Proving the Schwartz Inequality and Heisenberg's Uncertainty Principle

In this post we once again derive the Heisenberg uncertainty principle, but this time we make use of the Schwartz inequality and the position-momentum commutator. We begin our proof by defining the variables:

Below we have the Schwartz inequality:

Is it true? Let's prove it. At lines 2 and 3 we map inner products with multiple (n) dimensions to simpler 2D Pythagorean expressions. At 4 through 7 we define the bras and kets and their inner products in terms of a, b, c, d.

Next, we take the products of the inner products and derive 11 below:

At 12 and 13 we convert the variables ad and bc to x and x+h. You may recognize h from calculus texts. In this case, it is just any arbitrary number. At 14 and 15 we do a little algebra to get 16:

At 16 it is obvious the absolute value of h^2 is greater than or equal to zero. Thus, the Schwartz inequality is true.

Before we put it to work, we need to define energy (E) and time (t). E is the lowest possible energy (ground-state) and t is the reciprocal of frequency (f). We sandwich these between the normalized bras and kets at 20. That brings us to the energy-time uncertainty at 21.

At 22 and 23 we do a quick-and-dirty derivation of the momentum-position uncertainty:

We can also find the momentum-position uncertainty by making use of its commutator and a wave function (psi). At 24 we define the commutator; at 25 we define momentum (p). After making a substitution for p at 26, we do some more algebra until we get the desired outcome at 30.

Equation 30 is looking good, but there is a slight problem: it's an equation! We want an inequality, so we make use of the Schwartz inequality one more time:

Ah, that's better.