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# 物理代写|量子力学代写Quantum mechanics代考|PHYS2941 Classically Allowed Region to the Left

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## 物理代写|量子力学代写Quantum mechanics代考|Classically Allowed Region to the Left

Let us now examine the case in which we have a classically allowed region to the left of a turning point, with a classically forbidden region on the right, as illustrated in Fig. 7. The turning point is denoted $x_r$ (since it is to the right of the classically allowed region). We are only sketching part of the $x$-axis in this diagram, and we make no assumptions about what the potential does to the left or the right of the diagram. For example, to the left it may rise again, creating a potential well, or just asymptote to zero, making a scattering problem. Or, to the right, it may go down again, creating a barrier the particle can tunnel through.

First we treat region I, the classically allowed region, where $xV(x)$. Here $p(x)$ is real and positive as defined by Eq. (49). In this region we define $S$ by
$$S(x)=\int_{x_r}^x p\left(x^{\prime}\right) d x^{\prime}$$
which is the same as Eq. 47 except that now we are agreeing to measure the action from the turning point $x_r$. This is merely a matter of convenience, but it means that $S(x)$ is real and negative in region I, and increasing to the right since $d S / d x=p(x)>0$ . Taking $S(x)$ and $-S(x)$ as the two solutions of the Hamilton-Jacobi equation, the WKB solution is a general linear combination of two waves,
$$\psi_{\mathrm{I}}(x)=c_r \frac{e^{i S(x) / \hbar+i \pi / 4}}{\sqrt{p(x)}}+c_{\ell} \frac{e^{-i S(x) / \hbar-i \pi / 4}}{\sqrt{p(x)}},$$
where $c_r$ and $c_{\ell}$ are two generally complex constants and where we have introduced phase shifts of $\pm \pi / 4$ in the two terms for later convenience. Properly speaking, we should have used $-p(x)$ in the denominator of the second term, but we have absorbed the factor of $\sqrt{-1}=i$ into the second constant. The first term is a wave traveling to the right since $d S / d x>0$ , and the second, a wave traveling to the left; this is the meaning of the subscripts on the constants $c_r$ and $c_{\ell}$. The general solution is a linear combination of such waves.

## 物理代写|量子力学代写Quantum mechanics代考|The Connection Rules

We now summarize the connection rules, first for the case of the classically allowed region to the left of the turning point. This is the case analyzed in Sec. 7. The potential is sketched in Fig. 7, with turning point $x_r$.

In region I, the classically allowed region where $x\left\langle x_r\right.$ and $\left.E\right\rangle V(x), p(x)$ is real and positive and is given by Eq. (49). The action $S$ is given by Eq. (52), which we now write with a slight change of notation,
$$S\left(x, x_r\right)=\int_{x_r}^x p\left(x^{\prime}\right) d x^{\prime}$$
indicating both limits of the integral. The action $S\left(x, x_r\right)$ is real, negative, and increasing to the right in region I. The wave function (53) is now written as
$$\psi_{\mathrm{I}}(x)=\frac{1}{\sqrt{p(x)}}\left(c_r e^{i\left[S\left(x, x_r\right) / \hbar+\pi / 4\right]}+c_{\ell} e^{-i\left[S\left(x, x_r\right) / \hbar+\pi / 4\right]}\right)$$

## 物理代写|量子力学代写QUANTUM MECHANICS代 芸|CLASSICALLY ALLOWED REGION TO THE LEFT

$$S(x)=\int_{x_*}^x p\left(x^{\prime}\right) d x^{\prime}$$

$$\psi_{\mathrm{I}}(x)=c_r \frac{e^{i S(x) / \hbar+i \pi / 4}}{\sqrt{p(x)}}+c_{\ell} \frac{e^{-i S(x) / \hbar-i \pi / 4}}{\sqrt{p(x)}},$$

## 物理代写|量子力学代写QUANTUM MECHANICS代 考|THE CONNECTION RULES

$$S\left(x, x_r\right)=\int_{x_r}^x p\left(x^{\prime}\right) d x^{\prime}$$

$$\psi_{\mathrm{I}}(x)=\frac{1}{\sqrt{p(x)}}\left(c_r e^{i\left[S\left(x, x_r\right) / \hbar+\pi / 4\right]}+c_{\ell} e^{-i\left[S\left(x, x_{+}\right) / \hbar+\pi / 4\right]}\right)$$

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