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PHY456H1F: Quantum Mechanics II. Lecture 25 (Taught by Prof J.E. Sipe). Born approximation.

Posted by peeterjoot on December 7, 2011

[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

Born approximation.

READING: section 20 [1]

We’ve been arguing that we can write the stationary equation

\begin{aligned}\left( \boldsymbol{\nabla}^2 + \mathbf{k}^2\right) \psi_\mathbf{k}(\mathbf{r}) = s(\mathbf{r})\end{aligned} \hspace{\stretch{1}}(2.1)

with

\begin{aligned}s(\mathbf{r}) = \frac{2\mu}{\hbar^2} V(\mathbf{r}) \psi_\mathbf{k}(\mathbf{r})\end{aligned} \hspace{\stretch{1}}(2.2)

\begin{aligned}\psi_\mathbf{k}(\mathbf{r}) = \psi_\mathbf{k}^{\text{homogeneous}}(\mathbf{r}) + \psi_\mathbf{k}^{\text{particular}}(\mathbf{r})\end{aligned} \hspace{\stretch{1}}(2.3)

Introduce Green function

\begin{aligned}\left( \boldsymbol{\nabla}^2 + \mathbf{k}^2\right) G^0(\mathbf{r}, \mathbf{r}') = \delta(\mathbf{r}- \mathbf{r}')\end{aligned} \hspace{\stretch{1}}(2.4)

Suppose that I can find G^0(\mathbf{r}, \mathbf{r}'), then

\begin{aligned}\psi_\mathbf{k}^{\text{particular}}(\mathbf{r}) = \int G^0(\mathbf{r}, \mathbf{r}') s(\mathbf{r}') d^3 \mathbf{r}'\end{aligned} \hspace{\stretch{1}}(2.5)

It turns out that finding the Green’s function G^0(\mathbf{r}, \mathbf{r}') is not so hard. Note the following, for k = 0, we have

\begin{aligned}\boldsymbol{\nabla}^2 G^0_0(\mathbf{r}, \mathbf{r}') = \delta(\mathbf{r} - \mathbf{r}')\end{aligned} \hspace{\stretch{1}}(2.6)

(where a zero subscript is used to mark the k = 0 case). We know this Green’s function from electrostatics, and conclude that

\begin{aligned}G^0_0(\mathbf{r}, \mathbf{r}') = - \frac{1}{{4 \pi}} \frac{1}{{{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}}}\end{aligned} \hspace{\stretch{1}}(2.7)

For \mathbf{r} \ne \mathbf{r}' we can easily show that

\begin{aligned}G^0(\mathbf{r}, \mathbf{r}') = - \frac{1}{{4 \pi}} \frac{e^{i k{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}}}{{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(2.8)

This is correct for all \mathbf{r} because it also gives the right limit as \mathbf{r} \rightarrow \mathbf{r}'. This argument was first given by Lorentz. We can now write our particular solution

\begin{aligned}\psi_\mathbf{k}(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}}- \frac{1}{{4 \pi}} \int \frac{e^{i k{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}}}{{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}} s(\mathbf{r}') d^3 \mathbf{r}'\end{aligned} \hspace{\stretch{1}}(2.9)

This is of no immediate help since we don’t know \psi_\mathbf{k}(\mathbf{r}) and that is embedded in s(\mathbf{r}).

\begin{aligned}\psi_\mathbf{k}(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}}- \frac{2 \mu}{4 \pi \hbar^2} \int \frac{e^{i k{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}}}{{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert}} V(\mathbf{r}') \psi_\mathbf{k}(\mathbf{r}') d^3 \mathbf{r}'\end{aligned} \hspace{\stretch{1}}(2.10)

Now look at this for \mathbf{r} \gg \mathbf{r}'

\begin{aligned}{\left\lvert{\mathbf{r} - \mathbf{r}'}\right\rvert} &= \left( \mathbf{r}^2 + (\mathbf{r}')^2 - 2 \mathbf{r} \cdot \mathbf{r}'\right)^{1/2} \\ &=r \left( 1 + \frac{(\mathbf{r}')^2}{\mathbf{r}^2} - 2 \frac{1}{{\mathbf{r}^2}} \mathbf{r} \cdot \mathbf{r}'\right)^{1/2} \\ &=r \left( 1 - \frac{1}{{2}} \frac{2}{\mathbf{r}^2} \mathbf{r} \cdot \mathbf{r}'+ O\left(\frac{r'}{r}\right)^2\right)^{1/2} \\ &=r - \hat{\mathbf{r}} \cdot \mathbf{r}'+ O\left(\frac{{r'}^2}{r}\right)\end{aligned}

We get

\begin{aligned}\begin{aligned}\psi_\mathbf{k}(\mathbf{r}) &\rightarrow e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{2 \mu}{4 \pi \hbar^2} \frac{ e^{i k r}}{r} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') \psi_\mathbf{k}(\mathbf{r}') d^3 \mathbf{r}' \\ &=e^{i \mathbf{k} \cdot \mathbf{r}} + f_\mathbf{k}(\theta, \phi) \frac{ e^{i k r}}{r},\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.11)

where

\begin{aligned}f_\mathbf{k}(\theta, \phi) =- \frac{\mu}{2 \pi \hbar^2} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') \psi_\mathbf{k}(\mathbf{r}') d^3 \mathbf{r}' \end{aligned} \hspace{\stretch{1}}(2.12)

If the scattering is weak we have the Born approximation

\begin{aligned}f_\mathbf{k}(\theta, \phi) =- \frac{\mu}{2 \pi \hbar^2} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') e^{i \mathbf{k} \cdot \mathbf{r}'} d^3 \mathbf{r}',\end{aligned} \hspace{\stretch{1}}(2.13)

or

\begin{aligned}\psi_\mathbf{k}(\mathbf{r}) =e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{\mu}{2 \pi \hbar^2} \frac{ e^{i k r}}{r} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') e^{i \mathbf{k} \cdot \mathbf{r}'} d^3 \mathbf{r}'.\end{aligned} \hspace{\stretch{1}}(2.14)

Should we wish to make a further approximation, we can take the wave function resulting from application of the Born approximation, and use that a second time. This gives us the “Born again” approximation of

\begin{aligned}\begin{aligned}\psi_\mathbf{k}(\mathbf{r}) &=e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{\mu}{2 \pi \hbar^2} \frac{ e^{i k r}}{r} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') \left( e^{i \mathbf{k} \cdot \mathbf{r}'} - \frac{\mu}{2 \pi \hbar^2} \frac{ e^{i k r'}}{r'} \int e^{-i k \hat{\mathbf{r}}' \cdot \mathbf{r}''} V(\mathbf{r}'') e^{i \mathbf{k} \cdot \mathbf{r}''} d^3 \mathbf{r}''\right) d^3 \mathbf{r}' \\ &=e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{\mu}{2 \pi \hbar^2} \frac{ e^{i k r}}{r} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') e^{i \mathbf{k} \cdot \mathbf{r}'} d^3 \mathbf{r}' \\ &\quad +\frac{\mu^2}{(2 \pi)^2 \hbar^4}\frac{ e^{i k r}}{r} \int e^{-i k \hat{\mathbf{r}} \cdot \mathbf{r}'} V(\mathbf{r}') \frac{ e^{i k r'}}{r'} \int e^{-i k \hat{\mathbf{r}}' \cdot \mathbf{r}''} V(\mathbf{r}'') e^{i \mathbf{k} \cdot \mathbf{r}''} d^3 \mathbf{r}'' d^3 \mathbf{r}'.\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.15)

References

[1] BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.

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