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Chapter 9. Introduction to Relativistic Quantum Mechanics

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Chapter 9. Introduction to Relativistic Quantum Mechanics Essential Graduate Physics QM: Quantum Mechanics Chapter 9. Introduction to Relativistic Quantum Mechanics The brief introduction to relativistic quantum mechanics, presented in this chapter, consists of two very different parts. Its first part is a discussion of the basic elements of the quantum theory of the electromagnetic field (usually called quantum electrodynamics, QED), including the field quantization scheme, photon statistics, radiative atomic transitions, the spontaneous and stimulated radiation, and so-called cavity QED. We will see, in particular, that the QED may be considered as the relativistic quantum theory of particles with zero rest mass – photons. The second part of the chapter is a brief review of the relativistic quantum theory of particles with non-zero rest mass, including the Dirac theory of spin-½ particles. These theories mark the point of entry into a more complete relativistic quantum theory – the quantum field theory – which is beyond the scope of this course.1 9.1. Electromagnetic field quantization2 Classical physics gives us3 the following general relativistic relation between the momentum p and energy E of a free particle with rest mass m, which may be simplified in two limits – non-relativistic and ultra-relativistic: Free 2 2 particle’s 2 2 2 1/ 2 mc p / 2m, for p mc, relativistic E pc (mc ) (9.1) energy pc, for p mc. In both limits, the transfer from classical to quantum mechanics is easier than in the arbitrary case. Since all the previous part of this course was committed to the first, non-relativistic limit, I will now jump to a brief discussion of the ultra-relativistic limit p >> mc, for a particular but very important system – the electromagnetic field. Since the excitations of this field, called photons, are currently believed to have zero rest mass m,4 the ultra-relativistic relation E = pc is exactly valid for any photon energy E, and the quantization scheme is rather straightforward. As usual, the quantization has to be based on the classical theory of the system – in this case, the Maxwell equations. As the simplest case, let us consider the electromagnetic field inside a finite free- space volume limited by ideal walls, which reflect incident waves perfectly.5 Inside the volume, the Maxwell equations give a simple wave equation6 for the electric field 2 2 1 E E 0, (9.2) c 2 t 2 1 Note that some material covered in this chapter is frequently taught as a part of the quantum field theory. I will focus on the most important results that may be obtained without starting the heavy engines of that theory. 2 The described approach was pioneered by the same P. A. M. Dirac as early as 1927. 3 See, e.g., EM Chapter 9. 4 -22 By now this fact has been verified experimentally with an accuracy of at least ~10 me – see S. Eidelman et al., Phys. Lett. B 592, 1 (2004). 5 In the case of finite energy absorption in the walls, or in the wave propagation media (say, described by complex constants and ), the system is not energy-conserving (Hamiltonian), i.e. interacts with some dissipative environment. Specific cases of such interaction will be considered in Sections 2 and 3 below. 6 2 2 See, e.g., EM Eq. (7.3), for the particular case = 0, = 0, so that v 1/ = 1/00 c . © K. Likharev Essential Graduate Physics QM: Quantum Mechanics and an absolutely similar equation for the magnetic field B. We may look for the general solution of Eq. (2) in the variable-separating form E (r,t) p j (t)e j (r) . (9.3) j Physically, each term of this sum is a standing wave whose spatial distribution and polarization (“mode”) are described by the vector function ej(r), while the temporal dynamics, by the function pj(t). Plugging an arbitrary term of this sum into Eq. (2), and separating the variables exactly as we did, for example, in the Schrödinger equation in Sec. 1.5, we get 2 e j 1 p j 2 2 const k j , (9.4) e j c p j so that the spatial distribution of the mode satisfies the 3D Helmholtz equation: Equation 2e k 2e 0. (9.5) for spatial j j j distribution The set of solutions of this equation, with appropriate boundary conditions, determines the set of the functions ej, and simultaneously the spectrum of the wave number magnitudes kj. The latter values determine the mode eigenfrequencies, following from Eq. (4): 2 p j j p j 0, with j k jc . (9.6) There is a big philosophical difference between the quantum-mechanical approach to Eqs. (5) and (6), despite their single origin (4). The first (Helmholtz) equation may be rather difficult to solve in realistic geometries,7 but it remains intact in the basic quantum electrodynamics, with the scalar components of the vector functions ej(r) still treated (at each point r) as c-numbers. In contrast, the classical Eq. (6) is readily solvable (giving sinusoidal oscillations with frequency j), but this is exactly where we can make the transfer to quantum mechanics, because we already know how to quantize a mechanical 1D harmonic oscillator, which in classics obeys the same equation. As usual, we need to start with the appropriate Hamiltonian – the operator corresponding to the classical Hamiltonian function H of the proper set of generalized coordinates and momenta. The electromagnetic field’s Hamiltonian function (which in this case coincides with the field’s energy) is8 2 2 3 0E B H d r . (9.7) 2 20 Let us represent the magnetic field in a form similar to Eq. (3),9 7 See, e.g., various problems discussed in EM Chapter 7, especially in Sec. 7.9. 8 -2 See, e.g., EM Sec. 9.8, in particular, Eq. (9.225). Here I am using SI units, with 00 c ; in the Gaussian units, the coefficients 0 and 0 disappear, but there is an additional common factor 1/4 in the equation for energy. However, if we modify the normalization conditions (see below) accordingly, all the subsequent results, starting from Eq. (10), look similar in any system of units. 9 Here I am using the letter qj, instead of xj, for the generalized coordinate of the field oscillator, in order to emphasize the difference between the former variable, and one of the Cartesian coordinates, i.e. one of the arguments of the c-number functions e and b. Chapter 9 Page 2 of 36 Essential Graduate Physics QM: Quantum Mechanics B (r,t) j q j (t)b j (r) . (9.8) j Since, according to the Maxwell equations, in our case the magnetic field satisfies the equation similar to Eq. (2), the time-dependent amplitude qj of each of its modes bj(r) obeys an equation similar to Eq. (6), i.e. in the classical theory also changes in time sinusoidally, with the same frequency j. Plugging Eqs. (3) and (8) into Eq. (7), we may recast it as p 2 2 q 2 1 H j e 2 (r)d 3r j j b 2 r d 3r . (9.9) 0 j j j 2 2 0 Since the distribution of constant factors between two multiplication operands in each term of Eq. (3) is so far arbitrary, we may fix it by requiring the first integral in Eq. (9) to equal 1. It is straightforward to check that according to the Maxwell equations, which give a specific relation between vectors E and B,10 this normalization makes the second integral in Eq. (9) equal 1 as well, and Eq. (9) becomes 2 2 2 p j j q j H H j , H j . (9.10a) j 2 2 Note that that pj is the legitimate generalized momentum corresponding to the generalized coordinate qj, because it is equal to L / q j , where L is the Lagrangian function of the field – see EM Eq. (9.217): 2 2 2 2 2 3 E B p j j q j L d r 0 L , L . (9.10b) j j 2 20 j 2 2 Hence we can carry out the standard quantization procedure, namely declare Hj, pj, and qj the quantum-mechanical operators related exactly as in Eq. (10a), Electro- pˆ 2 2 qˆ 2 magnetic ˆ j j j mode’s H j . (9.11) Hamiltonian 2 2 We see that this Hamiltonian coincides with that of a 1D harmonic oscillator with the mass mj formally 11 equal to 1, and the eigenfrequency equal to j. However, in order to use Eq. (11) in the general Eq. (4.199) for the time evolution of Heisenberg-picture operators pˆ j and qˆ j , we need to know the commutation relation between these operators. To find them, let us calculate the Poisson bracket (4.204) for the functions A = qj’ and B = pj”, taking into account that in the classical Hamiltonian mechanics, all generalized coordinates qj and the corresponding momenta pj have to be considered independent arguments of H, only one term (with j = j’ = j”) in only one of the sums (12) (namely, with j’ = j”), gives a non-zero value (-1), so that q p q p q ,p j' j" j' j" . (9.12) j' j" P j'j" j p j q j q j p j Hence, according to the general quantization rule (4.205), the commutation relation of the operators corresponding to qj’ and pj” is 10 See, e.g., EM Eq. (7.6). 11 Selecting a different normalization of the functions ej(r) and bj(r), we could readily arrange any value of mj, and the choice corresponding to mj = 1 is the best one just for the notation simplicity.
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