130_notes.dvi

5.7.6 Estimate the Hydrogen Ground State Energy

The reason the Hydrogen atom (and other atoms) is so large is the essentially uncertainty principle.
If the electron were confined to a smaller volume, ∆pwould increase, causingpto increase on
average. The energy would increase not decrease.

We can use the uncertainty principle to estimate the minimum energy for Hydrogen. This is not a
perfect calculation but it is more correct than the Bohr model. The idea is that the radius must be
larger than the spread in position, and the momentum must be largerthan the spread in momentum.

r ≥ ∆x

p ≥ ∆p=

̄h

2 r

This is our formula for the potential energy in terms of the dimensionless fine structure constantα.

V(r) =−

e^2

r

=−

α ̄hc

r

Lets estimate the energy

E=

p^2

2 m

−

α ̄hc
r
and put in the effect of the uncertainty principle.

pr = ̄h

E =

p^2

2 m

−

α ̄hcp

̄h

E =

p^2

2 m

−αcp

Differentiate with respect topand set equal to zero to get the minimum.

dE

dp

=

p

m

−αc= 0

p = αmc

E =

α^2 mc^2

2

−α^2 mc^2 =−

α^2 mc^2

2

=− 13 .6 eV

Note that the potential energy is just (-2) times the kinetic energy (as we expect from the Virial
Theorem). The ground state energy formula is correct.

We can also estimate the radius.

r=

̄h

p

=

̄h

αmc

=

̄hc

αmc^2

=

1973 eV ̊A(137)

511000 eV

= 0. 53 ̊A

The ground state of Hydrogen has zero (orbital) angular momentum. It is not moving in a circular
orbit as Bohr hypothesized. The electron just has a probability distribution that is spread out over
about 1 ̊A. If it were not spread out, the energy would go up.