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# 物理代写|原子物理代考Atomic and Molecular Physics代考|PHYS40500 The dual nature of physical phenomena

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## 物理代写|原子物理代考Atomic and Molecular Physics代考|Wave–matter duality

It is a matter of common experience that macroscopic physical phenomena reveal either as waves or as particles. By contrast, the Einstein hypothesis about the nature of e.m. waves as a flux of photons challenges this (pre)conception, which we now understand to be only due to the limits of our sensorial experience of the physical word. Light can in fact manifest either as a wave or as a beam of (pseudo)particles, according to the actual phenomenon we are addressing. Certainly the physics of our ocular vision or the propagation of light in vacuum are phenomena very well described by wave equations. On the other hand, the absorption/emission of light by an atomic system or the photoemission of electrons from a metal plate can only be explained by invoking the concept of photon.

In principle, we could speculate that this duality is similarly valid for massive particles, as first discussed by L de Broglie in 1924. If for a photon we can relate wave-like and particle-like properties through such relations as $E=h \nu$ or $\mathbf{p}=\hbar \mathbf{k}$ (where $\mathbf{p}$ is the photon momentum and $\mathbf{k}$ is the wavevector of the corresponding e.m. wave), then we could guess that a matter wave of wavelength $\lambda$ is associated with any particle with mass $m$ and moving with velocity $\mathbf{v}$ according to
$$\lambda=\frac{h}{p}=\frac{h}{m v}=\frac{h}{2 m E_{\text {kin }}}$$
where $\mathbf{p}=m v$ is of course the particle momentum and $E_{\text {kin }}=m v^{2} / 2$ is its kinetic energy. This statement is nothing other than a speculation if no experimental evidence is supplied to support it.

## 物理代写|原子物理代考Atomic and Molecular Physics代考|A constitutive equation for matter waves

Once we acknowledge that any microscopic particle, say an electron, behaves as a matter wave, we must duly feel committed in the search for the constitutive equation ruling over the physics of such unfamiliar waves. This is indeed a very subtle problem that, as a matter of fact, is still an open issue of intense fundamental research. We can nevertheless draw a semi-empirical picture which could be the conceptual guideline in developing a more satisfying formal theory.

As a first step we recognise that any wave, of whichever nature, is described by the d’Alembert equation. Accordingly, if we name $\Psi(x, t)$ the wavefunction of the matter wave describing an electron in one-dimensional motion with speed $v$, we can write
$$\frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}-\frac{1}{v^{2}} \frac{\partial^{2} \Psi(x, t)}{\partial t^{2}}=0$$
where $x$ indicates the direction of motion. By assuming an harmonic time dependence $\Psi(x, t)=\psi(x) \exp (i \omega t)$ we easily get
$$\frac{d^{2} \psi(x)}{d x^{2}}+\left(\frac{\omega}{v}\right)^{2} \psi(x)=0$$
where $\omega=v k=2 \pi v / \lambda$ and $k=2 \pi / \lambda$ is the wavenumber of the matter wave with de Broglie wavelength $\lambda=h / m_{\mathrm{e}} v$. Equation (1.30) is easily rewritten in the more useful form
$$\frac{d^{2} \psi(x)}{d x^{2}}+\left(\frac{m_{\mathrm{e}} v}{\hbar}\right)^{2} \psi(x)=0$$

## 物理代写|原子物理代考Atomic and Molecular Physics代考|Wave-matter duality

$$\lambda=\frac{h}{p}=\frac{h}{m v}=\frac{h}{2 m E_{\text {kin }}}$$

## 物理代写|原子物理代考Atomic and Molecular Physics代考|A constitutive equation for matter waves

$$\frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}-\frac{1}{v^{2}} \frac{\partial^{2} \Psi(x, t)}{\partial t^{2}}=0$$

$$\frac{d^{2} \psi(x)}{d x^{2}}+\left(\frac{\omega}{v}\right)^{2} \psi(x)=0$$

$$\frac{d^{2} \psi(x)}{d x^{2}}+\left(\frac{m_{\mathrm{e}} v}{\hbar}\right)^{2} \psi(x)=0$$

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