物理代考| Classical Optics量子力学代写代写 - 代考代写：100%准时可靠您的作业代写专家

Consider a non-dispersive wave which is the real part of
$$
\Psi(x, t)=e^{i(k x-\omega t)}=e^{i k(x-c t)} \quad ; \omega=k c
$$
Here $c$ is the velocity of the wave, and the frequency and wavelength are related by
$$
\omega=2 \pi \nu=k c=2 \pi \frac{c}{\lambda}
$$
As we have seen, this could be an electromagnetic wave in vacuum, a transverse wave on a string under tension, or the sound wave in a medium. This wave satisfies the wave equation
$$
\frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}=\frac{1}{c^{2}} \frac{\partial^{2} \Psi(x, t)}{\partial t^{2}} \quad ; \text { wave equation }
$$
We have also seen that a linear combination of two such waves with slightly different wavenumbers $k$, produces an amplitude modulated signal. A more general linear combination can produce a localized wave packet, or pulse.

Huygen’s principle states that each point on a wavefront acts as a source of an outgoing spherical wave. From this, and its generalizations, one derives single-slit diffraction, two-slit and multi-slit interference, and most of classical wave optics. ${ }^{1}$

Planck Distribution

Early in the twentieth century, Planck was studying the distribution of energy as a function of frequency for the electromagnetic radiation in a cavity. Normal modes are uncoupled simple harmonic oscillators. The classical equipartition theorem says that the energy of a simple harmonic oscillator at an absolute temperature $T$ is
$$
\langle\varepsilon(\nu)\rangle=k_{B} T \quad ; \text { equipartition for s.h.o. }
$$
where $k_{B}$ is Boltzmann’s constant
$$
k_{B}=1.381 \times 10^{-23} \mathrm{~J} /{ }^{\circ} \mathrm{K} \quad ; \text { Boltzmann’s constant }
$$
Since there is no limit to how small the wavelength can be, or how high the frequency, this classical result says there should be an ever-increasing energy as a function of frequency for the radiation in a cavity, the so-called ultraviolet catastrophe. ${ }^{2}$

To fit his data, Planck employed an empirical expression of the form
$$
\langle\varepsilon(\nu)\rangle=\frac{h \nu}{e^{h \nu / k_{B} T}-1}
$$
where $h$ is a constant obtained from the fit, now known as Planck’s constant
$$
\frac{h}{2 \pi} \equiv \hbar=1.055 \times 10^{-34} \mathrm{Js} \quad ; \text { Planck’s constant }
$$
Note that at low frequency, the Planck distribution reproduces the equipartition result
$$
\frac{h \nu}{e^{h \nu / k_{B} T}-1} \rightarrow k_{B} T \quad ; h \nu \ll k_{B} T
$$
while at high frequency, it now disappears exponentially
$$
\frac{h \nu}{e^{h \nu / k_{B} T}-1} \rightarrow h \nu e^{-h \nu / k_{B} T} \quad ; h \nu \gg k_{B} T
$$
One can ask where this empirical Planck distribution might come from. Suppose that in each mode with frequency $\nu$ in the cavity it is possible to have any number $n$ of photons, each with energy
$$
\varepsilon=h \nu \quad ; \text { photon energy }
$$

Then the mean energy in the mode at the temperature $T$ follows from an elementary statistical calculation with the Boltzmann weighting factor $e^{-n h \nu / k_{B} T}$ as
$$
\begin{aligned}
\langle\varepsilon(\nu)\rangle &=\frac{\sum_{n=0}^{\infty}(n h \nu) e^{-n h \nu / k_{B} T}}{\sum_{n=0}^{\infty} e^{-n h \nu / k_{B} T}} \
&=-\frac{d}{d\left(1 / k_{B} T\right)} \ln \left(\sum_{n=0}^{\infty} e^{-n h \nu / k_{B} T}\right)
\end{aligned}
$$
The sum is just a geometric series $^{3}$
$$
\sum_{n=0}^{\infty} x^{n}=\frac{1}{1-x}
$$
It follows that
$$
\begin{aligned}
\langle\varepsilon(\nu)\rangle &=\frac{d}{d\left(1 / k_{B} T\right)} \ln \left(1-e^{-h \nu / k_{B} T}\right) \
&=\frac{h \nu}{e^{h \nu / k_{B} T}-1}
\end{aligned}
$$
This reproduces Planck’s distribution.

Photons

The fact that light waves actually consist of photons, which manifest particle properties, was demonstrated by Einstein in his examination of the photoelectric effect, where light shining on various solids ejects electrons. The photons of light each have an energy
$$
\varepsilon=h \nu \quad ; \text { photon }
$$
We know the momentum flux in an electromagnetic wave is $1 / c$ times the energy flux, and hence each photon in light also has a momentum
$$
p=\frac{h \nu}{c} \quad ; \text { photon }
$$
Photons are now observed every day in the laboratory as single events in low-intensity radiation detectors.

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考虑一个非色散波，它是
$$
\Psi(x, t)=e^{i(k x-\omega t)}=e^{i k(x-c t)} \quad ; \omega=k c
$$
这里$c$是波的速度，频率和波长的关系为
$$
\omega=2 \pi \nu=k c=2 \pi \frac{c}{\lambda}
$$
正如我们所见，这可能是真空中的电磁波、受拉弦上的横波或介质中的声波。该波满足波动方程
$$
\frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}=\frac{1}{c^{2}} \frac{\partial^{2} \Psi (x, t)}{\partial t^{2}} \quad ; \text { 波动方程 }
$$
我们还看到，具有略微不同波数 $k$ 的两个此类波的线性组合会产生幅度调制信号。更一般的线性组合可以产生局部波包或脉冲。

惠更斯原理指出，波前的每个点都充当出射球面波的源。从这一点及其概括，可以得出单缝衍射、双缝和多缝干涉以及大多数经典波动光学。 ${ }^{1}$