Finite Potential Well in Quantum Mechanics (Step by Step Derivation)

In this video, I discuss the Finite Potential Well Problem in ID. I use the Schrodinger Equation to derive the nature of the wavefunction solutions, use boundary conditions to obtain the Transcendental equations, solve them graphically, obtain an expression for number of bound states in a finite well, and show you how to calculate the Energy level of those bound states.

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A finite potential well is a fundamental concept in quantum physics that describes a region where particles, such as electrons, are confined by a potential energy barrier that has a limited height. Unlike an infinite well, where the barriers are insurmountable, a finite well allows for some interaction between the confined region and the surrounding space.

In this scenario, particles within the well possess specific, discrete energy levels, much like electrons orbiting an atom. These energy states are determined by the depth and width of the well. Importantly, because the potential barriers are not infinitely high, the particles’ quantum wavefunctions extend slightly beyond the well’s boundaries. This phenomenon, known as quantum tunneling, means there is a probability that particles can exist outside the well, even if their energy is less than the height of the barriers.

00:00 Introduction
03:08 Schrodinger Equation Solutions
17:09 Boundary Conditions
22:53 Transcendental Equations
32:03 Bound State Solutions (Graphical Analysis)
47:05 Energy Calculation (Numeriacal)

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