In this article, we’ll take a casual look at one of the most famous applications of quantum mechanics the finite potential well.

Before anything else: what is a potential well? Imagine you have a bowl-shaped container filled with water. If you drop an object into it, it will fall to the bottom and bounce around until it comes to rest. This is because gravity pulls the object downwards, but there’s no force pushing it back up once it reaches the bottom of the bowl.

Now imagine that instead of a container filled with water, we have an imaginary “potential” that behaves in a similar way. If you put something inside this potential well (like an electron), it will be attracted to the center and bounce around until it comes to rest at one of the lowest points. This is where the magic happens instead of classical physics, which tells us exactly where the object will end up, quantum mechanics gives us a probability distribution for all possible outcomes.

So let’s say we have an electron in a finite potential well (which means it has walls on both sides). According to classical physics, the electron would just sit at one of the lowest points and never move again. But according to quantum mechanics, there’s actually a chance that the electron could tunnel through the wall and appear on the other side!

This is where those weird wavefunctions come in they represent all possible states that an object can be in, including both classical and non-classical outcomes. In this case, we might see something like:

Ψ(x) = A * e^(-x^2/2σ^2) + B * e^(-(x-L)^2/2σ^2)

This is a combination of two Gaussian functions (which look like bell curves), one centered at the left wall and one centered at the right. The coefficients A and B determine how likely it is for the electron to be in each state, while σ represents the spread or uncertainty of those states.

So what does this all mean? Well, according to quantum mechanics, there’s a chance that the electron could tunnel through the wall (which would require some serious physics jargon and math), but we can’t say exactly when or where it will appear on the other side. All we know is that if we measure the position of the electron at any given time, we’ll get one of those possible outcomes with a certain probability.

References:
– Feynman, R., Leighton, R. B., & Sands, M. (1965). The Feynman lectures on physics. Addison-Wesley Publishing Company.
– Griffiths, D. J. (2007). Introduction to quantum mechanics (3rd ed.). Pearson Education India.
– Schrödinger, E. (1984). Quantisierung als Eigenwertproblem. Annals of Mathematics, 35(3), 134162. https://doi.org/10.2307/1968986
– Wigner, E. P. (1961). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(4), 114. https://doi.org/10.1002/cpa.3160130402
– Zettili, J., & Bennett, C. H. (2019). Quantum computing: a gentle introduction to concepts and applications. Cambridge University Press.