Double delta-function potential

by mattmatt321
Tags: deltafunction, double, potential
 P: 7 Hello, recently I had a problem in QM involving a double delta-function potential. A bunch of qualitative questions were asked, some of which were obvious to me immediately, some of which I made an educated guess on, and others I totally guessed. I'm following Griffiths' textbook in studying QM, and luckily I found another problem analogous to the one I had before. Any guidance/explanation would be appreciated. 1. The problem statement, all variables and given/known data The potential is V(x) = -α[$$\delta$$(x + a) + $$\delta$$(x - a)]. The problem specifies that we are only conserved about bound states, E < 0. 2. Relevant equations In the end, the even wave function solution yields (ħk/mα) = e-2ka + 1, and the odd wave function solution yields (ħk/mα) = 1 - e-2ka. 3. The attempt at a solution Based on these solutions, I'm asked a few questions: 1) What is the approximate energies of both the even and odd bound states in the limit 2maα/ħ2 >> 1? 2) Show that as 2maα/ħ2 --> $$\infty$$, both energies converge to the energy of the bound state of the single delta-function potential. 3) Show that there is only one bound state in the limit 2maα/ħ2 << 1. 4) Show that the wave function behaves like the bound-state wave function of a single delta-function potential for |x| >> a. Any guidance regarding these questions is appreciated. I might be able to provide more information if anything is unclear.
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P: 11,776
 Quote by mattmatt321 In the end, the even wave function solution yields (ħk/mα) = e-2ka + 1, and the odd wave function solution yields (ħk/mα) = 1 - e-2ka.
$\hbar$ should be squared in both equations.
 3. The attempt at a solution Based on these solutions, I'm asked a few questions: 1) What is the approximate energies of both the even and odd bound states in the limit 2maα/ħ2 >> 1? 2) Show that as 2maα/ħ2 --> $$\infty$$, both energies converge to the energy of the bound state of the single delta-function potential. 3) Show that there is only one bound state in the limit 2maα/ħ2 << 1. 4) Show that the wave function behaves like the bound-state wave function of a single delta-function potential for |x| >> a. Any guidance regarding these questions is appreciated. I might be able to provide more information if anything is unclear.
You can write the relations above as

$$\frac{2ka}{2ma\alpha/\hbar^2} = 1\pm e^{-2ka}$$

Let $c=2ma\alpha/\hbar^2$ and $x=2ka$. Then you get

$$\frac{x}{c} = 1 \pm e^{-x}$$

You might find it illuminating to plot both sides of the equation to see where the solutions are and what the effect of varying the parameter c is.

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