Applications of the Heisenberg Uncertainty Principle: The Radius of Bohr’s First Orbit

In one of my earlier articles, I have discussed the one the applications of the Heisenberg uncertainty principle that is non-existence of electron in the nucleus. Let us discuss today the one more application of the Heisenberg uncertainty principle that is the determination of the radius of the Bohr’s first orbit. Let us start:

If ∆x and ∆px are the uncertainties in the simultaneous measurements of position and momentum of the electron in the first orbit, then from uncertainty principle

∆x∆px = Ћ

Where Ћ = h/2∏

Or    ∆px = Ћ /∆x                                                                 (1)

As kinetic energy is given as

K = p2/2m

Then uncertainty in K.E is

∆K =∆p2x/2m

Put equation (i) in above equation

∆K= Ћ2 /2m(∆x)2 (2)

As potential energy is given by

∆V= -1/4∏ε0 Ze2/∆x                                                      (3)

The uncertainty in total energy is given by adding equations (2) and (3), that is Continue reading “Applications of the Heisenberg Uncertainty Principle: The Radius of Bohr’s First Orbit”