### 物理代写|理论力学作业代写Theoretical Mechanics代考|PHYS386

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

Other important hypothesis of quantum mechanics is the correspondence principle. We assume the following suitable statement: the wave-function $\Psi(\mathbf{q}, t)$ can be approximated in the quasi-classic limit $\hbar \rightarrow 0$ as follows (12):
$$\Psi(\mathbf{q}, t) \sim \exp [i S(\mathbf{q}, t) / \hbar],$$
where $S(\mathbf{q}, t)$ is the classical action of the system associated with the known Hamilton-Jacobi theory of classical mechanics. Physically, this principle expresses that quantum mechanics contains classical mechanics as an asymptotic theory. At the same time, it states that quantum mechanics should be formulated under the correspondence with classical mechanics. Physically speaking, it is impossible to introduce a consistent quantum mechanics formulation without the consideration of classical notions. Precisely, this is a very consequence of the complementarity between the dynamical description performed in terms of the wave function $\Psi$ and the space-time classical description associated with the results of experimental measurements. The completeness of quantum description performed in terms of the wave function $\Psi$ demands both the presence of quantum statistical ensemble and classical objects that play the role of measuring instruments.

Historically, correspondence principle was formally introduced by Bohr in 1920 (16), although he previously made use of it as early as 1913 in developing his model of the atom (17). According to this principle, quantum description should be consistent with classical description in the limit of large quantum numbers. In the framework of Schrödinger’s wave mechanics, this principle appears as a suitable generalization of the so-called optics-mechanical analogy (18). In geometric optics, the light propagation is described in the so-called rays approximation. According to the Fermat’s principle, the ray trajectories extremize the optical length $\ell[\mathbf{q}(s)]$ :
$$\ell[\mathbf{q}(s)]=\int_{s_{1}}^{s)} n[\mathbf{q}(s)] d s \rightarrow \delta \ell[\mathbf{q}(s)]=0,$$
which is calculated along the curve $\mathbf{q}(s)$ with fixed extreme points $\mathbf{q}\left(s_{1}\right)-p$ and $\mathbf{q}\left(s_{2}\right)-Q$. Here, $n(\mathbf{q})$ is the refraction index of the optical medium and $d s=|d \mathbf{q}|$. Equivalently, the rays propagation can be described by Eikonal equation:
$$|\nabla \varphi(\mathbf{q})|^{2}=k_{0}^{2} n^{2}(\mathbf{q}),$$
where $\varphi(\mathbf{q})$ is the phase of the undulatory function $u(\mathbf{q}, t)=a(\mathbf{q}, t) \exp [-i \omega t+i \varphi(\mathbf{q})]$ in the wave optics, $k_{0}=\omega / c$ and $c$ are the modulus of the wave vector and the speed of light in vacuum, respectively. The phase $\varphi(\mathbf{q})$ allows to obtain the wave vector $\mathbf{k}(\mathbf{q})$ within the optical medium:
$$\mathbf{k}(\mathbf{q})=\nabla \varphi(\mathbf{q}) \rightarrow k(\mathbf{q})=|\mathbf{k}(\mathbf{q})|=k_{0} n(\mathbf{q})$$
which provides the orientation of the ray propagation:
$$\frac{d \mathbf{q}(s)}{d s}=\frac{\mathbf{k}(\mathbf{q})}{|\mathbf{k}(\mathbf{q})|}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Operators of physical observables and Schrödinger equation

Physical interpretation of the wave function $\Psi(\mathbf{q}, t)$ implies that the expectation value of any arbitrary function $A(\mathbf{q})$ that is defined on the space coordinates $q$ is expressed as follows:
$$\langle A\rangle=\int|\Psi(\mathbf{q}, t)|^{2} A(\mathbf{q}) d \mathbf{q} .$$
For calculating the expectation value of an arbitrary physical observable $O$, the previous expression should be extended to a bilinear form in term of the wave function $\Psi(\mathbf{q}, t)(19)$ :
$$\langle O\rangle=\int \Psi^{}(\mathbf{q}, t) O(\mathbf{q}, \tilde{\mathbf{q}}, t) \Psi(\tilde{\mathbf{q}}, t) d \mathbf{q} d \tilde{\mathbf{q}},$$ where $O(\mathbf{q}, \tilde{\mathbf{q}}, t)$ is the kernel of the physical observable $O$. As already commented, there exist some physical observables, e.g.: the momentum $\mathbf{p}$, whose determination demands repetitions of measurements in a finite region of the space sufficient for the manifestation of wave properties of the function $\Psi(\mathbf{q}, t)$. Precisely, this type of procedure involves a comparison or correlation between different points of the space $(\mathbf{q}, \tilde{\mathbf{q}})$, which is accounted for by the kernel $O(\mathbf{q}, \tilde{\mathbf{q}}, t)$. Due to the expectation value of any physical observable $O$ is a real number, the kernel $O(\mathbf{q}, \tilde{\mathbf{q}}, t)$ should obey the hermitian condition: $$O^{}(\tilde{\mathbf{q}}, \mathbf{q}, t)=O(\mathbf{q}, \tilde{\mathbf{q}}, t)$$
As commented before, superposition principle (5) has naturally introduced the linear algebra on a Hilbert space $\mathcal{H}$ as the mathematical apparatus of quantum mechanics. Using the decomposition of the wave function $\Psi$ into a certain basis $\left{\Psi_{\alpha}\right}$, it is possible to obtain the following expressions:
$$\langle O\rangle=\sum_{\alpha \beta} a_{\hat{\alpha}}^{} O_{\alpha \beta} a_{\beta},$$ where: $$O_{\alpha \beta}=\int \Psi_{a}^{}(\mathbf{q}, t) O(\mathbf{q}, \tilde{\mathbf{q}}, t) \Psi_{\beta}(\tilde{\mathbf{q}}, t) d \tilde{\mathbf{q}} d \mathbf{q}$$

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

$$\Psi(\mathbf{q}, t) \sim \exp [i S(\mathbf{q}, t) / \hbar]$$

$$\ell[\mathbf{q}(s)]=\int_{s_{1}}^{s)} n[\mathbf{q}(s)] d s \rightarrow \delta \ell[\mathbf{q}(s)]=0$$

$$|\nabla \varphi(\mathbf{q})|^{2}=k_{0}^{2} n^{2}(\mathbf{q})$$

$$\mathbf{k}(\mathbf{q})=\nabla \varphi(\mathbf{q}) \rightarrow k(\mathbf{q})=|\mathbf{k}(\mathbf{q})|=k_{0} n(\mathbf{q})$$

$$\frac{d \mathbf{q}(s)}{d s}=\frac{\mathbf{k}(\mathbf{q})}{|\mathbf{k}(\mathbf{q})|}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Operators of physical observables and Schrödinger equation

$$\langle A\rangle=\int|\Psi(\mathbf{q}, t)|^{2} A(\mathbf{q}) d \mathbf{q} .$$

$$\langle O\rangle=\int \Psi(\mathbf{q}, t) O(\mathbf{q}, \tilde{\mathbf{q}}, t) \Psi(\tilde{\mathbf{q}}, t) d \mathbf{q} d \tilde{\mathbf{q}}$$

$$O(\tilde{\mathbf{q}}, \mathbf{q}, t)=O(\mathbf{q}, \tilde{\mathbf{q}}, t)$$
$\mathrm{~ 如 前 所 述 ， 喗 加 原 理 ~ ( 5 ) ~ 自 然 地 在 希 尔 伯 特 空 间 上 引 入 了 线 性 代 数}$ 解 $\Psi$ 进入一定的基础 Ueft{{Psi_{lalpha}\right }, 可以得到以下表达式:
$$\langle O\rangle=\sum_{\alpha \beta} a_{\hat{\alpha}} O_{\alpha \beta} a_{\beta},$$

$$O_{\alpha \beta}=\int \Psi_{a}(\mathbf{q}, t) O(\mathbf{q}, \tilde{\mathbf{q}}, t) \Psi_{\beta}(\tilde{\mathbf{q}}, t) d \tilde{\mathbf{q}} d \mathbf{q}$$

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