Quantum Tunneling Out of the Black Hole

A particle is trapped inside a black hole. What is the probability it will escape via quantum tunneling? To figure this out, we begin by defining the variables involved:

The diagram below shows two regions: I and O. Region I is inside the event horizon, within the Schwarzshild radius (rs). Region O is outside the event horizon. The question we ask is what is the probability the particle (red dot) will reach region O? And, we could also ask what is the probability the particle will fail to reach O and remain trapped in region I? These probabilities, when added together should equal 1.

Region I covers a distance (r) from zero to rs. Region O is from rs to infinity. These will be the boundaries we will be using in the integrals below.

To find a probability in quantum mechanics we are told to square the wave-function amplitude. To see how this works, let's consider finding the probability without squaring the wave function. (See equations 1 and 2 below.)

We can model the wave function using right triangles. This is appropriate, since sine and cosine represent waves.

At the second triangle above we substitute some dummy wave functions for demonstration purposes. The trigonometric proof below shows why it is important to square a wave function to get a probability:

At equations 3 and 4 we didn't square the wave functions--they rarely add up to one, so they can't be probabilities. However once they are squared, they add up to one and could be probabilities (see equations 5 through 9). You may have noticed the wave functions have negative exponents. This feature prevents an exponential blow up to infinity.

Now, to get the right values for the probabilities we also need to include a normalization factor (A). At equations 10 to 15 below, we calculate the value of A and see why we need it. When we take the sum of all probabilities, from zero to infinity, we want the grand total to be one.

Equation 16 below is our new-and-improved wave function. Equation 17 is the one we use to calculate the probability densities of regions I and O. Equations 18 through 22 yield our desired result: equation 23, the probability density of region O; i.e., the probability that the particle will successfully escape the black hole.

Equation 24 should be the probability density for the particle's failure to escape. Let's check this:

Equation 28 confirms and matches 24. Below we use Schrodinger's equation to find the value of the wave number k:

Equation 34 shows that k is a function of V--the black hole's potential. The bigger V is, the bigger k is, the smaller the wave function and the probability that the particle will escape.

For more on the topic of quantum tunneling and QM, I highly recommend Robert Eagle's (aka: DrPhysicsA) video series: