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物理代考| Perturbation Theory 量子力学代写

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3.7 Perturbation Theory
Suppose the hamiltonian has an additional small piece $\delta V(x)$, which makes the Schrödinger equation difficult to solve analytically
We return to Eq. (3.6) $$\mid E=\frac{\int d x \psi^{}(x) H \psi(x)}{\int d x|\psi(x)|^{2}}=\frac{\int d x \psi^{}(x)\left[H_{0}+\delta V(x)\right] \psi(x)}{\int d x|\psi(x)|^{2}}$$
Let us use the eigenfunction $\psi_{n}(x)$ of $H_{0}$ in this expression to obtain $$E_{n}=E_{n}^{0}+\frac{\int d x \psi_{n}^{}(x)[\delta V(x)] \psi_{n}(x)}{\int d x\left|\psi_{n}(x)\right|^{2}}$$ the eigenvalue $$\delta E_{n}=\frac{\int d x \psi_{n}^{}(x)[\delta V(x)] \psi_{n}(x)}{\int d x\left|\psi_{n}(x)\right|^{2}} \quad ; \text { perturbation theory }(3.46)$$
The small shift in the eigenvalue is the integral of the perturbation over the eigenfunction.
Include Potential $V(x)$
As an example, suppose that with the particle in the box in Fig. $3.1$ there is a small, narrow potential step at the midpoint
$$\delta V(x)=\nu_{0} \quad ;\left|x-\frac{L}{2}\right|<l$$
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where $l \ll L$. The eigenfunctions are $\sqrt{2 / L} \sin (n \pi x / L)$. For odd $n$, the magnitude of the sine is unity at the midpoint where $x=L / 2$. For even $n$, it vanishes there. ${ }^{4}$ Hence, for $l \ll L$, one has
\begin{aligned} \delta E_{n} &=4 \nu_{0} \frac{l}{L} & & ; n=1,3,5, \cdots \ &=0 & & ; n=2,4,6, \cdots \end{aligned}
3.7.1 Non-Degenerate Perturbation Theory
Let us make the analysis more general. We want to solve for the eigenfunctions and eigenvalues in the Schrödinger equation
$H \psi(x)=\left[H_{0}+\delta V(x)\right] \psi(x)=E \psi(x)$
$$\psi(x)=\sum_{m} c_{m} \psi_{m}(x)$$ $$H_{0} \psi_{m}(x)=E_{m}^{0} \psi_{m}(x)$$ Substitute this in the above equation $$\sum_{m}\left(E-E_{m}^{0}\right) c_{m} \psi_{m}(x)=\delta V(x) \psi(x)$$ Now multiply by $\psi_{n}^{}(x)$ on the left, integrate over $x$, and me now the eigenfunctions $$c_{n}=\frac{1}{E-E_{n}^{0}} \int d x \psi_{n}^{}(x) \delta V(x) \psi(x)$$
Let us discuss this:

• This choice is for a given $n$;
4 See Figs. $3.2$ and $3.3 .$

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3.7 微扰理论

$$\delta V(x)=\nu_{0} \quad ;\left|x-\frac{L}{2}\right|<l$$
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$$\开始{对齐} \delta E_{n} &=4 \nu_{0} \frac{l}{L} & & ; n=1,3,5, \cdots \ &=0 & & ; n=2,4,6, \cdots \end{对齐}$$
3.7.1 非退化微扰理论

$H \psi(x)=\left[H_{0}+\delta V(x)\right] \psi(x)=E \psi(x)$
$$\psi(x)=\sum_{m} c_{m} \psi_{m}(x)$$ $$H_{0} \psi_{m}(x)=E_{m}^{0} \psi_{m}(x)$$ 代入上式 $$\sum_{m}\left(E-E_{m}^{0}\right) c_{m} \psi_{m}(x )=\delta V(x) \psi(x)$$ 现在乘以左边的 $\psi_{n}^{}(x)$，积分超过 $x$，我现在得到特征函数 $$c_ {n}=\frac{1}{E-E_{n}^{0}} \int dx \psi_{n}^{}(x) \delta V(x) \psi(x)$$

• 此选择适用于给定的 $n$；
4 见图。 3.2 美元和 3.3 美元 .\$

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