### 物理代写|理论力学作业代写Theoretical Mechanics代考|PHY300

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## 物理代写|理论力学作业代写Theoretical Mechanics代考|The wave function Ψ and its physical relevance

Dynamical description of a quantum system is performed in terms of the so-called the wave function $\Psi$ (12). For example, such as the frequency $\omega$ and wave vector $\mathbf{k}$ observed in electron diffraction experiments are related to dynamical variables as energy $E$ and momentum $p$ in terms of de Broglie’s relations (2). Accordingly, the wave function $\Psi(\mathbf{q}, t)$ associated with a free microparticle (as the electrons in a beam with very low intensity) behaves as follows:
$$\Psi(\mathbf{q}, t)=C \exp [-i(E t-\mathbf{p} \cdot \mathbf{q}) / \hbar]$$
Historically, de Broglie proposed the relations (2) as a direct generalization of quantum hypothesis of light developed by Planck and Einstein for any kind of microparticles (14). The experimental confirmation of these wave-particle duality for any kind of matter revealed the unity of material world. In fact, wave-particle duality is a property of matter as universal as the fact that any kind of matter is able to produce a gravitational interaction.

While the state of a system in classical mechanics is determined by the knowledge of the positions $\mathbf{q}$ and momenta $\mathbf{p}$ of all its constituents, the state of a system in the framework of quantum mechanics is determined by the knowledge of its wave function $\Psi(\mathbf{q}, t)$ (or its generalization $\Psi\left(\mathbf{q}^{1}, \mathbf{q}^{2}, \ldots, \mathbf{q}^{n}, t\right)$ for a system with many constituents, notation that is omitted hereafter for the sake of simplicity). In fact, the knowledge of the wave function $\Psi\left(\mathbf{q}, t_{0}\right)$ in an initial instant $t_{0}$ allows the prediction of its future evolution prior to the realization of a measurement (12). The wave function $\Psi(\mathbf{q}, t)$ is a complex function whose modulus $|\Psi(\mathbf{q}, t)|^{2}$ describes the probability density, in an absolute or relative sense, to detect a microparticle at the position $\mathrm{q}$ as a result of a measurement at the time $t$ (15). Such a statistical relevance of the wave function $\Psi(\mathbf{q}, t)$ about its relation with the experimental results is the most condensed expression of complementarity of quantum phenomena.

Due to its statistical relevance, the reconstruction of the wave function $\Psi(\mathbf{q}, t)$ from a given experimental situation demands the notion of statistical ensemble (12). In electron diffraction experiments, each electron in the beam manifests undulatory properties in its dynamical behavior. However, the interaction of this microparticle with a measuring instrument (a classical object as a photographic plate) radically affects its initial state, e.g.: electron is forced to localize in a very narrow region (the spot). In this case, a single measuring process is useless to reveal the wave properties of its previous quantum state. To rebuild the wave function $\Psi$ (up to the precision of an unimportant constant complex factor $e^{i \phi}$ ), it is necessary to perform infinite repeated measurements of the quantum system under the same initial conditions. Abstractly, this procedure is equivalent to consider simultaneous measurements over a quantum statistical ensemble: such as an infinite set of identical copies of the quantum system, which have been previously prepared under the same experimental procedure ${ }^{2}$. Due to the important role of measurements in the knowledge state of quantum systems, quantum mechanics is a physical theory that allows us to predict the results of certain experimental measurements taken over a quantum statistical ensemble that it has been previously prepared under certain experimental criteria (12).

## 物理代写|理论力学作业代写Theoretical Mechanics代考|The superposition principle

To explains interference phenomena observed in the double-slit experiments, the wave function $\Psi(\mathbf{q}, t)$ of a quantum system should satisfy the superposition principle (12):
$$\Psi(\mathbf{q}, t)=\sum_{a} a_{i t} \Psi_{a t}(\mathbf{q}, t) .$$
Here, $\Psi_{a}(\mathbf{q}, t)$ represents the normalized wave function associated with the $\alpha$-th independent state. As example, $\Psi_{a}(\mathbf{q}, t)$ could represent the wave function contribution associated with each slit during electron interference experiments; while the modulus $\left|a_{i c}\right|^{2}$ of the complex amplitudes $a_{\alpha}$ are proportional to incident beam intensities $I_{\alpha}$, or equivalently, the probability $p_{\alpha}$ that a given electron crosses through the $\alpha$-th slit.

Superposition principle is the most important hypothesis with a positive content of quantum theory. In particular, it evidences that dynamical equations of the wave function $\Psi(\mathbf{q}, t)$ should exhibit a linear character. By itself, the superposition principle allows to assume linear algebra as the mathematical apparatus of quantum mechanics. Thus, the wave function $\Psi(\mathbf{q}, t)$ can be regarded as a complex vector in a Hilbert space $\mathcal{H}$. Under this interpretation, the superposition formula (5) can be regarded as a decomposition of a vector $\Psi$ in a basis of independent vectors $\left{\Psi_{\alpha}\right}$. The normalization of the wave function $\Psi$ can be interpreted as the vectorial norm:
$$|\Psi|^{2}=\int \Psi^{}(\mathbf{q}, t) \Psi(\mathbf{q}, t) d \mathbf{q}=\sum_{\alpha \beta} g_{\alpha \beta} a_{i}^{} a_{\beta}=1 .$$
Here, the matrix elements $g_{\alpha \beta}$ denote the scalar product (complex) between different basis elements:
$$g_{\alpha \beta}=\int \Psi_{\alpha}^{}(\mathbf{q}, t) \Psi_{\beta}(\mathbf{q}, t) d \mathbf{q}$$ which accounts for the existence of interference effects during the experimental measurements. As expected, the interference matrix, $g_{\alpha \beta}$, is a hermitian matrix, $g_{\alpha \beta}=g_{\beta a}^{}$. The basis $\left{\Psi_{\alpha}(\mathbf{q}, t)\right}$ is said to be orthonormal if their elements satisfy orthogonality condition:
$$\int \Psi_{\alpha}^{}(\mathbf{q}, t) \Psi_{\beta}(\mathbf{q}, t) d \mathbf{q}=\delta_{\alpha \beta}$$ where $\delta_{\alpha \beta}$ represents Kroneker delta (for a basis with discrete elements) or a Dirac delta functions (for the basis with continuous elements). The basis of independent states is complete if any admissible state $\Psi \in \mathcal{H}$ can be represented with this basis. In particular, a basis with independent orthogonal elements is complete if it satisfies the completeness condition: $$\sum_{\alpha} \Psi_{a}^{}(\tilde{\mathbf{q}}, t) \Psi_{\alpha}(\mathbf{q}, t)=\delta(\tilde{\mathbf{q}}-\mathbf{q})$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|The wave function Ψ and its physical relevance

$$\Psi(\mathbf{q}, t)=C \exp [-i(E t-\mathbf{p} \cdot \mathbf{q}) / \hbar]$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|The superposition principle

$$\Psi(\mathbf{q}, t)=\sum_{a} a_{i t} \Psi_{a t}(\mathbf{q}, t) .$$

$$|\Psi|^{2}=\int \Psi(\mathbf{q}, t) \Psi(\mathbf{q}, t) d \mathbf{q}=\sum_{\alpha \beta} g_{\alpha \beta} a_{i} a_{\beta}=1 .$$

$$g_{\alpha \beta}=\int \Psi_{\alpha}(\mathbf{q}, t) \Psi_{\beta}(\mathbf{q}, t) d \mathbf{q}$$

$$\int \Psi_{\alpha}(\mathbf{q}, t) \Psi_{\beta}(\mathbf{q}, t) d \mathbf{q}=\delta_{\alpha \beta}$$

$$\sum_{\alpha} \Psi_{a}(\tilde{\mathbf{q}}, t) \Psi_{\alpha}(\mathbf{q}, t)=\delta(\tilde{\mathbf{q}}-\mathbf{q})$$

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