alexvas
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Hello,
I am trying to understand the idea of using analytic continuation to find bound states in a scattering problem. What do the poles of the reflection coefficient have to do with bound states? In a problem that my quantum professor did in class (from a previous final), we looked at the 1D potential
$$ V(x) = \begin{cases} \infty & x \le 0 \\ V_0<0 & 0<x\le a \\ 0 & x > a \end{cases} $$
After some work, we found that the transmission coefficient near $x=a$ has to look like
$$ S(k) = e^{-2ika} \frac{1+i\tan(k'x)\frac{k}{k'}}{1-i\tan(k'a)\frac{k}{k'}} $$
with
$$k'=\frac{\sqrt{|V_0|-E}}{\hbar} \quad\text{and}\quad k=\frac{\sqrt{2mE}}{\hbar}$$ .
(Notice that k' is real but k is imaginary.) He then said that to find the bound states we want to analytically continue and then look for poles in S(k) (why?!), so we take $$k\to i\kappa$$ (why?!) where $$\kappa = \frac{\sqrt{2m|E|}}{\hbar}$$, so the denominator of S(k) (at the simple pole) becomes
$$ 0=1-i\tan (k'a)\frac{(i\kappa)}{k'} = 1+\tan (k'a)\frac{\kappa}{k'} $$
Therefore (why?!) the bound states are given by solutions to $$\tan(k'a) = -\frac{k'}{\kappa}$$.
This is the last part of a four-part problem, so it could be that I'm not including a critical detail. The full problem statement is problem 3 here:
http://www.phys.washington.edu/~karch/517/2012/fin11.pdf
Thanks.
I am trying to understand the idea of using analytic continuation to find bound states in a scattering problem. What do the poles of the reflection coefficient have to do with bound states? In a problem that my quantum professor did in class (from a previous final), we looked at the 1D potential
$$ V(x) = \begin{cases} \infty & x \le 0 \\ V_0<0 & 0<x\le a \\ 0 & x > a \end{cases} $$
After some work, we found that the transmission coefficient near $x=a$ has to look like
$$ S(k) = e^{-2ika} \frac{1+i\tan(k'x)\frac{k}{k'}}{1-i\tan(k'a)\frac{k}{k'}} $$
with
$$k'=\frac{\sqrt{|V_0|-E}}{\hbar} \quad\text{and}\quad k=\frac{\sqrt{2mE}}{\hbar}$$ .
(Notice that k' is real but k is imaginary.) He then said that to find the bound states we want to analytically continue and then look for poles in S(k) (why?!), so we take $$k\to i\kappa$$ (why?!) where $$\kappa = \frac{\sqrt{2m|E|}}{\hbar}$$, so the denominator of S(k) (at the simple pole) becomes
$$ 0=1-i\tan (k'a)\frac{(i\kappa)}{k'} = 1+\tan (k'a)\frac{\kappa}{k'} $$
Therefore (why?!) the bound states are given by solutions to $$\tan(k'a) = -\frac{k'}{\kappa}$$.
This is the last part of a four-part problem, so it could be that I'm not including a critical detail. The full problem statement is problem 3 here:
http://www.phys.washington.edu/~karch/517/2012/fin11.pdf
Thanks.
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