### 物理代写|量子力学代写quantum mechanics代考|PHYS3040

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子力学代写quantum mechanics代考|Classical Optics

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

## 物理代写|量子力学代写quantum mechanics代考|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
$$\langle\varepsilon(\nu)\rangle=\frac{h \nu}{e^{h \nu / k_{B} T}-1} \quad ; \text { Planck distribution }$$
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
$$\frac{h \nu}{e^{h \nu / k_{B} T}-1} \rightarrow h \nu e^{-h \nu / k_{B} T} \quad ; h \nu \gg k_{B} T$$

## 物理代写|量子力学代写quantum mechanics代考|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.Sound waves in materials also regularly exhibit particle properties through phonons, which satisfy analogous relations to the above. ${ }^{4}$

## 物理代写|量子力学代写quantum mechanics代考|Classical Optics

Ψ(X,吨)=和一世(ķX−ω吨)=和一世ķ(X−C吨);ω=ķC

ω=2圆周率ν=ķC=2圆周率Cλ

∂2Ψ(X,吨)∂X2=1C2∂2Ψ(X,吨)∂吨2; 波动方程

## 物理代写|量子力学代写quantum mechanics代考|Planck Distribution

⟨e(ν)⟩=ķ乙吨; sho均分

ķ乙=1.381×10−23 Ĵ/∘ķ; 玻尔兹曼常数

⟨e(ν)⟩=Hν和Hν/ķ乙吨−1; 普朗克分布

H2圆周率≡⁇=1.055×10−34Ĵs; 普朗克常数

Hν和Hν/ķ乙吨−1→ķ乙吨;Hν≪ķ乙吨

Hν和Hν/ķ乙吨−1→Hν和−Hν/ķ乙吨;Hν≫ķ乙吨

e=Hν; 光子

p=HνC; 光子

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## MATLAB代写

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