# Quantization of the Electromagnetic Field

• MisterX
In summary, quantization of the electromagnetic field is a fundamental concept in quantum mechanics that explains the discrete nature of electromagnetic radiation. It is important because it helps us understand the behavior of light and other forms of electromagnetic radiation at the atomic and subatomic level, and has various practical applications in modern technology. This concept was first introduced by Max Planck in 1900 and has been further developed by other scientists such as Albert Einstein and Niels Bohr. It is closely related to the wave-particle duality of light and has the potential to greatly impact our daily lives in the future through the development of new technologies such as quantum computing and quantum cryptography.
MisterX
We have recently begun learning about quantization of the electromagnetic field and I would to understand more. It is tempting to want to connect the number states of a definite momentum and polarization to the concept of a classical plane wave. However it seems less straightforward than I might have thought.
$$\vec{E}_{classical} = C\cos\left(\mathbf{k}\cdot \mathbf{r}-\omega t + \phi\right)$$
For the quantized fields we have been using
$$\mathbf{A}\left(\mathbf{r}, t\right) \propto \int \frac{d^3k}{\sqrt{\omega}} \sum_\alpha \boldsymbol{\epsilon}_\alpha a_\alpha\left(\mathbf{k}\right)e^{i\mathbf{k}\cdot \mathbf{r}}e^{-i\omega t} + \boldsymbol{\epsilon}^*_\alpha a^\dagger_\alpha\left(\mathbf{k}\right)e^{-i\mathbf{k}\cdot \mathbf{r}}e^{i\omega t}$$
$$\mathbf{E}\left(\mathbf{r}, t\right) = -\frac{\partial}{\partial t} \mathbf{A} \;\;\;\;\;\;\;\; \mathbf{B}\left(\mathbf{r}, t\right) = \boldsymbol{\nabla}\times \mathbf{A}$$
I notice that these operators all change the occupation numbers by $\pm 1$. What is the significance of this? I would like to somehow see an oscillating, measurable quantity with frequency $\omega$ , perhaps akin to the classical plane wave. I suppose we might be interested in the above operators, and perhaps also the average number of photons with some particular $\mathbf{k}$ .
I have read of the "coherent states," which would be eigenvectors of $a_\alpha\left(\mathbf{k}\right)$ .
$$\mid \alpha \rangle = e^{\left|\alpha\right|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}\mid n \rangle \;\;\;\;\;\;\;\;a\mid \alpha \rangle = \alpha\mid \alpha \rangle$$
Which evolve in time as
$U\left(t, 0\right)\mid \alpha \rangle = \mid \alpha e^{-i\omega t} \rangle$. However, $a$ is not an observable. I am not sure how to interpret the coherent states, or the action $\mathbf{A}$ upon them. I worked out that $$a^\dagger\mid \alpha \rangle = e^{\left|\alpha\right|^2/2}\sum_n \frac{n\alpha^{n-1}}{\sqrt{n!}}$$
I guess my questions are relating to how to interpret the operators $\mathbf{A}, \mathbf{E},\, \text{and}\, \mathbf{B}$ . How do we obtain something resembling the classical oscillating quantities? What states correspond most closely with a classical plane wave? What are the eigenvectors of $\mathbf{A}$?

MisterX said:
I guess my questions are relating to how to interpret the operators $\mathbf{A}, \mathbf{E},\, \text{and}\, \mathbf{B}$ . How do we obtain something resembling the classical oscillating quantities? What states correspond most closely with a classical plane wave?

The coherent states are the states that are most similar to classical plane waves.

Recall the 1D harmonic oscillator. The coherent states of the 1D harmonic oscillator have exactly the same form as the ones you wrote out for the electromagnetic field. They are the states in which both the position and and velocity of the particle have small uncertainties, so that the particle "looks classical," at least if the parameter ##\alpha## is ##\gg 1## (which is the condition for the amplitude of the motion to be much larger than the uncertainty in the position). The coherent states of the simple harmonic oscillator are the states most similar to classical simple harmonic motion.

Similarly, the coherent states of the EM field are the states for which both the fields and their time derivatives have small uncertainties, so that the field "looks classical" if ##\alpha \gg 1## (which is the condition for the uncertainty in the fields to be much smaller than the magnitude of the fields).

Try evaluating the expectation value

##\langle \alpha | \vec{E}(\vec{r}, t) | \alpha \rangle##

in some coherent state ##|\alpha\rangle##. This will give you the expectation value of the electric field in the coherent state; it should look like a plane wave with an amplitude proportional to ##\alpha## and a wave number, frequency, and polarization determined by which creation operator ##a^{\dagger}_{\lambda}## you used to construct the coherent state ##|\alpha\rangle##.

MisterX said:
What are the eigenvectors of $\mathbf{A}$?

They are analogous to the eigenvectors of ##x## in the case of the 1D harmonic oscillator. Eigenvectors of ##x## have definite position; eigenvectors of ##\vec{A}## have definite values of ##\vec{A}##. But in each case, these states are rather unphysical. For instance, they have infinite energy; a quantum particle with 0 position uncertainty has infinite kinetic energy because there is an uncertainty relation between the position and momentum. Similarly a quantum field has an uncertainty relation between its value and its time derivative, so that if there is 0 uncertainty in the field value the time derivative has infinite uncertainty, which gives an infinite energy.

More physical (and more interesting) states include the eigenstates of the Hamiltonian and the coherent states.

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1 person

## 1. What is quantization of the electromagnetic field?

Quantization of the electromagnetic field is a fundamental concept in quantum mechanics that explains the discrete nature of electromagnetic radiation. It states that light and other forms of electromagnetic radiation can only be emitted or absorbed in discrete packets of energy called photons.

## 2. Why is quantization of the electromagnetic field important?

Quantization of the electromagnetic field is important because it helps us understand the behavior of light and other forms of electromagnetic radiation at the atomic and subatomic level. It also plays a crucial role in fields such as quantum optics, quantum information processing, and quantum computing.

## 3. How was quantization of the electromagnetic field discovered?

The concept of quantization of the electromagnetic field was first introduced by Max Planck in 1900 when he proposed that energy is not continuous, but rather comes in discrete packets. This idea was further developed by Albert Einstein in 1905 when he explained the photoelectric effect, and later by Niels Bohr in 1913 when he proposed the quantization of electron energy levels in atoms.

## 4. What is the relationship between quantization of the electromagnetic field and the wave-particle duality of light?

Quantization of the electromagnetic field is closely related to the wave-particle duality of light, which states that light can behave as both a wave and a particle. The quantization of the electromagnetic field explains this behavior by showing that light can act as a wave when propagating, but is emitted and absorbed in discrete packets of energy (photons) like particles.

## 5. How does quantization of the electromagnetic field impact our everyday lives?

Quantization of the electromagnetic field has many practical applications in modern technology, such as in lasers, solar cells, and electronic devices. It also plays a crucial role in the development of new technologies such as quantum cryptography and quantum computing, which have the potential to greatly impact our daily lives in the future.

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