## 物理代写|电动力学代写electromagnetism代考|PHYS3040

2023年3月29日

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## 物理代写|电动力学代写electromagnetism代考|Coherent States

In this section we confine attention initially to a single mode of the field with definite polarisation; then we may drop the labels $k$ and $\lambda$. We now wish to investigate further the properties of the adjoint pair of operators $\mathrm{c}$ and $\mathrm{c}^{+}$for an oscillator of frequency $\omega$. As just discussed, the number operator for the mode is
$$\mathbf{n}=\mathrm{c}^{+} \mathrm{c}$$ with eigenstates given by
$$\mathrm{n}|n\rangle=n|n\rangle$$
The coherent states of the oscillator are the eigenstates of the annihilation operator $\mathrm{c}$,
$$c|\alpha\rangle=\alpha|\alpha\rangle$$
Using (7.13), this is explicitly
$$\left(\frac{i}{\sqrt{2 \hbar \omega}} \mathrm{P}+\sqrt{\frac{\omega}{2 \hbar}} \mathrm{X}\right)|\alpha\rangle=\alpha|\alpha\rangle$$
so that in the coordinate $(X)$ representation for the oscillator the probability amplitude $\langle X \mid \alpha\rangle$ satisfies a first-order differential equation,
$$\left(\sqrt{\frac{\hbar}{2 \omega}} \frac{\mathrm{d}}{\mathrm{d} X}+\sqrt{\frac{\omega}{2 \hbar}} X\right)\langle X \mid \alpha\rangle=\alpha\langle X \mid \alpha\rangle .$$
Since $\mathrm{c}$ is not self-adjoint, the eigenvalues $\alpha$ will in general be complex numbers. The normalised solution is ${ }^2$
$$\langle X \mid \alpha\rangle=\left(\frac{\omega}{\pi \hbar}\right)^{\frac{1}{4}} \exp \left(-\left(\sqrt{\frac{\omega}{2 \hbar}} X-\alpha_R\right)^2+i \sqrt{\frac{2 \omega}{\hbar}} \alpha_l X\right),$$
where we have separated $\alpha$ into real $\left(\alpha_R\right)$ and imaginary $\left(\alpha_l\right)$ parts.
Since $\langle\alpha|\mathrm{P}| \alpha\rangle$ and $\langle\alpha|\mathrm{X}| \alpha\rangle$ are both real, we can put
\begin{aligned} & \alpha_R=\sqrt{\frac{\omega}{2 \hbar}}\langle\alpha|\mathrm{X}| \alpha\rangle \equiv \sqrt{\frac{\omega}{2 \hbar}} u, \ & \alpha_l=\frac{1}{\sqrt{2 \hbar \omega}}\langle\alpha|\mathrm{P}| \alpha\rangle \equiv \sqrt{\frac{\hbar}{2 \omega}} v . \end{aligned}

## 物理代写|电动力学代写electromagnetism代考|Coherence of the Electromagnetic Field

The coherence properties of the electromagnetic field are described in terms of the properties of the mean values of polynomials of the field strength operators. These operators will involve products of the annihilation and creation operators $\left(\mathrm{c}i, \mathrm{c}_j^{+}\right)$for the modes of the field which can be brought to normal ordered form. Thus the study of coherence from the theoretical point of view can be based on the evaluation of the correlation functions of the annihilation and creation operators, $$\left\langle\mathrm{c}{i_1}^{+} \ldots \mathrm{c}{i_n}^{+} \mathrm{c}{i_{n+1}} \ldots . \mathrm{c}{i{n+m}}\right\rangle_\rho=G^{(n, m)}\left(i_1, \ldots i_n, i_{n+1}, \ldots i_{n+m}\right)$$
where $\rho$ is a density operator for the field and $\langle\Lambda\rangle_\rho=\operatorname{Tr}(\rho \Omega)$ according to (5.18); the annihilation and creation operators in (7.78) are all taken at the same instant in time.
The state of the electromagnetic field described by the density operator $\rho$ is said to be fully coherent if there exists a sequence of complex numbers $\left{z_i: z_1, z_2, \ldots\right}$ such that for every value of $n$ and for every set of indices $i_1, \ldots i_n, i_{n+1}, \ldots i_{2 n}$ we have
$$G^{(n, n)}\left(i_1, \ldots i_n, i_{n+1}, \ldots i_{2 n}\right)=\prod_{k=1}^n z_{i_k}^* \prod_{m=n+1}^{2 n} z_{i_m} .$$
If the correlation functions (7.78) possess this property only for $n \leq M$, we say that the state of the field has only $M$ th order coherence [7]. The lowest-order correlation function, $G^{(1,1)}$, is the most familiar since it determines the intensity of scattered light in a scattering experiment, and hence cross sections.

An alternative formulation, which parallels the usual description of coherence in classical electromagnetism, utilises an average over products of the electric field operators,
$$G^{(n+m)}\left(t_1, \ldots t_n ; t_{n+1}, \ldots t_{n+m}\right)=\left\langle E\left(t_1^{-} \ldots E\left(t_n\right)^{-} E\left(t_{n+1}\right)^{+} \ldots E\left(t_{n+m}\right)^{+}\right\rangle\right.$$ such that $G^{(1,1)}(t ; t)$ is again proportional to the scattered intensity. The power spectrum $S(\omega)$ of fluorescence is the Fourier transform of the two-point correlation function $G^{(1,1)}(t ; t+\tau)$. These remarks apply equally to classical and quantum descriptions of the field; the difference between the two accounts lies in the interpretation of the averaging implied by $\langle\ldots\rangle$ in $(7.78)$, $(7.80)$. Once one goes beyond simple intensity measurements $\left(G^{(1,1)}\right)$, differences arise in these correlation functions according to classical and quantum electrodynamics; measurements of the properties of the electromagnetic field in, for example, spontaneous emission and resonance fluorescence experiments confirm the quantum nature of the field [8].

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Coherent States

$$\mathbf{n}=\mathrm{c}^{+} \mathrm{c}$$

$$\mathrm{n}|n\rangle=n|n\rangle$$

$$c|\alpha\rangle=\alpha|\alpha\rangle$$

$$\left(\frac{i}{\sqrt{2 \hbar \omega}} \mathrm{P}+\sqrt{\frac{\omega}{2 \hbar}} \mathrm{X}\right)|\alpha\rangle=\alpha|\alpha\rangle$$这样在坐标 $(X)$ 表示振荡器的概率幅度 $\langle X| \alpha$ 满足一 阶微分方程，
$$\left(\sqrt{\frac{\hbar}{2 \omega}} \frac{\mathrm{d}}{\mathrm{d} X}+\sqrt{\frac{\omega}{2 \hbar}} X\right)\langle X \mid \alpha\rangle=\alpha\langle X \mid \alpha\rangle$$

$$\langle X \mid \alpha\rangle=\left(\frac{\omega}{\pi \hbar}\right)^{\frac{1}{4}} \exp \left(-\left(\sqrt{\frac{\omega}{2 \hbar}} X-\alpha_R\right)^2+i \sqrt{ }\right.$$

$$\alpha_R=\sqrt{\frac{\omega}{2 \hbar}}\langle\alpha|\mathrm{X}| \alpha\rangle \equiv \sqrt{\frac{\omega}{2 \hbar}} u, \quad \alpha_l=\frac{1}{\sqrt{2 \hbar \omega}}\langle c$$

## 物理代写|电动力学代写electromagnetism代考|Coherence of the Electromagnetic Field

$$G^{(n, n)}\left(i_1, \ldots i_n, i_{n+1}, \ldots i_{2 n}\right)=\prod_{k=1}^n z_{i_k}^* \prod_{m=n+1}^{2 n} z_{i_m}$$

$$G^{(n+m)}\left(t_1, \ldots t_n ; t_{n+1}, \ldots t_{n+m}\right)=\left\langleE \left( t_1^{-} \ldots E\left(t_n\right)\right.\right.$$

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## MATLAB代写

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