Abstract
In this chapter we show that the electromagnetic field can be considered as an infinite set of harmonic oscillators, each corresponding to a particular value of the frequency, wave vector, and a particular state of polarization. Comparing with the quantum mechanical treatment of harmonic oscillators, we replace the generalized coordinates and generalized momenta by operators. By imposing the commutation relations between the canonical variables, it is shown that the energy of each oscillator can increase or decrease by integral multiples of a certain quantum of energy; this quantum of energy is known as the photon. Having quantized the field, we show that the state which corresponds to a given number of photons (also referred to as the number state) for a particular mode does not correspond to the classical plane wave. Indeed, we show that the eigenstates of the annihilation operator (which are known as the coherent states) resemble the classical plane wave for large intensities.
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Notes
- 1.
This is because \(\nabla \times \left( {\nabla \chi } \right) = 0\) for arbitrary χ.
- 2.
Notice that
$$\frac{{\partial {\textbf{H}}_\lambda }}{{\partial Q_\lambda }} = \omega _\lambda ^2 Q_\lambda = \omega _\lambda ^2 \left( {\varepsilon _0 V} \right)^{1/2} \left( {q_\lambda + q_\lambda ^{\ast}} \right) = i\left( {\varepsilon _0 V\omega _\lambda ^2 } \right)^{1/2} \left( {\dot q_\lambda - \dot q_\lambda ^{\ast}} \right) = - \dot P_\lambda $$Similarly
$$\frac{{\partial {\textbf{H}}_\lambda }}{{\partial P_\lambda }} = \dot Q_\lambda $$which are nothing but Hamilton’s equations of motion (see, e.g., Goldstein (1950)). Thus \(Q_\lambda \) and \(P_\lambda \) are the canonical coordinates.
- 3.
We say it loosely because it is not possible to define a phase operator which is real (see Section 9.7).
- 4.
In the Heisenberg representation, the expectation value of \(\hat a\) would have been \(\left\langle {\Psi \left( 0 \right)} \right|\hat a\left( t \right)\left| {\Psi \left( 0 \right)} \right\rangle = \left\langle \alpha \right|\hat a{e}^{{i}\omega t} \left| \alpha \right\rangle = a^{\ast}{e}^{{i}\omega t} \), which is the same as expressed by Eq. (9.111).
- 5.
It is of interest to point out that even in nuclear counting, the uncertainty in the actual count is \(N^{1/2} \) (see, e.g., Bleuler and Goldsmith (1952)).
- 6.
The appearance of the term \(\left( {n + 1} \right)\) is because of the relation
$$\hat a^{\dag} \left| n \right\rangle = \left( {n + 1} \right)^{1/2} \left| {n + 1} \right\rangle $$ - 7.
This can be shown by noting that repeated application of Eq. (9.200) gives
$$N_{{\textrm{op}}} \phi ^m - \phi ^m N_{{\textrm{op}}} = - {i}m\phi ^{m - 1} $$Now
$$\left[ {N_{{\textrm{op}}} ,{e}^{{i}\phi } } \right] = \sum\limits_m {\frac{{{i}^m }}{{m!}}} \left[ {N_{{\textrm{op}}} ,e^m } \right] = \sum {\frac{{{i}^{m - 1} \phi ^{m - 1} }}{{\left( {m - 1} \right)!}}} = {e}^{{i}\phi } $$ - 8.
References
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Thyagarajan, K., Ghatak, A. (2011). Quantum Theory of Interaction of Radiation Field with Matter. In: Lasers. Graduate Texts in Physics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-6442-7_9
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DOI: https://doi.org/10.1007/978-1-4419-6442-7_9
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