Simple Nature - Light and Matter

We now solve the equationke^2 /r ∼ h^2 n^2 / 2 mr^2 forrand throw
away numerical factors we can’t hope to have gotten right, yielding

[4] r∼

h^2 n^2

mke^2

.

Pluggingn= 1 into this equation givesr= 2 nm, which is indeed

on the right order of magnitude. Finally we combine equations [4]

and [1] to find

E∼−

mk^2 e^4

h^2 n^2

,

which is correct except for the numerical factors we never aimed to
find.

Exact treatment of the ground state

The general proof of the Bohr equation for all values ofnis

beyond the mathematical scope of this book, but it’s fairly straight-

forward to verify it for a particularn, especially given a lucky guess

as to what functional form to try for the wavefunction. The form

that works for the ground state is

Ψ =ue−r/a,

wherer=

√

x^2 +y^2 +z^2 is the electron’s distance from the pro-

ton, anduprovides for normalization. In the following, the result

∂r/∂x= x/rcomes in handy. Computing the partial derivatives

that occur in the Laplacian, we obtain for thexterm

∂Ψ

∂x

=

∂Ψ

∂r

∂r

∂x

=−

x

ar

Ψ

∂^2 Ψ

∂x^2

=−

1

ar

Ψ−

x

a

(

∂

dx

1

r

)

Ψ +

(x

ar

) 2

Ψ

=−

1

ar

Ψ +

x^2

ar^3

Ψ +

(x

ar

) 2

Ψ,

so

∇^2 Ψ =

(

−

2

ar

+

1

a^2

)

Ψ.

The Schr ̈odinger equation gives

E·Ψ =−

~^2

2 m

∇^2 Ψ +U·Ψ

=

~^2

2 m

(

2

ar

−

1

a^2

)

Ψ−

ke^2

r

·Ψ

If we require this equation to hold for allr, then we must have

equality for both the terms of the form (constant)×Ψ and for those

Section 13.4 The atom 931