Calculating Normalization, Eigenenergies and Probabilities in the 1D Particle-In-A-Box Problem

Quantum Chemistry Problem [Q21-01-00]
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Question:
Calculate the probability of finding the particle in the box in the region between L/4 and 3L/4.
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What is the #Particle-In-A-Box problem in quantum mechanics? Why are the eigenfunctions of the 1-dimensional particle in a box sine and cosine functions and what are their corresponding eigenenergies? How do you easily calculate the #normalization constants for the particle-in-a-box wavefunctions without doing integrals? What coordinate transformation can you use to simplify the calculation of the #probability of finding the particle in a 1D box?

In this video, we explain the fundamental concepts of the particle in a box problem in 1D. We show you a mathematical trick to quickly deduce the proper normalization coefficients for each eigenfunction of the particle-in-a-box problem. Using the functional form of the eigenfunctions, we walk through step-by-step how to calculate their corresponding eigenvalues. We explain how to compute the probability of finding the particle within any region inside the box and show you how to use Mathematica® to easily perform the necessary integrals.