form uses a matrix: ct x y z ct x y z = γ βγ 0 0 βγ γ 0 0 0 0 1 0 0 0 0 1 ct x y z (2) To clarify what is meant by 4-vector, let us define it clearly (albeit mathematically)
A contravariant 4-vector a µ is something that transforms in the same way as the space-time transformations given above, i.e. as a µ a µ = x µ x ν aν (3) Meanwhile, a covariant 4-vector a µ is something that transforms as a µ a µ = xν x µ a ν (4) Note the use of the summation convention repeated indices are summed over. (The repeated index can be referred to as a dummy variable.) It is conventional to use a superscript for contravariant 4-vectors and a subscript for covariant 4-vectors. Examples of contravariant 4-vectors are 4-position: x µ = (ct, x, y, z) 4-momentum: p µ = ( 1 c E, p x, p y, p z ) 4-potential: A µ = ( 1 c φ, A x, A y, A z ) (where φ and A are the electric and magnetic potentials, respectively see page 4) and an example of a covariant 4-vector is differential vector operator: µ = ( 1 c, x, y, z ) Using this notation, the transformation of Eq. (1) (and Eq. (3)) can be written where in matrix form, as in Eq. (2) Λ µ ν = x µ x µ = Λ µ ν x ν (5) γ βγ 0 0 βγ γ 0 0 0 0 1 0 0 0 0 1 Meanwhile, the covariant transform of Eq. (4) can be written (6) a µ a µ = a ν (Λ µ ν ) 1 (7) where (Λ µ ν ) 1 is the inverse Lorentz transformation, and can be represented by a matrix that is the inverse of that given in Eq. (6). Here are some example questions to help revise four vectors. Q1 A J/ψ particle decays at rest to a µ + µ pair. What are the energy and momenta of the muons? (The J/ψ and µ masses are 3.1 GeV and 106 MeV, respectively.) Q2 An electron travelling with energy 9.0 GeV in the +z direction collides with a positron travelling with energy 3.1 GeV in the z direction. What is the centre-of-mass energy of the system? What are the Lorentz boost factors β and γ of this system in the laboratory frame? Q3 Satisfy yourself that x µ, p µ and A µ transform as contravariant 4-vectors. Show that µ transforms as a covariant 4-vector. Q4 Calculate the elements of (Λ µ ν ) 1 (in its matrix representation). Show that the inverse Lorentz transformation corresponds to a Lorentz boost with velocity β.
Q5 Show that Eq. (7) correctly describes the covariant transformation of Eq. (4) Q6 Show that the matrix of Eq. (6) can be written Λ µ ν = cosh θ sinh θ 0 0 sinh θ cosh θ 0 0 0 0 1 0 0 0 0 1 (8) where θ is called the rapidity. Write θ in terms of the Lorentz factors β and γ. Note that not all 4 component vectors are either covariant or contravariant. [While we will always write a contravariant 4-vector with a superscript, and a covariant 4-vector with a subscript, just because something has a superscript does not necessarily mean that it is a contravariant 4-vector!] The product of any covariant 4-vector and any contravariant 4-vector is invariant. This can easily be shown using the transformation properties given above (Eqs. 5 and 7). a µ b µ a µb µ = a ν (Λ µ ν ) 1 Λ µ ν b ν = a ν b ν (9) since (Λ µ ν ) 1 Λ µ ν is by definition the identity. Examples of Lorentz invariant quantities that are obtained in this way include the separation squared (x µ x µ ) and the invariant mass squared (p µ p µ ). We can form a covariant 4-vector from a contravariant 4-vector using the lowering operator g µν : x µ = g µν x ν (10) Explicitly, we can write g µν as a 4 4 matrix (the so-called metric tensor): g µν = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (11) Similarly, we can convert from a covariant 4-vector to a contravariant 4-vector, using the raising operator g µν : x µ = g µν x ν (12) Written out as a matrix, g µν has the same elements as g µν. Yet they are not the same thing, since as operators they have different effects. Q7 What is g µλ g λν? How should it be represented, in terms of subscripts and superscripts? Q8 The separation squared (x 1 ) µ (x 2 ) µ can be either time-like (i.e. > 0) or space-like (i.e. < 0). What is the meaning of the equivalent possibilities for (p 1 ) µ (p 2 ) µ?
Electromagnetism Quantum electrodynamics is the theory describing the interactions between light and matter. To begin to construct such a (relativistic, quantum mechanical) theory, we should understand relativistic effects not only on matter, but also on light. In one sense this is trivial light is always relativistic since it travels at the speed of light (i.e. photons are massless). In fact, Maxwell s equations, which describe classical electrodynamics, are indeed relativistically invariant..e = ρ em E = B.B = 0 B = j em + E Gauss Faraday Lenz no magnetic monopoles Ampère Maxwell [n.b. here we are using units of µ 0 = ɛ 0 = 1, which goes a bit beyond our definition of natural units; in fact these are sometimes referred to as Heaviside-Lorentz units.] These equations imply the vitally important continuity equation ρ em (13) +.j em = 0. (14) In fact, it was the desire to satisfy this relation that led Maxwell to modify Ampère s law (adding the E/ term), and hence complete the set of Eq. (13). Note that Eq. (13) (and also Eq. (14)) contain only first-order derivatives. Maxwell s equations are hugely important as nicely encapsulated in this quote by Feynman: From a long view of the history of mankind, seen from, say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell s discovery of the laws of electrodynamics. Q9 Show explicitly that Maxwell s equations (Eq. (13)) lead to the continuity equation (Eq. (14)). We would like to make the Lorentz invariance of Maxwell s equations manifest. This can be done by using the electromagnetic 4-vector potential, A µ = (φ, A), where B = A and E = φ A. (φ and A are just the electric and magnetic potentials, respectively.) We next define the electromagnetic field strength tensor Q10 Show that F µν is antisymmetric. Q11 How many independent elements does F µν have? F µν = µ A ν ν A µ, (15) Q12 Show explicitly that F µν = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 (16) Using Eq. (15), we can write Maxwell s equations in a convenient form which makes the Lorentz invariance manifest: µ F µν = j ν em, (17) where j ν em = (ρ em, j). This automatically embodies the continuity equation, which can be written ν j ν em = 0. (18)
Q13 Relate Eq. (17) to the conventional form of Maxwell s equations (Eq. (13)) as well as the continuity equation (Eq. (14)). The presence of a conserved current j ν em is physically highly significant, and will continue to be so in relativistic quantum mechanics (and in quantum field theory). Quantum Mechanics Important topics from quantum theory that will be relevant include: Wave-particle duality quantization of light waves into photons the de Broglie wavelength of particles The wavefunction interpretation: the state of a system is described by a complex valued wavefunction, with the square of the magnitude of the wavefunction giving the probability density (Born interpretation) Heisenberg s uncertainty principle: x p (19) 2 The above is the most common form for the uncertainty principle, though it can be written in many different ways. One convenient and rather general formulation gives that for an operator ˆx and its conjugate momenta ˆp, their commutator is given by [ˆx, ˆp] = i. Pauli s exclusion principle: The exclusion principle states that no two particles can simultaneously occupy the same quantum state. Devised to explain the spectra of the hydrogen atoms (why all the electrons do not fall down to the lowest available energy state), it applies to fermions (spin 1/2) but not to bosons (integer spin). Note that the quantum numbers labelling the different states available at each given energy were presented by Sommerfeld prior to Pauli s work. These include a mysterious fourth quantum number that Pauli referred to as two-valued-ness. Spin An explanation for the fourth quantum number, two-valued-ness, as being due to the intrinsic spin of the electron, was presented by Uhlenbeck and Goudsmit. The idea was not well-received at first, as two-valued-ness was seen as being an essentially quantum phenomena (as indeed it is), which could not be related to classical concepts such as angular momenta. Nonetheless, due to the success of spin in explaining (inter alia) the results of the Stern-Gerlach experiment, it rapidly gained credence. (The Stern-Gerlach experiment, performed in 1922 using silver atoms, was originally (incorrectly) thought to demonstrate quantization of orbital angular momentum. The correct interpretation as due to electron spin followed the 1927 experiments of Phipps and Taylor using hydrogen atoms.) Of great importance in QM is Schrödinger s equation, which we can write H ψ = i ψ (20) Here we have used the so-called bra-ket notation (introduced by Dirac) wherein we treat ψ as a state (also sometimes called vector). Formally, this is different from treating ψ as a wavefunction, but we will gloss over such distinctions. The operator H that appears in Eq. (20) is the Hamiltonian. It will be familiar (at least) to those who have studied Hamiltonian mechanics. The Lagrange- Hamiltonian formalism is of great use in the study of relativistic quantum mechanics. However, we will defer revision of this topic until such time as it is needed.
Q14 The Hamiltonian is named in honour of Mark Hamill, who played Luke in the original Star Wars trilogy (and again in Episode 7). True or false? For now, it is sufficient to state that the Hamiltonian represents the total energy of the system, so using E = T + V, and with the nonrelativistic kinetic energy T = p 2 /2m, and quantizing by p i, we obtain the familiar form 2 2m 2 ψ + V ψ = i ψ (21) Unsurprisingly, the Schrödinger equation is not relativistically invariant. This is clear since (i) we used the non-relativistic expression for the kinetic energy in its derivation; (ii) the equation contains second-order derivatives with respect to the space-like co-ordinates, but only a first-order derivate with respect to time. Thus space and time are not on the same footing, and the equation cannot be relativistically invariant. In fact, Schrödinger was aware of this drawback when he first formulated the equation (moreover he had previously rejected an equation which is a valid relativistic quantum mechanical wave equation, and which we will later encounter, named the Klein-Gordon equation). However, the Schrödinger equation works well in describing the energy levels of electron orbits in the hydrogen atom (the Klein-Gordon equation does not). Problems with Quantum Mechanics Why do we need to go beyond quantum mechanics? Some of the reasons are: We know that the theory of special relativity exists and is well verified experimentally. We should not be satisfied with a non-relativistic theory to describe fundamental particles, since we know that it will fail when the particles become relativistic. A particle can be considered relativistic when its kinetic energy approaches its rest-mass energy, i.e. when p 2 /2m mc 2 /2. Using de Broglie (p h/λ) or Heisenberg (x /p) this allows us to obtain the appropriate distance scale, the Compton wavelength λ c 1 m mc (22) where the former is in natural units. We therefore expect relativistic effects to become important when studying either (i) high energies or (ii) short distances. The concept of spin is an ad-hoc addition to non-relativistic quantum theory. Where does it come from? Similarly, Pauli s exclusion principle must be treated as a new law of nature in non-relativistic quantum mechanics. Perhaps it has some explanation in a more fundamental theory? If light can be treated as particles, why don t these particles collide when two beams of light are shone at each other? There are nowadays very many experimental observations that cannot be understood without relativistic quantum mechanics. This is partly due to the use of particle accelerators, which accelerate particles such as electrons and protons to energies at which relativistic effects cannot be ignored (or even dominate). Relativistic electrons are also used in arguably less exotic machines such as high resolution electron microscopes. Various astronomical observations (cosmic rays, etc.) also involve relativistic particles sometimes at energies much higher than can be achieved in man-made experiments. One of the most striking new phenomena that cannot be explained non-relativistically is that of (electron-positron) pair production and annihilation. This can occur at energies E > 2m e c 2.
Figure 1: Induced (atomic) positron production probability P e + for 240 kev < E e + < 430 kev as a function of Z u, for collisions at 5.9 MeV/u and scattering angular range 38 < Θ ion < 52. Background due to nuclear excitation is subtracted. The solid line is a fit to P e + = cz n u with n = 16. From P. Kienle Phys. Scr. T23, 123 (1988). See also Greiner, Müller and Rafelski, Quantum Electrodynamics of Strong Fields (1985). In fact, effects due to this process had been observed in the 1920s (in experiments on scattering of high energy gamma rays), but their origin was not understood until later. One consequence of this process is the so-called vacuum instability that can lead to nuclear screening effects in heavy ion collisions. When a nucleus with very high Z is created, the binding energy of the innermost electronic orbit becomes sufficient to create e + e pairs. (Nonrelativistically the binding energy is given by E = Z 2 e 4 m/(2 2 ), though corrections due to relativistic effects and the finite size of the heavy nucleus are necessary for a proper calculation.) An e + e pair can therefore be created out of the vacuum the electron then fills the lowest available energy level and the positron is emitted. Consequently the rate of positron emission increases rapidly once a certain threshold Z (near Z = 160) is passed, as can be seen in Fig. 1. Q15 At what energies/lengths are the following relativistic: (i) photons; (ii) electrons; (iii) protons? Q16 If light can be treated as particles, why don t these particles collide when two beams of light are shone at each other? Q17 Using a non-relativistic approximation, estimate the threshold Z for pair-production due to the vacuum instability. The theories of special relativity and quantum mechanics can be made self-consistent in relativistic quantum mechanics. This theory will clearly address the first problem discussed above, and will also resolve issues related to spin, which arises naturally in the equations. However, relativistic quantum mechanics per se does not tell us how particles interact. Problems related to particle interactions are therefore not solved simply by making quantum mechanics relativistically invariant (or by making relativity quantum mechanical). They can, however be addressed by further developing the theory, which is done by application of the gauge principle to RQM, i.e. in gauge theory. A full treatment of this theory requires the quantization of the field, i.e. quantum field theory. The quantum field theory that is relevant for interactions of light with matter is called quantum electrodynamics. Some of these more involved theories will be touched on in this module, with further development in PX430:
Gauge Theories of Particle Physics, though this will still stop short of a full treatment of quantum field theory. You will need to do some extensive further reading (or a Ph.D.) to go further. The Pauli Hamiltonian Since spin is central to discussions of relativistic (and non-relativistic) quantum mechanics, it is worthwhile spending some time looking at how it is handled in the non-relativistic case. The starting point is the so-called Pauli Hamiltonian H = 1 2m (p qa)2 + qφ q 1 σ.( A) (23) 2m where we have used natural units and (φ, A) is the electromagnetic field potential the magnetic field is given by B = A σ is a vector of the Pauli matrices ( ) 0 1 σ 1 = 1 0 σ 2 = ( 0 i i 0 ) σ 3 = ( 1 0 0 1 ) (24) The following may be useful to revise the Pauli matrices: Q18 Show that σ 2 1 = σ2 2 = σ2 3 = I Q19 Show that det(σ i ) = 1 for i = 1, 2, 3 Q20 Show that Tr(σ i ) = 0 for i = 1, 2, 3 Q21 Prove the commutation relations [σ i, σ j ] = 2iɛ ijk σ k (where ɛ ijk is the Levi-Civita symbol) Q22 Prove the anti-commutation relations {σ i, σ j } = 2δ ij I It may be unfamiliar to encounter a vector with elements that are anything other than scalar numbers. However, there is no problem with this mathematically, it just takes some getting used to. To be explicit, σ.b = σ x B x + σ y B y + σ z B z, therefore this quantity is a 2 2 matrix. Eq. (23) now appears unbalanced, since the right-hand side contains three terms that are a scalar, a scalar and a matrix, respectively. There is no mathematical rule to add a scalar to a matrix, so this does not make sense, unless we understand the presence of unwritten identity matrices multiplying the first two terms. With these included, the Hamiltonian is therefore a 2 2 matrix. It is logical to now ask what form the solutions to the generalised Schrödinger equation H ψ = ψ will take. It should be clear that ψ should be a two-component column vector. i Q23 Show that solutions to the Schrödinger-Pauli equation should be two-component column vectors. The Pauli Hamiltonian of Eq. (23) can be rewritten as follows where H = 1 2m (p qa)2 + qφ gµ B ( 1 σ).( A) (25) 2 µ B is the Bohr magneton µ B = q 2m = q 2mc g is the so-called gyro-magnetic ratio
the magnetic moment of the electron is given by µ = gµ B s where the spin vector is given by s = 1 2 σ The first two terms appear in the normal Schrödinger equation. The third is due to the spin of the electron and is sometimes called the Stern-Gerlach term. This, however, only works with the choice g = 2. Classically, one would expect the gyro-magnetic ratio to take the value g = 1. While g = 2 leads to some simplification in the non-relativistic equation (as we shall see), it arises naturally in the relativistic version of the theory. In fact, there are small corrections from 1 2 (g 2) = 0 due to quantum electrodynamics. These were first calculated by Schwinger, and the consistency between the theoretical prediction and the experimentally measured value had now been tested to an extraordinary degree of precision: the CODATA recommended value for the so-called anomalous magnetic moment of the electron is 1 2 (g 2) = (1159.65218091 ± 0.00000026) 10 6. The theoretical calculation relies on the value of the fine-structure constant α. Since the measurement is so precise, this is now usually turned around and used to measure α. Nevertheless, with the choice g = 2, we can rewrite, again, the Pauli Hamiltonian to obtain H = 1 2m (σ. (p qa))2 + qφ (26) or to obtain the Schrödinger-Pauli equation in the form (recall that p = i ). Q24 Verify the relation 1 2m (σ. ( i qa))2 ψ = where a and b are arbitrary vectors (of scalar quantities). (i qφ ) ψ (27) (σ.a)(σ.b) = (a.b)i + iσ.(a b) (28) Q25 Considering the case that the vectors a and b may contain quantities other than scalars, what conditions on a and b are required so that the above expression still holds? Q26 Give an expression for (σ.a) 2. Q27 Rearrange Eq. (25) or Eq. (23) to obtain Eq. (26). Alternatively, rearrange Eq. (26) to obtain Eq. (25) or Eq. (23). Eq. (26) is useful to illustrate the so-called minimal substitution with which electromagnetic fields can be introduced into (free-field) wave equations. The substitution is p = i i qa E = i i qφ (29) or, in 4-vector form ( ) µ =, D µ = µ + iqa µ (30) P µ = (E, p) Π µ = P µ qa µ (31) This will turn out to be highly important.
Historical Development of Quantum Mechanics Some may find this timeline useful and/or interesting to put the development of relativistic quantum mechanics into historical context. Much of this is taken from Quantum by Manjit Kumar which, although not particularly helpful for this module, is recommended as an enjoyable read. 1897 Discovery of the electron by J.J. Thomson Discovery of the Zeeman effect 1900 Planck s blackbody radiation law 1905 Einstein s explanation of the photoelectric effect Einstein s special theory of relativity 1911 Discovery of the atomic nucleus by Rutherford 1913 Bohr s theory of the atom Stark effect discovered 1916 Sommerfeld begins to explain fine structure 1920... continuing until a fourth quantum number is added 1922 Stern-Gerlach experiment 1923 Discovery of Compton effect de Broglie extends wave-particle duality to matter 1924 Stoner studies splitting of energy levels in magnetic fields 1925 Pauli develops exclusion principle Heisenberg develops matrix mechanics Goudsmit and Uhlenbeck propose spin 1926 Schrödinger publishes his wave equation Heisenberg shows matrix and wave mechanics to be equivalent... later extended by Dirac using transformation theory Born interpretation of the wavefunction as being related to probability density 1927 Davisson and Germer and (separately) G. Thomson diffract electrons Heisenberg discovers the uncertainty principle Bohr develops the idea of complementarity (later the Copenhagen interpretation ) Klein and Gordon present their relativistically invariant wave equation 1928 The Dirac equation 1932 Discovery of positrons by Anderson Fermi sets out a quantum theory of radiation (building on work by Dirac, 1927) 1940 Pauli derives spin-statistics theorem 1943 Tomonaga begins work on what will later become known as renormalization 1946 Tomonaga publishes a theory of Quantum Electrodynamics 1947 Schwinger calculates the anomalous magnetic moment of the electron The Lamb shift is discovered An explanation of the Lamb shift is presented by Bethe Kusch measures the anomalous magnetic moment of the electron 1948 Schwinger publishes a theory of Quantum Electrodynamics 1949 Feynman publishes a theory of Quantum Electrodynamics Dyson shows that Tomonaga, Schwinger and Feynman s theories are self-consisent