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# 物理代考| Reduction of the Basis量子力学代写

## 物理代写

8.2 Reduction of the Basis
Let us try to formalize this measurement theory. Suppose we are looking at a single particle, and we have a complete set of the eigenfunctions of some hermitian operator with real eigenvalues at our disposal
$$F \psi_{f}(x)=f \psi_{f}(x) \quad ; \text { eigenfunctions }$$
Order the eigenvalues $f_{0} \leq f_{1} \leq f_{2} \cdots$, and expand the wave function $\Psi(x, t)$ in this complete set of eigenfuctions
$$\Psi(x, t)=\sum_{f} c_{f}(t) \psi_{f}(x) \quad ; \text { complete set }$$
The state is normalized, so that
$$\sum_{f}\left|c_{f}(t)\right|^{2}=1$$
Measurement theory then assumes the following:
(1) If we make a precise measurement of the quantity $F$, we will observe one of the eigenvalues $f$;
(2) If we perform a pure pass measurement at a time $t_{0}$ that lets the eigenvalue $f$ through, then the wave function is reduced to $^{4}$
\begin{aligned} \Psi(x, t) &=c_{f}(t) \psi_{f}(x) \quad ; t \geq t_{0} \ \left|c_{f}(t)\right|^{2} &=1 \end{aligned}
The measurement is reproducible and the basis is reduced.
(3) If the measurement simply lets the eigenvalues in the set $f_{1} \leq f \leq f_{2}$ through, then the basis is reduced to
\begin{aligned} \Psi(x, t) &=\sum_{f}^{\prime} c_{f}(t) \psi_{f}(x) & & ; t \geq t_{0} \ \sum_{f}^{\prime}\left|c_{f}(t)\right|^{2} &=1 & & \end{aligned}
where the sum $\sum_{f}^{\prime}$ goes over $f_{1} \leq f \leq f_{2}$.
${ }^{4}$ Note that the coefficient $c_{f}(t)$ must be rescaled to achieve this norm (see Prob. 10.2).

## 物理代考

8.2 降低基数

$$F \psi_{f}(x)=f \psi_{f}(x) \quad ; \text { 特征函数 }$$

$$\Psi(x, t)=\sum_{f} c_{f}(t) \psi_{f}(x) \quad ; \text { 完整集 }$$

$$\sum_{f}\left|c_{f}(t)\right|^{2}=1$$

(1) 如果我们对数量 $F$ 进行精确测量，我们将观察到一个特征值 $f$；
(2) 如果我们在时间 $t_{0}$ 执行纯通过测量，让特征值 $f$ 通过，那么波函数将简化为 $^{4}$
$$\开始{对齐} \Psi(x, t) &=c_{f}(t) \psi_{f}(x) \quad ; t \geq t_{0} \ \left|c_{f}(t)\right|^{2} &=1 \end{对齐}$$

(3) 如果测量只是让集合 $f_{1} \leq f \leq f_{2}$ 中的特征值通过，则基简化为
$$\开始{对齐} \Psi(x, t) &=\sum_{f}^{\prime} c_{f}(t) \psi_{f}(x) & & ; t \geq t_{0} \ \sum_{f}^{\prime}\left|c_{f}(t)\right|^{2} &=1 & & \end{对齐}$$

${ }^{4}$ 注意，必须重新调整系数 $c_{f}(t)$ 才能达到这个标准（见 Prob. 10.2）。

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