Required math: algebra, calculus (partial derivatives and integration by parts), complex numbers
Required physics: Schrödinger equation, probability density
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.9.
Suppose a particle is in the quantum state
where is the normalization constant and is a constant with dimensions of 1/time. We can find from normalization:
The spatial component of the wave function is
and it must satisfy the time-independent Schrödinger equation in one dimension
The energy can be found from the time equation:
From 7 we have
We can work out a few average values:
since is even.
The standard deviations are
and the uncertainty principle is
so in this case, the uncertainty is the minimum possible.