# Equation for probability of quantum tunneling

I am looking at fusion reactions in stars and came across how particles will bypass the Coulomb barrier through quantum tunneling. I was wondering if there is an equation for the probability of a particle to tunnel when it encounters an energy barrier.

It depends on the potential you are working with. The mathematical concept behind quantum tunneling is quite simple: when solving Schrödinger's equation, in those regions where your particle's energy is lower than the potential, the solution to said equation will be a linear combination of positive and negative real exponentials. Of course, you need to impose the coefficient preceding the positive exponential to be zero, since otherwise the wavefunction would diverge when $$r\to\infty$$. Classically, the particle cannot surpass the obstacles we place on its way (i.e: a potential higher than its energy). Nonetheless, in QM things are different, and the wavefunction is nonzero even in those regions where the potential is higher than the particle's energy. This translates into the probability density associated to the particle to being nonzero in said regions, and therefore there is a slight chance that the particle will indeed be found in the classically forbidden regions.
There is no such thing as a formula for the probability of a particle to quantum tunnel. Instead, what you have is the particle's wavefunction in space, and you can then take its modulus to the second power (that's the probability density of finding the particle in a given position when you measure said observable, i.e: $$P(\vec{r})=|\Psi|^2$$). If you are interested in the probability of finding the particle in the classically forbidden regions (CFRs) (let's say this region is in the interval $$x\in[a,\infty)$$), then you would do:
$$P(\text{particle in CFR})=\int_a^\infty P(x)dx=\int_a^\infty |\Psi|^2dx$$