- #1
AntiStrange
- 20
- 1
Homework Statement
Consider the harmonic scillator potential perturbed by a small cubic term, so that
[tex]V(x) = \frac{1}{2}m\omega^{2} (x^{2} - \frac{x^{3}}{a})[/tex]
if a is large compared to the characteristic dimension [itex](\hbar /m\omega)^{1/2}[/itex], the states will all me metastable, since there can be no lowest energy state (as [itex]x\rightarrow\infty[/itex], the energy gets arbitrarily negative). Estimate the probability of tunneling from the ground state to the region on the far right.
Homework Equations
probability = [tex]|T|^{2} = e^{-2\int^{a}_{0} dx \sqrt{2m(V-E)/\hbar^{2}}}[/tex]
The Attempt at a Solution
A rough sketch of what the potential should look like when graphed is attached. However, I cut out the [itex]\frac{1}{2}m\omega^{2}[/itex] part at the front (is that bad?) but anyway that is what [itex]x^{2}-x^{3}/a[/itex] looks like. it crosses the x-axis at x=0 and x=a, and the maximum on the right side is at (2/3)a.
I have tried several things. Just substituting the potential given in the problem and I'm using E = (1/2)*h-bar*omega as the energy (ground state of the harmonic oscillator), into the equation but I can't solve the integral, even an online automatic integrator doesn't figure it out.
I have also tried equating the potential to the energy using the E = (1/2)*h-bar*omega and setting that equal to the potential and solving for omega. even though I'm not sure I am allowed to do this, it simplifies things a little bit, but the integral still seems impossible to solve.
I have also tried to approximate the curve on the right side by a negative parabola, perhaps it would work but I am having some trouble finding a parabola that fits close enough.
Any help would be very appreciated, or a point in the right direction.