Van der Waals Forces Between Ground-State Atoms
Perturbation theory within the multipolar coupling scheme is used to calculate the van der Waals potential of two electric ground-state atoms. With the aid of Feynman diagrams, the interaction is shown to depend on the classical Green’s tensor as well as
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Van der Waals Forces Between Ground-State Atoms
As we will see in this chapter, vdW forces between two atoms sensitively react to the presence of bodies or media. Our general treatment of vdW forces will be a generalisation of the single-atom case, where higher-order perturbation theory is required to account for the coupling of both atoms to the electromagnetic field. In order keep the calculations manageable, we work exclusively within the multipolar coupling scheme and rely on Feynman diagrams. After applying the general theory to the free-space case, we investigate how the presence of a plate or sphere may alter the vdW interaction. To begin with, let us briefly outline how the perturbative approach can be generalised to the two-atom case, in agreement with the ideas of Casimir and Polder [1]. Starting point is the multipolar Hamiltonian Hˆ =
pˆ 2 pˆ 2A + B + E nA |n A n A | + E nB |n B n B | + Hˆ F 2m A 2m B n n
(5.1)
+ Hˆ AF + Hˆ BF , recall (2.388) with (2.369), which is the two-atom generalisation of the Hamiltonian (4.1) used in Chap. 4 to derive the CP potential. Since we use the multipolar coupling scheme throughout this chapter, we have dropped the primes distinguishing multipolar variables from the minimal-coupling ones. Applying the Born– Oppenheimer approximation by integrating the internal atomic dynamics for given centre-of-mass positions rˆ A , rˆ B and momenta pˆ A , pˆ B , we obtain the effective Hamiltonian pˆ 2 pˆ 2 (5.2) Hˆ eff = A + A + E + ΔE . 2m A 2m A Here, E denotes the energy of the two uncoupled atoms and the field; and the energy shift (0)
(0)
ΔE = ΔE A + ΔE B + ΔE (1) (ˆr A ) + ΔE (1) (ˆr B ) + ΔE(ˆr A , rˆ B ) , S. Y. Buhmann, Dispersion Forces I, Springer Tracts in Modern Physics 247, DOI: 10.1007/978-3-642-32484-0_5, © Springer-Verlag Berlin Heidelberg 2012
(5.3) 209
210
5 Van der Waals Forces Between Ground-State Atoms
can be separated into a position-independent part that contains the Lamb shifts of both atoms, two parts depending only on the positions of one atom and a genuine two-atom part. Recalling the commutation relations (2.310), the above Hamiltonian generates the following equations of motion for atom A: 1 m A rˆ A , Hˆ eff = pˆ A , i 1 ˙ ˆ A. m A rˆ A , Hˆ eff = F m A r¨ˆ A = i m A r˙ˆ A =
(5.4) (5.5)
The total force F A = F(r A ) + F(r A , r B )
(5.6)
on the atom consists of the CP force F(r A ), as given by (4.6) and discussed in the previous Chap. 4 and a vdW force F(r A , r B ) = −∇ A U (r A , r B )
(5.7)
which is due to the additional atom B. The associated vdW potential is given by the two-atom part of the energy shift U (r A , r B ) = ΔE(r A , r B )
(5.8)
As we will see later on, both the vdW force and potential can be further separated according to F(r A , r B ) = F (0) (r A , r B ) + F (1) (r A , r B )
(5.9)
U (r A , r B ) = U (0) (r A , r B ) + U (1) (r A , r B )
(5.10)
into a pure atom–atom part and an atom–atom–body part. The latter component F (1) (r A , r B ) represents the modification of the atom–atom vdW force du
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