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The specific question that I'm working on is "If I have a particle in the bound state of a 1-D delta function potential at $t = - \infty$, and I apply a harmonic perturbation $V(x,t) = V_0xcos(\omega t)$. After a long time, T, what is the probability that the particle will be found to no longer be in the bound state of the potential?

My attempt at a solution, and the more general questions I have as a result:

I'm currently studying from Griffiths' QM textbook. In it, he derives an equation for the probability for a particle in state "a" to be found in state "b" at a later time, "t" under the influence of a harmoinc pertubation as being given by, $ P_{ab}(t) = \frac{|V_{ab}|^2}{\hbar ^2} \frac{sin^2 [(\omega_0 - \omega)\frac{t}{2}]}{(\omega_0 - \omega)^2}$.

In this particular case, I know that state "a" is the single bound state for a delta potential, $\psi_a = \frac{\sqrt{m \alpha}}{\hbar}e^{\frac{-m \alpha |x|}{\hbar}}$, and that state "b" is the free particle, $\psi_b = e^{ikx}$. Using these, I calculate $|V_{ab}|^2 = \frac{16p^2m^3 \alpha^3}{(m^2 \alpha^2 + p^2)^4}$, and $\omega = \frac{E_b - E_a}{\hbar} = \frac{\hbar^2 p^2 + m^2 \alpha^2}{2m\hbar^3}$.

With this, I find that $P_{ab}(T) = \frac{16V_{0}^2m^3 \alpha^3p^2 sin^2[(\omega_0 - \omega(p))\frac{t}{2}]}{\hbar^2(m^2 \alpha^2 + p^2)^4 (\omega_0 - \omega(p))^2}$. I believe that this is the probability of finding the particle in the unbound state with a specific momentum p. As such, I attempted to integrate this function from $0$ to $\infty$ with respect to p in hopes of finding the probability of finding the particle outside the bound state with any momentum. However, I could find no way to evaluate that integral.

Is my approach correct, and the integral is simply nasty, or did I make a mistake in my application?

I also attempted to use Fermi's golden rule to find the rate of transition, which I would then integrate to find probability at $t=T$. However, I do not understand what the density of states would be for a free particle, since its smooth continuum. Does anyone have any idea how to advise on that approach as well?

-sorry for the long question, but thank you for taking time to read it and help!

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