# 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

∆E= ∆K+∆V

= Ћ2 /2∏(∆x)2 –Ze2/4∏ε0∆x

If ∆x = r= radius of Bohr’s orbit, then

∆E= Ћ2 /2mr2 –Ze2/4∏ε0r                                                   (4)

The Uncertainty in total energy will be minimum if

d(∆E)/dr=0 and d2((∆E)/dr2 is positive

Differentiating equation (4) w.r.t. r, we get

d(∆E)/dr=0= – Ћ 2/mr3+Ze2/4π ε0r2 (5)

For minimum value of ∆E

d(∆E)/dr=0= – Ћ 2/mr2+Ze2/4π ε0r2

or                  Ze2/4π ε0r2= Ћ 2/mr3

Or                       r=4π ε0 Ћ 2/me2 (6)

Further differentiating equation (5), we get

d2(∆E)/dr2=3 Ћ 2/mr4-2Ze2/4π ε0r3

By putting value of r from equation (6) in above equation, we get positive value of

d2(∆E)/dr2

Therefore equation (4) represents the condition of minimum in the first orbit.

Hence, the radius of first orbit is given by

r=4π ε0Ћ 2/me2=0.53 angstrom                                  (For H atom Z=1)

Put value of r in equation (4), we get

Emin= -13.6 e V

This value is same as determined by using Bohr’s theory.

Therefore, with the help of Heisenberg’s uncertainty principle, one can determine the radius of the Bohr’s first orbit.