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物理代考| Ehrenfest’s Theorem 量子力学代写

物理代写

Ehrenfest’s Theorem

The formal solution to the Schrödinger equation in abstract Hilbert space can be written as
$$|\Psi(t)\rangle=e^{-i \hat{H} t / \hbar}|\Psi(0)\rangle \quad ; \text { formal solution }$$
where the exponential of the operator has a well-defined meaning in terms of its power-series expansion. If we consider the hermitian operator $\hat{F}$ then what we would measure for the time development of this quantity if the system is in the state $|\Psi(t)\rangle$ is the expectation value $^{8}$
$$F(t)=\langle\Psi(t)|\hat{F}| \Psi(t)\rangle=\left\langle\Psi(0)\left|e^{i \hat{H} t / \hbar} \hat{F} e^{-i \hat{H} t / \hbar}\right| \Psi(0)\right\rangle$$
Differentiate this with respect to time

$$\frac{H F(t)}{d t}=\left\langle\Psi(t)\left|\frac{i}{\hbar}[\hat{H}, \hat{F}]\right| \Psi(t)\right\rangle$$
The operator whose expectation value then yields the time development of $F(t)$ is given by the commutator with the hamiltonian

$$\left(\frac{d F}{d t}\right){\mathrm{op}}=\frac{i}{\hbar}[\hat{H}, \hat{F}]$$ If $\hat{F}$ should have an additional explicit time dependence, this relation becomes $$\left(\frac{d F}{d t}\right){\mathrm{op}}=\frac{\partial \hat{F}}{\partial t}+\frac{i}{\hbar}[\hat{H}, \hat{F}] \quad ; \text { Ehrenfest’s theorem }(9.30)$$

This is known as Ehrenfest’s theorem. We give three consequences:
(1) If the operator $\hat{F}=\hat{1}$, then we are simply investigating the time development of the norm of the state. It is evident that
$$\frac{i}{\hbar}[\hat{H}, \hat{1}]=0$$

and ${ }^{8}$ The state is normalized (see below); see also Prob. $9.5$.
Formal Structure of Quantum Mechanics
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Thus if the state is originally normalized, it will continue to be normalized as time progresses;
(2) Suppose we have a time-independent operator $\hat{O}$ that commutes with the hamiltonian. Then
$$\left(\frac{d O}{d t}\right)_{\mathrm{op}}=\frac{i}{\hbar}[\hat{H}, \hat{O}]=0$$
The expectation value of this operator does not change with time; hence, the operator $\hat{O}$ represents a constant of the motion;
(3) With a hamiltonian of the form
$$\hat{H}=\frac{\hat{p}^{2}}{2 m}+V(\hat{x})$$
one has
\begin{aligned} \frac{i}{\hbar}[\hat{H}, \hat{x}] &=\frac{\hat{p}}{m} \ \frac{i}{\hbar}[\hat{H}, \hat{p}] &=\frac{i}{\hbar}[V(\hat{x}), \hat{p}] \end{aligned}
In the coordinate representation, the last expression becomes
$$\frac{i}{\hbar}\left\langle x^{\prime}|[V(\hat{x}), \hat{p}]| x\right\rangle=-\frac{\partial V(x)}{\partial x} \delta\left(x-x^{\prime}\right)$$
Equations (9.28)-(9.35) then become the quantum analogs of Hamilton’s equations in classical mechanics.

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艾伦费斯特定理

$$|\Psi(t)\rangle=e^{-i \hat{H} t / \hbar}|\Psi(0)\rangle \quad ; \text { 正式解决方案 }$$

$$F(t)=\langle\Psi(t)|\hat{F}| \Psi(t)\rangle=\left\langle\Psi(0)\left|e^{i \hat{H} t / \hbar} \hat{F} e^{-i \hat{H} t / \hbar}\右| \Psi(0)\右\rangle$$

$$\frac{HF(t)}{dt}=\left\langle\Psi(t)\left|\frac{i}{\hbar}[\hat{H}, \hat{F}] \对| \Psi(t)\右\rangle$$

$$\left(\frac{d F}{d t}\right){\mathrm{op}}=\frac{i}{\hbar}[\hat{H}, \hat{F}]$$ 如果 $\hat{F}$ 应该有一个额外的显式时间依赖性，这个关系就变成 $$\left(\frac{d F}{dt}\right){\mathrm{op}}=\frac{\partial \hat{F}}{\partial t}+\frac{i}{\hbar} [\hat{H}, \hat{F}] \quad ; \text { Ehrenfest 定理 }(9.30)$$

(1) 如果算子$\hat{F}=\hat{1}$，那么我们只是在考察状态范数的时间发展。很明显，
$$\frac{i}{\hbar}[\hat{H}, \hat{1}]=0$$
${ }^{8}$ 状态被归一化（见下文）；另见概率。 9.5 美元。

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(2) 假设我们有一个与时间无关的算子$\hat{O}$，它与hamiltonian 交换。然后
$$\left(\frac{d O}{d t}\right)_{\mathrm{op}}=\frac{i}{\hbar}[\hat{H}, \hat{O}]=0$$

(3) 具有以下形式的汉密尔顿式
$$\hat{H}=\frac{\hat{p}^{2}}{2 m}+V(\hat{x})$$

$$\开始{对齐} \frac{i}{\hbar}[\hat{H}, \hat{x}] &=\frac{\hat{p}}{m} \ \frac{i}{\hbar}[\hat{H}, \hat{p}] &=\frac{i}{\hbar}[V(\hat{x}), \hat{p}] \end{对齐}$$

$$\frac{i}{\hbar}\left\langle x^{\prime}|[V(\hat{x}), \hat{p}]| x\right\rangle=-\frac{\partial V(x)}{\partial x} \delta\left(x-x^{\prime}\right)$$

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