Supplemental Lecture 16 Introduction to the Foundations of Quantum Field Theory For Physics Students III. The Casimir Effect Abstract Uncharged grounded conducting parallel plates experience a mutual attractive force which is a quantum, relativistic effect: the force is proportional to Planck’s constant and the speed of light. It was calculated in the early days of relativistic quantum field theory (1948). In fact, it was originally analyzed as a limiting case of the retarded van der Waal’s force between dielectric plates. Casimir made the fascinating observation that if the plates had sufficiently high dielectric constants, then for some physical effects the plates’ effect on the electromagnetic field could be replaced by boundary conditions and the force between the plates could be calculated just by considering the quantum zero point fluctuations of the electromagnetic field in this environment. In this case the attractive force is called the Casimir effect. These considerations indicated that the force can be calculated either from 1. the direct electromagnetic interactions between the plates (van der Waals), or 2. the spatial dependence of the energy stored in the vacuum fluctuations of the electromagnetic field. The heuristic derivation of the Casimir effect is presented here. The derivation is analyzed, critically assessed and it’s physical and unphysical elements are discussed. The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays. This Essay supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work. 1 Keywords: Casimir effect, van der Waals forces, Casimir-Polder, F. London, polarizable materials, electromagnetic field, zero point fluctuations. --------------------------------------------------------------------------------------------------------------------- Contents Van der Waals Forces .................................................................................................................2 Heuristic Derivation of Casimir Effect for Parallel Conducting Plates .........................................8 Appendix. The Euler-Maclaurin Formula .................................................................................. 17 References ................................................................................................................................ 17 Van der Waals Forces The ultimate goal of this Essay is to improve our understanding of quantum field theory by computing the force between two parallel plates of neutral, grounded conductors. We shall take two approaches to this problem. The first is to use quantum mechanics and attack the problem as an example of a “van der Waals” interaction, familiar from discussions of the quantum force between neutral hydrogen atoms in non-relativistic quantum mechanics. In fact, we have to do somewhat better than non-relativistic quantum mechanics here: in order to calculate the long range van der Waals force in this case, the relativistic, quantum nature of the electromagnetic field must be accounted for. When this is done, one speaks of the “retarded van der Waals force”. The second is to use field theoretic methods and calculate the change in the electromagnetic energy stored in vacuum fluctuations between the conducting plates. The second method is simple in the idealized limiting case where the conductor is “perfect” and its effect on the electromagnetic field is idealized as a boundary condition, that the electromagnetic field vanishes inside the conductor. This idealization is not completely accurate or physical, as we will discuss below, but it is adequate to calculate the long range force between the plates. Let’s begin by thinking about forces explicitly and review the non-relativistic van der Waals force between two neutral hydrogen atoms in their ground state. We show the geometry of the protons and electrons of the two hydrogen atoms in Fig. 1, 2 Fig.1 The coordinates of the electrons and protons of two distant but interacting hydrogen atoms. The vector 푅⃗ between the protons is held fixed while the electrons propagate around each proton according to the laws of non-relativistic quantum mechanics. We suppose that |푅⃗ | is large, |푅⃗ | ≫ 푎0, the Bohr radius of the hydrogen atom, so the multipole expansion of the potentials between the constituents of the two hydrogen atoms is useful and accurate. In isolation the dipole moment of each has a vanishing expectation value. In other words, for atom #1, 푑1 = 푒푟⃗⃗ 1 and 2 <1,0,0|푑1|1,0,0 > = 0 because < 푟 1 > = ∫|휓100| 푟 1 푑푟 1 = 0 by spherical symmetry. Here 휓100 means the hydrogen state with quantum numbers 푛 = 1, 푙 = 0, 푚 = 0 and the spins of the e and p are ignored and non-relativistic quantum mechanics is employed. Before doing any calculation, let’s understand why there is a force between the two neutral atoms. At any moment 푡1 atom #1 has a dipole moment given by 푑1 = 푒푟 1. It generates an electrostatic potential at the position of the second atom #2 which effects the second atom’s instantaneous dipole moment. The electrostatic potential at the location of the second atom is, to leading order in |푅⃗ | for |푅⃗ | ≫ 푎0, ⃗ 푈(푅⃗ ) = 푑1∙푅 1.1a 4휋푅3 3 which produces the electric field, 퐸⃗ (푅⃗ ) = −∇⃗⃗ 푈 = − 푒 (푟 − 3푅̂ 푅̂ ∙ 푟 ) 1.1b 푅 4휋푅3 1 1 So, the instantaneous dipole-dipole interaction energy reads, 2 푈 = −푑 ∙ 퐸⃗ (푅⃗ ) = 푒 (푟 ∙ 푟 − 3푅̂ ∙ 푟 푅̂ ∙ 푟 ) 1.1c 12 2 4휋푅3 1 2 2 1 We see that 푈12 depends intricately on the orientation of the dipoles and their orientations to the large vector 푅⃗ between the atoms. So far this discussion is just non-relativistic Newtonian mechanics. But now we want to treat 푈12 as a perturbation in the total energy (Hamiltonian) of the two atoms. We can calculate the energy of the ground state of the two interacting atoms using perturbation theory. The first order shift in the energy is given by the matrix element of the perturbation 푈12 in the unperturbed ground state, (1) (1) (2) (1) (2) ∆퐸 =< 휓100휓100|푈12|휓100휓100 > 1.2 (1) But 휓100(푟 1) does not support a permanent dipole moment, in other words, < 푟 1 > = 2 (1) (1) (2) ∫|휓100(푟 1)| 푟 1 푑 푟 1 = 0, so ∆퐸 vanishes identically. But the second order energy shift ∆퐸 involves a sum over intermediate states, ′ ′ ′ ′ (2) <100;100|푈12|푛푙푚;푛 푙 푚 ><푛푙푚 ;푛′푙′푚′|푈12|100;100> ∆퐸 = ∑푛푙푚;푛′푙′푚′ 1.3a 퐸100+퐸100−퐸푛푙푚−퐸푛′푙′푚′ Let’s write Eq. 1.3a out. Choose the 푧 −axis in the 푅̂ direction, so that 푈12 can be written, 2 2 푈 = 푒 (푥 푥 + 푦 푦 − 2푧 푧 ) = 푒 푃 1.3b 12 4휋푅3 1 2 1 2 1 2 4휋푅3 12 where 푃12 = 푥1푥2 + 푦1푦2 − 2푧1푧2. Then Eq 1.3a becomes, 2 2 2 (2) 푒 |<푛푙푚;푛′푙′푚′|푃12|100;100>| ∆퐸 = ( 3) ∑푛푙푚;푛′푙′푚′ 1.3c 4휋푅 2퐸100−퐸푛푙푚−퐸푛′푙′푚′ ′ ′ ′ (1) (2) ′ ′ ′ where < 푟 1푟 2| 푛푙푚; 푛 푙 푚 > is 휓푛푙푚(푟 1)휓푛′푙′푚′(푟 2) and < 푛 푙 푚 |푟 1|푛푙푚 > = (1)∗ ( ) (1) ( ) ∫ 휓푛′푙′푚′ 푟 1 푟 1휓푛푙푚 푟 1 푑푟 1. First note two structural properties of Eq. 1.3c: 1. it is necessarily negative, because the energy denominator is negative for all 푛푙푚; 푛′푙′푚′, and 2. the change in 4 (2) −6 (2) energy ∆퐸 falls as 푅 , which produces an attractive force law, 퐹 (푅⃗ ) = −∇⃗⃗ 푅 (∆퐸 ), which falls as −푅−7, a high power of the distance between the atoms. The sum over states in Eq. 1.3c is quite a challenge. However, many terms vanish because < 푛푙푚|푟 1|100 > = 0 for 푙 ≥ 2. This should be familiar because it is a dipole radiation selection rule and it follows from the general properties of quantum angular momentum eigenstates. (2) −1 So much for exact results. Next, we can estimate ∆퐸 by replacing (2퐸100 − 퐸푛푙푚 − 퐸푛′푙′푚′) −1 in Eq. 1.3c with (2퐸100 − 2퐸210) and do the sum over intermediate states using completeness, ∑푛푙푚|푛푙푚 >< 푛푙푚| = 1 1.4a Then Eq. 1.3c becomes, 2 2 (2) 푒 1 2 2 2 2 2 2 ∆퐸 ≅ ( 3) < 100; 100|푥1 푥2 + 푦1 푦2 + 4푧1 푧2 |100; 100 > 1.4b 4휋푅 2퐸100−2퐸210 2 where we have kept the quadratic terms in the product 푃12 that contribute to the matrix element and dropped the ones that vanish. To finish the evaluation of Eq. 1.4b we look up various 2 2 2 integrals involving the stationary states of the hydrogen atom [1], < 푥푖 > = < 푟 ⁄3 > = 푎0 −1 where 푎0 is the Bohr radius which is given by 푎0 = (훼 푚) where 훼 is the fine structure constant and 푚 is the mass of the electron and we are using units in which ℏ = 푐 = 1. Finally, ∆퐸(2) is the change in the potential energy 푉(푅) of the two atom system, 5 ∆퐸(2) = 푉(푅) ≅ − 6훼푎0 1.4c 푅6 and we used the Rydberg formula for the energies of the stationary states of hydrogen, 퐸푛 = 2 −훼⁄2푛 푎0. There is another way to write Eq. 1.4c which is particularly useful in applications. The electric polarizability of the hydrogen atom expresses its induced dipole moment when it is subjected to a uniform electric field 퐸⃗ 0, 푑 = 4휋훼퐸퐸⃗ 0. Choose the external electric field in the z-direction, (2) 퐸⃗ 0 = (0,0, 퐸0), then calculate ∆퐸 in second order perturbation theory with the perturbation 푈 = −푑 ∙ 퐸⃗ = −푒푧퐸 and identify 훼 from the expression ∆퐸(2) = − 1 4휋훼 퐸⃗ 2, 0 0 퐸 2 퐸 0 2 |<푛푙푚|푧|100>| 3 훼퐸 = 2훼 ∑푛푙푚 ≅ 푎0 1.4d 퐸100−퐸푛푙푚 5 Then the potential between the two hydrogen atoms can be written, 6 2 훼 푎0 휔0훼퐸 푉(푅) ≅ − 6 ≅ 6 1.4e 푎0 푅 푅 where 휔0 is the ground state energy of hydrogen, given by the Rydberg-Balmer formula, and Eq.