## Mean of powers of r for hydrogen wave functions (Pt. 2)

**Problems:**

1.) Show that

2.) Show that

**Solutions:**

(1) Given that

,

.

But note that ,

so .

Now, recall the radial equation

.

But . Substitution and rearranging gives:

.

To find , treat the electron charges as continuous variable

Multiplying both sides by and integrating over r,

Note that:

and

So,

.

Recall that the Bohr radius, ,

hence, can be written as:

.

(2) Similarly,

But note that ,

so

Now recall the radial equation

To find , treat the orbital quantum number l as continuous variable.

So,

Multiplying both sides by and integrating over r,

Note that:

and

Hence,