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# 物理代写|量子力学代写Quantum mechanics代考|KYA321 Semiclassical expansion

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## 物理代写|量子力学代写Quantum mechanics代考|Semiclassical expansion

We will discuss only the problem of a particle in a potential, or the interacting two particle problem that can be reduced to the one particle problem in the center of mass. Similar methods would apply to more complicated systems as well. Thus, consider the Schrödinger equation with potential energy $V(\mathbf{r})$
$$i \hbar \partial_t \psi(\mathbf{r}, t)=\left(-\frac{\hbar^2}{2 m} \nabla^2+V(\mathbf{r})\right) \psi(\mathbf{r}, t) .$$
Without loss of generality one may write
$$\psi(\mathbf{r}, t)=A \exp (i W(\mathbf{r}, t) / \hbar),$$
as long as $W(\mathbf{r}, t)$ is a complex function. The normalization constant $A$ could be absorbed into $W$, but it is convenient to keep it separate, so that $W$ may be defined up to an additive complex constant. Substituting this form into the Schrödinger equation one gets
$$\partial_t W+\frac{1}{2 m}(\boldsymbol{\nabla} W)^2+V-\frac{i \hbar}{2 m} \nabla^2 W=0 .$$

## 物理代写|量子力学代写Quantum mechanics代考|Extrapolation through turning points

Before proceeding further, let us investigate the conditions under which the approximation is valid. The obvious criterion is that all neglected terms must be small. Formally, the higher order terms in $\hbar$ are small, however one must take care that the coefficients that multiply these powers are not large. Therefore, the correct criterion is
$$\hbar\left|S_2\right|<<\left|S_1\right| .$$
It is clear from (11.17) that this condition cannot be met in the vicinity of $x \sim x_i(E)$ where the local momentum vanishes $p\left(x_i\right)=\tilde{p}\left(x_i\right)=0$, or
$$V\left(x_i\right)=E .$$
At these points the classical momentum changes sign, thus such points are the classical “turning points”. Hence the expression for the wavefunction given above is certainly not valid near the classical turning points. By examining the expressions for $S_1, S_2$ one notices that, as long as $p(x), \tilde{p}(x)$ are not near zero, and they are slowly varying functions, the condition is met in the domains of $x$ where
$$\frac{\hbar\left|p^{\prime}\right|}{p^2}<<1,$$
and similarly for $\tilde{p}(x)$. The physical meaning of this criterion is understood by defining the local deBroglie wavelength $\lambda(x)=\hbar / p(x)$ and writing
$$\lambda(x)\left|\frac{p^{\prime}(x)}{p(x)}\right|<<1 .$$

## 物理代写|量子力学代写|量子力学代考|经典扩展

$$i \hbar \partial_t \psi(\mathbf{r}, t)=\left(-frac{hbar\^2}{2 m}) \nabla^2+V(\mathbf{r})\right) \psi(\mathbf{r}, t)$$

$$\psi(\mathbf{r}, t)=A\exp (i W(\mathbf{r}, t) / \hbar)$$

$$\partial_t W+frac{1}{2 m}(\nabla W)^2+V-frac{i \hbar}{2 m}。\nabla^2 W=0$$

## 物理代写|量子力学代写|通过转折点的推断

$$\hbar\left|S_2\right|<<<left|S_1\right| 。$$

$$V\left(x_i\right)=E$$

$$\frac{\hbar\left|p^{\prime}\right|}{p^2}<<1$$

$$\λ(x)\left|\frac{p^{\prime}(x)}{p(x)}\right|<<1$$

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