Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.8.
As another example of an explicit case of a particle in the infinite square well, we can consider a particle that starts off in a state where it is equally likely to be found anywhere in the left half of the well. This means that the wave function is:
and zero everywhere else.
Normalizing, we require
To find the probability that the particle is in the ground state (with energy ), we need to find the coefficient in the expansion of in terms of the orthonormal function set. Thus:
The probability of this energy is then .