Abstract
In this chapter, we introduce the basic principles and applications of quantum electrodynamics while describing the motion of a particle within the nonrelativistic approximation. We start by showing how classical electrodynamics may be cast into the Hamilton formalism, the subsequent transition from classical to quantum electrodynamics being then straightforward. Next, we focus on the quantization of a free electromagnetic field wherein plays the abstract solution for harmonic oscillator based on the introduction of non-Hermitian ladder operators a decisive role.
This is a preview of subscription content, log in via an institution.
Notes
- 1.
- 2.
Mr. Coulomb died 25 years prior to the birth of Mr. Maxwell, but the reader will certainly not be confused.
- 3.
One usually claims that the physical meaning is carried by the Hermitian operators, whereas the non-Hermitian ones are merely useful mathematical tools. However, we see that in the case of the quantized EM field, the situation is in fact the very opposite: while the canonical coordinates and momenta bear no obvious physical meaning, their non-Hermitian combinations \(\hat{\mathsf{a}}\) and \(\hat{\mathsf{a}}^{+}\) do.
- 4.
Strictly speaking, this is true only if the gravitation is excluded. Let us also note that this energy of the vacuum represents a part of the cosmological constant problem, see for instance [16].
- 5.
The only assumption we have made is that the initial and final states can be identified with the eigenstates of the free Hamiltonian. This assumption holds approximately, not exactly, see Sect. 6.5.3, and relates to the issue of a so-called “renormalization” which we will discuss in great detail in Chap. 7
- 6.
Recall that \(\left \vert I_{\mathrm{at}}\right> = \left \vert n_{I},l_{I},m_{I}\right>\) and \(\left \vert F_{\mathrm{at}}\right> = \left \vert n_{F},l_{F},m_{I} - 1\right>\).
- 7.
The distinction between individual colors is not sharp, of course. Similarly, the exact wavelength range for visible light is a subject to an endless debate. The reader should thus not be surprised to find slightly different values in the literature.
- 8.
As already emphasized in the previous section, the dipole approximation can be used whenever the nonrelativistic approximation has been invoked.
- 9.
The following exposition concerning the wave functions of the continuous part of the spectrum and transitions from the discrete to continuous parts is inspired by the exposition given in [11].
- 10.
A mathematician would most certainly not proceed this way. However, for the purpose of this book, we do not wish to systematically explain the method of solution of the second order linear differential equations nor the method of the Laplace transform.
- 11.
This is because
$$\displaystyle\begin{array}{rcl} \int _{C_{0}}{ \mathrm{d}t \over 2\pi \mathrm{i}} \mathrm{e}^{t}t^{-c}& =&{ 1 \over 2\pi \mathrm{i}}\left [\int _{0}^{\infty }\mathrm{d}x\mathrm{e}^{-x}(x\mathrm{e}^{-\mathrm{i}\pi })^{-c} -\int _{ 0}^{\infty }\mathrm{d}x\mathrm{e}^{-x}(x\mathrm{e}^{\mathrm{i}\pi })^{-c}\right ] {}\\ & =&{ 1 \over \pi } \int _{0}^{\infty }\mathrm{d}x\mathrm{e}^{-x}x^{-c}{\mathrm{e}^{\mathrm{i}\pi c} -\mathrm{ e}^{-\mathrm{i}\pi c} \over 2\mathrm{i}} ={ 1 \over \pi } \varGamma (1 - c)\sin (\pi c) ={ 1 \over \varGamma (c)}\,, {}\\ \end{array}$$where we used the defining formula for Γ-function, Γ(z) = ∫ 0 ∞dxe−x x z−1, and Eq. (6.155) in the last two steps.
- 12.
In literature, for example in the already mentioned course [11], one can find the notation \(\psi _{\mathbf{k}_{e}}^{(-)}\) for this function. The sign is related to the choice of the boundary condition; here the outgoing plane wave, see Eq. (6.113). The plane wave yields a positive value of the probability current \(J =\oint \mathbf{j} \cdot \mathrm{ d}\mathbf{S}\), where j is given by Eq. (6.116). Similarly, one can encounter the function \(\psi _{\mathbf{k}_{e}}^{(+)}\) with the boundary condition of an ingoing plane wave. These functions are nearly identical; they differ here and there merely by signs. For instance, it holds that \(\psi _{\mathbf{k}_{e}}^{(-)} =\psi _{ -\mathbf{k}_{e}}^{(+){\ast}}\).
- 13.
De Broglie electron wavelength is related to the electron energy via the formula \(\lambda = 2\pi /p = 2\pi /\sqrt{2mE}\). By means of Eqs. (1) and (5), this translates to \(\lambda \approx { 10^{-9} \over E(\mathrm{eV})}[\mathrm{m}]\). Since the “size” of an atom is of the order 10−10 m, we see that the overlap is in fact negligible only for energies above 102 eV.
- 14.
The use of the Fermi golden rule for scattering problems is usually referred to as the first Born approximation.
- 15.
- 16.
This kind of resonance is called the Feshbach resonance.
- 17.
Strictly speaking, this is not entirely correct. Firstly, a photon with a definite wave vector k 1 is completely delocalized in space and time. One thus cannot talk about the times of emission, interaction and absorption. Nonetheless, taking the appropriate linear combination of states with wave vectors close to k 1, we can obtain a state localized in space and time. This also corresponds to the photons the experimenter is able to create. Secondly, the interaction between the atom and the EM field can never be “switched off,” see Sect. 6.5. Even if no photon is present, the atom still interacts with fluctuations of the EM field. However, we ignore both reservations. The first one is merely a technicality; when considering the appropriate linear combination, we arrive at the very same result, but complicate the calculation leading to it. The second one leads only to a small correction to the result, see the discussion in Sect. 6.4.3.
- 18.
Having developed our considerations to such an extent, we should also mention that for the very same reason of interaction of the discrete state with continuum of the states, the physical initial and final states do not match exactly the eigenstates (6.178) of the “free” Hamiltonian. If we take the interaction into account, the energy of atom initial state E I at is also slightly shifted, see Sect. (6.5). For a more systematic treatment, we refer the reader to [4].
- 19.
- 20.
One third appears in Eq. (6.202) for the following reason. In Eq. (6.190), we have schematically
$$\displaystyle{M_{ij} =\delta _{ij} - Q_{ij}\,.}$$Setting i = j in this equation leads to (note that the Einstein summation convention is used)
$$\displaystyle{M_{ii} = 3 - Q_{ii}\,.}$$Setting i = k in M ik = δ ik M leads to
$$\displaystyle{M_{ii} = 3M\,.}$$From the last two equations, we find
$$\displaystyle{M = 1 -{ 1 \over 3}Q_{ii}\,.}$$ - 21.
In case of the calculation of (6.239), we numerically integrate an expression which behaves for the s-states for large k e as k e −4ln(k e ). On the other hand, in case of (6.244), we integrate an expression which behaves for the s-states for large k e as k e −2ln(k e ), and thus is on the very border of convergence. These statement will become clear to the reader once he attempts the following exercise.
- 22.
Clearly, \(\hat{\mathsf{A}} = \hat{\mathsf{p}}_{1i}\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot \mathbf{r}_{1}}\), \(\hat{\mathsf{B}} = \hat{\mathsf{p}}_{2j}\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot \mathbf{r}_{2}}\) and \(\hat{\mathsf{C}} = \hat{\mathsf{H}}_{\mathrm{at}} - E_{0}\). In addition, \(\hat{\mathsf{p}}_{i}\) and eik⋅ r can be freely commuted. Their commutator is k i eik⋅ r, but the whole expression is multiplied by the projector P ij and hence the commutator vanishes, see Eq. (6.33). This can be traced back to the Coulomb gauge \(\nabla \cdot \hat{\mathsf{\boldsymbol{A}}} = 0\).
- 23.
Of course, an impatient reader will not worry at all.
References
J. Benda, K. Houfek, Comput. Phys. Commun. 185, 2903 (2014)
I. Bray, A.T. Stelbovics, Adv. At. Mol. Opt. Phys. 35, 209 (1995)
M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996); S. Haroche, Rev. Mod. Phys. 85, 1083 (2013); S. Haroche, M. Brune, J.M. Raimond, Phys. Today 66, 27 (2013); S. Haroche, J.M. Raimond, Exploring the Quantum, Atoms, Cavities, Photons (Oxford University Press, Oxford, 2006)
C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon Inter Actions: Basic Processes and Applications. Wiley Science Paperback Series (Wiley, London, 1992)
G. Drake, Phys. Rev. A 34, 2871 (1986)
I. Duck, E.C.G. Sudarsahan, Pauli and the Spin-Statistics Theorem (World Scientific, Singapore, 1997)
R.P. Feynman, R.B. Leighton, M. Sands, Feynman Lectures on Physics 3 (Addison-Wesley, Reading, 1971)
R.P. Feynman, R.B. Leighton, M. Sands, Feynman Lectures on Physics 1, 2 (Addison-Wesley, Reading, 1977)
C.J. Foot, Atomic Physics. Oxford Master Series in Physics (Oxford University Press, Oxford, 2007)
S.S. Hodgman et al., Phys. Rev. Lett. 103, 053002 (2009); G.W.F. Drake, Phys. Rev. A 3, 908 (1971)
L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Non-relativistic Theory (Elsevier/Butterworth-Heinemann, Amsterdam/London, 1976)
A. Messiah, Quantum Mechanics (North Holland, Amsterdam, 1961)
J. Schwinger et al., Classical Electrodynamics (Perseus Books, Cambridge, 1998)
J.R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions (Dover Publications, New York, 2006)
S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008)
A. Zee, Einstein Gravity in a Nutshell (Princeton University Press, Princeton, 2013)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Zamastil, J., Benda, J. (2017). Dynamics: The Nonrelativistic Theory. In: Quantum Mechanics and Electrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-65780-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-65780-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65779-0
Online ISBN: 978-3-319-65780-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)