Thought you all might like to know, say, the probability of a human passing through a wall. Yes? Yes, because #QuantumMechanics is fun! Let's begin!

The set up: a barrier with a higher energy than an electron. Think a tall wall, and a ball that didn't get thrown high enough to go over. Classically, the electron scatters back, like the ball bounces back. But in quantum, that electron has a probability of passing through!

With the above case for the electron, we can find something called the transmission coefficient, which is literally the probability of a particle to pass through a barrier. It looks like this: (Sakurai P. 526)

What those things stand for:
V: energy of potential barrier
E: energy of particle
a: width of barrier
m: mass of particle
ћ (or h_bar): Planck constant divided by 2π (just a constant we need in quantum).
Sinh is the hyperbolic sine function that absorbs the imaginary part of sin

So, using the tiny mass of an electron and a really tiny barrier, we get substantial probabilities that the electron will tunnel through the barrier. Things that decrease the probabilities significantly: using more massive particles, and thicker barriers.

Let's see how this works for a human. In this case we'll use the equation that has the k and κ in it, where these are just wavenumbers, which measure properties of a wave. We'll focus on the sinh part of equation, where κ = p/h_bar, where p is the momentum = mass times velocity.

Let's have a person with a mass of, say, 70 kg, run towards a wall that's about 10 cm, or 0.1 m thick, with a speed of 4 m/s. Now, let's look again at that sinh part, and note that sinh^2(κa) is approximately e^2κa, so we'll just use the exponential.

Plugging in the numbers gives e^5.3*10^35 which is about e^10^35. That number is so incomprehensible big that the remaining factors of the equation are insignificant! But what we got is in the denominator, so the probability of a human of tunnelling is e^-10^35.

So, while there's *kinda* is a nonzero probability of a human tunnelling through a wall, it's so vanishingly small that we can safely say it's not gonna happen. So, let's save our human friend that unnecessary thump with that wall and let particles do the tunnelling

#SaturdayNightThoughts

Note: when I say the probability of a human tunneling through a wall is e^-10^35, I mean the exponential raised to -10^35, as in, the negative 1 followed by 35 zeros, as in e^-100,000,000,000,000,000,000,000,000,000,000,000. So, "vanishingly small" is an understatement!

This quantum tunneling thread was such a hit that I went ahead and drew a cute cartoon thingy of a tennis ball and an electron trying to tunnel for you all, and I can't even draw