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.