Question

Determination of enrgy of hydrogn atom

Answer 1

When you apply the quantum mechanical Schrodinger Equation for hydrogen atom, the quantization condition for the wave function of r to remain finite as r goes to infinity is:

                                       nπ-me²÷h²=0 , n=1,2,3

Where,

                                            π=(-2mE)^1/2÷h

Substituting,

                                                       π

into the quantization-condition equation gives you the following:

following:

So here’s the energy, E (Note: Because E depends on the principal quantum number, you rename it En):

Physicists often write this result in terms of the Bohr radius — the orbital radius that Niels Bohr calculated for the electron in a hydrogen atom, r0. The Bohr radius is

And in terms of r0, here’s what En equals:

The ground state, where n = 1, works out to be about E = –13.6 eV.

Notice that this energy is negative because the electron is in a bound state — you’d have to add energy to the electron to free it from the hydrogen atom. Here are the first and second excited states:

First excited state, n = 2: E = –3.4 eV

Second excited state, n = 3: E = –1.5 eV

So you’ve now used the quantization condition, which

to determine the energy levels of the hydrogen atom.