# Approximate the probability of tunneling

1. Oct 14, 2009

### AntiStrange

1. The problem statement, all variables and given/known data
Consider the harmonic scillator potential perturbed by a small cubic term, so that
$$V(x) = \frac{1}{2}m\omega^{2} (x^{2} - \frac{x^{3}}{a})$$
if a is large compared to the characteristic dimension $(\hbar /m\omega)^{1/2}$, the states will all me metastable, since there can be no lowest energy state (as $x\rightarrow\infty$, the energy gets arbitrarily negative). Estimate the probability of tunneling from the ground state to the region on the far right.

2. Relevant equations
probability = $$|T|^{2} = e^{-2\int^{a}_{0} dx \sqrt{2m(V-E)/\hbar^{2}}}$$

3. The attempt at a solution
A rough sketch of what the potential should look like when graphed is attached. However, I cut out the $\frac{1}{2}m\omega^{2}$ part at the front (is that bad?) but anyway that is what $x^{2}-x^{3}/a$ 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.

#### Attached Files:

• ###### graph.jpg
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2. Oct 16, 2009

### gabbagabbahey

That doesn't look right....Shouldn't the probability that a particle tunnels over the barrier into the far right be given by

$$T=\int_{\frac{2a}{3}}^{\infty} |\psi(x)|^2 dx$$

?