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# 物理代考| Component Form 量子力学代写

## 物理代写

$9.2$ Component Form
Let us start with the component form of our relations. In order to do this we need to introduce the concept of a continuous component, and we also need to make use of the Dirac delta function that was introduced when discussing transition rates. In fact, the concepts and notation for what we are doing here were originally introduced by Dirac in his fundamental work. ${ }^{1}$

We first label the components of the abstract state vectors that we are studying with a subscript $x$, and calculate the inner product of two of those vectors as $^{2}$
$$\left\langle\psi_{m} \mid \psi_{n}\right\rangle=\sum_{x}\left(\psi_{m}\right){x}^{}\left(\psi{n}\right){x}$$ We now have to define the continuous sum, as well as the components in the $x$-direction. We do this by writing the sum as an integral and using our coordinate-space wave functions for the components $$\left\langle\psi{m} \mid \psi_{n}\right\rangle=\sum_{x}\left(\psi_{m}\right){x}^{}\left(\psi{n}\right){x} \equiv \int d x \psi{m}^{}(x) \psi_{n}(x)$$ This defines the continuous sum. It is just the integral over the appropriate interval of two of our previous wave functions. ${ }^{1}$ See [Dirac (1930)]. antum Mechanics Formal Structure of Quantum Mechanics introduce the eigenstates of the hermitian position oper- Formal Structure of Quantum We can then rewrite the above as $$\left\langle\psi_{m} \mid \psi_{n}\right\rangle=\sum_{x}\left(\psi_{m}\right){x}^{}\left(\psi{n}\right){x}=\sum{x}\left\langle\psi_{m} \mid x\right\rangle\left\langle x \mid \psi_{n}\right\rangle$$
where we have identified the wave functions as
\begin{aligned} \left\langle x \mid \psi_{n}\right\rangle &=\psi_{n}(x) \ \left\langle\psi_{m} \mid x\right\rangle &=\psi_{m}^{*}(x) \end{aligned}
We now have the consistent physical interpretation that the probability for finding the particle in the interval $d x$ at the position $x$ if it is in the state $\left|\psi_{n}\right\rangle$ is the absolute square of the probability amplitude obtained from the inner product $\left\langle x \mid \psi_{n}\right\rangle$
$$\left|\left\langle x \mid \psi_{n}\right\rangle\right|^{2} d x=\left|\psi_{n}(x)\right|^{2} d x \quad ; \text { probability }$$ Equation (9.7) allows us to identify the completeness relation for the eigenstates of position $$\sum_{x}|x\rangle\langle x|=\hat{1}$$
What about the inner product of these states? Here we are forced to deal with the fact that the eigenvalues of position are truly continuous, and we write $^{4}$

## 物理代考

$9.2$ 组件表格

$$\left\langle\psi_{m} \mid \psi_{n}\right\rangle=\sum_{x}\left(\psi_{m}\right){x}^{}\left(\psi {n}\右）{x}$$ 我们现在必须定义连续和，以及 $x$ 方向的分量。我们通过将总和写成积分并使用我们的坐标空间波函数来实现这一点 $$\left\langle\psi{m} \mid \psi_{n}\right\rangle=\sum_{x}\left(\psi_{m}\right){x}^{}\left(\psi {n}\right){x} \equiv \int dx \psi{m}^{}(x) \psi_{n}(x)$$ 这定义了连续总和。它只是我们之前的两个波函数在适当区间上的积分。 ${ }^{1}$ 见 [狄拉克 (1930)]。 安腾力学 量子力学的形式结构介绍了厄米位置运算的本征态 量子的形式结构 然后我们可以将上面的内容重写为 $$\left\langle\psi_{m} \mid \psi_{n}\right\rangle=\sum_{x}\left(\psi_{m}\right){x}^{}\left(\psi {n}\right){x}=\sum{x}\left\langle\psi_{m} \mid x\right\rangle\left\langle x \mid \psi_{n}\right\rangle$$

$$\开始{对齐} \left\langle x \mid \psi_{n}\right\rangle &=\psi_{n}(x) \ \left\langle\psi_{m} \mid x\right\rangle &=\psi_{m}^{*}(x) \end{对齐}$$

$$\left|\left\langle x \mid \psi_{n}\right\rangle\right|^{2} dx=\left|\psi_{n}(x)\right|^{2} dx \四边形； \text { probability }$$ 方程（9.7）允许我们识别位置 $$\sum_{x}|x\rangle\langle x|=\hat{1}$$ 的特征态的完整性关系

Matlab代写