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# 物理代写|热力学代写Thermodynamics代考|MECH3720

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## 物理代写|热力学代写Thermodynamics代考|The de Broglie Hypothesis

When Einstein modeled radiation as photons, he essentially postulated that light can act as a particle as well as a wave. In an analogous fashion, de Broglie suggested that matter can act as a wave as well as a particle. In other words, he postulated that wave-particle duality should hold for both matter and electromagnetic radiation. Therefore, for macroscopic systems, light and matter would display their traditional wave and particle properties, respectively. In contrast, for microscopic systems, light would behave as a particle while matter would behave as a wave.

From classical electromagnetic (or special relativity) theory, the linear momentum carried by a beam of parallel light is
$$p=\frac{\varepsilon}{c},$$

where $\varepsilon$ is the energy of the beam and $c$ is the speed of light. Substituting Eq. (5.1) into Eq. (5.15), we find that, for a single photon,
$$p=\frac{h v}{c}=\frac{h}{\lambda},$$
where we have again recognized that $\lambda v=c$. Therefore, according to Eq. (5.16), the wavelength of an electromagnetic wave can be linked to its momentum, although the latter concept is normally associated with particles. Similarly, de Broglie reasoned, the momentum of a particle, as imaginatively affiliated with “matter waves,” can be linked in a reverse manner to the wavelength via a simple transformation of Eq. (5.16) to
$$\lambda=\frac{h}{p} .$$
Because Planck’s constant is miniscule $\left(h=6.6261 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)$, Eq. (5.17) suggests that a large mass will always produce matter waves having a wavelength much too small to affect the dynamics of classical mechanical systems. In a similar fashion, fundamental particles are more likely to be associated with much larger wavelengths approaching atomic dimensions; in this case, the behavior of the particle will be strongly influenced by its accompanying wave characteristics. For this reason, the prediction of particle behavior within atomic and molecular systems requires a probabilistic rather than deterministic approach, particularly when evaluating particle location or momentum. Such behavior comports well with our previous notion that fundamental particles are normally indistinguishable.

We end our introduction to matter waves by pointing out an important relation between de Broglie’s hypothesis and the Bohr model for atomic hydrogen. Combining Eqs. (5.8) and (5.17), we obtain
$$\lambda=\frac{h}{m_e v}=\frac{2 \pi r}{n},$$
so that the wavelength of a matter wave affiliated with any electronic orbit of atomic hydrogen must be an integer fraction of its orbital circumference. In short, the assigned wavelength will conform to an electronic orbit only if its associated matter wave remains in phase around the nucleus. This phase condition avoids destructive interference, which would ultimately destroy any matter waves inherently prescribing electronic behavior at atomic dimensions.

## 物理代写|热力学代写Thermodynamics代考|A Heuristic Introduction to the Schrödinger Equation

We know from standard electromagnetic theory that macroscopic radiation can be modeled successfully via a classical wave equation. If, for atomic dimensions, matter behaves as a wave, should not a similar wave formulation hold for matter displaying microscopic behavior? Indeed, if an analogous expression could be developed for matter waves, might we then have a consistent rubric for quantum behavior, unlike the partly classic and partly quantum tactic used to model atomic hydrogen? By fostering such queries, de Broglie’s hypothesis eventually set the stage for the mathematical prowess of Erwin Schrödinger (1887-1961). The resulting Schrödinger wave equation is now considered to be a fundamental law of quantum mechanics, similar to the primary laws of classical mechanics, thermodynamics or electromagnetics. Hence, our upcoming presentation should not be considered a derivation of the Schrödinger wave equation, but rather a heuristic rationale for its formulation. As for other fundamental laws in science, its truth must rest solely on its ultimate capability for both explaining and predicting experimental behavior.

Since we have presumed an analogy between matter waves and electromagnetic waves, we begin by considering the wave equation for electromagnetic radiation in a homogeneous, uncharged, and nonconducting medium. For a single Cartesian dimension, the electric field, $E$, is governed by
$$\frac{\partial^2 E}{\partial x^2}=\frac{1}{\mathrm{v}^2} \frac{\partial^2 E}{\partial t^2},$$
where $v$ is the wave velocity and $t$ is the time. Schrödinger reasoned that this wave equation should apply to matter waves if account is taken of the potential energy of the particle. On this basis, he defined a wave function, $\Psi$, for matter waves in analogy to $E$, so that
$$\frac{\partial^2 \Psi}{\partial x^2}=\frac{1}{\mathrm{v}^2} \frac{\partial^2 \Psi}{\partial t^2} .$$
Depending on the specific boundary conditions, many solutions are possible for this one-dimensional wave equation. For simplicity, however, we consider only the well-known solution given by
$$\Psi(x, t)=C e^{i(k x-\omega t)},$$

where $C$ is a constant and the negative sign indicates wave propagation in the positive $x$-direction. From wave theory, the propagation number, $k$, is related to the wavelength, $\lambda$, by
$$k=\frac{2 \pi}{\lambda}$$
and, similarly, the angular velocity, $\omega$, is related to the frequency, $v$, by
$$\omega=2 \pi v .$$
Because the wave velocity $v=v \lambda$, from Eqs. (5.20) and (5.21) we also have
$$v=\frac{\omega}{k} .$$

## 物理代写|热力学代写Thermodynamics代考|The de Broglie Hypothesis

$$p=\frac{\varepsilon}{c},$$

$$p=\frac{h v}{c}=\frac{h}{\lambda},$$

$$\lambda=\frac{h}{p} .$$

$$\lambda=\frac{h}{m_e v}=\frac{2 \pi r}{n},$$

## 物理代写|热力学代写Thermodynamics代考|A Heuristic Introduction to the Schrödinger Equation

$$\frac{\partial^2 E}{\partial x^2}=\frac{1}{\mathrm{v}^2} \frac{\partial^2 E}{\partial t^2},$$

$$\frac{\partial^2 \Psi}{\partial x^2}=\frac{1}{\mathrm{v}^2} \frac{\partial^2 \Psi}{\partial t^2} .$$

$$\Psi(x, t)=C e^{i(k x-\omega t)},$$

$$k=\frac{2 \pi}{\lambda}$$

$$\omega=2 \pi v .$$

$$v=\frac{\omega}{k} .$$

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