Using the uncertainty principle determine the radius of the

Using the uncertainty principle determine the radius of the Bohr\'s first orbit. Using the uncertainty principle find the ground state energy of the hydrogen atom.

Solution

From Heisenberg uncertainity principle,

xp_x =

where x and p_x are the uncertainties in the simultaneous measurements of position and momentum of the electron

Where = h/2

Or    p_x = /x

If V is the uncertainity in the potential energy then

V = -KZe²/x

If K is the uncertainity in kinetic energy then

K = (p_x)²/2m

K= ² /2m(x)²

If E is the uncertainity in total energy (E) then

E = V+K

E = -KZe²/x + ² /2m(x)²

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

E= – KZe²/r + ² /2mr²

For minimum value of E   

d(E)/dr = 0; d²((E)/dr² >0

d(E)/dr=0= KZe²/r² – ²/mr³ = 0

KZe²/r² = ²/mr³

^{r = 2}/KmZe²

For H atom Z =1

Hence, the radius of first orbit is given by

r=²/Kme²  

= 1.05e-34; m = 9.1e-31; e = 1.6e-19

r = 0.529 Angstorm

Substituting value of r in expression of E one can get value of E_min

Hence E_min = -13.59 eV [Ground state energy]