# Is the wave function normalized?

1. Jan 27, 2009

### jg370

1. The problem statement, all variables and given/known data

The ground state wave function for the electron in a hydrogen atom is:

$$\psi(r) = \frac{1}{\sqrt (\pi a_o^3)} e^\frac{-r}{a_o}$$

where r is the radial coordinate of the electron and a_o is the Bohr radius.

Show that the wave function as given is normalized.

2. Relevant equations

Any wave function satisfying the following equation is said to be normalized:

$$\int_{-\infty}^{+\infty} |\psi|^2\dx = 1$$

3. The attempt at a solution

Because the sum of all probabilities over all values of r must be 1,

$$\int_{-\infty}^{+\infty} (\frac{1}{\sqrt (\pi a_o^3)} e^\frac{-r}{a_o})^2 dr = \frac{1}{\pi a_o^3} \int_{-\infty}^{+\infty} e^\frac{-2r}{a_o^2} dr = 1$$

Since the integral can be expressed as the sum of two integrals, we have,

$$\frac{1}{\pi a_o^3} \int_{-\infty}^{+\infty} e^\frac{-2r}{a_o^2} dr = \frac{2}{\pi a_o^3} \int_0^{+\infty} e^\frac{-2r}{a_o^2} dr = 1$$

After integrating, I obtain,

$$\frac{1}{\pi a_o^3} = 1$$

which is definitely incorrect. However, I do not see any other way to proceed. could someone give some assitance.

jg370

2. Jan 27, 2009

### latentcorpse

you integrated wrong -
$\int_0^{\infty} e^{-\frac{2r}{a_0^2}} dr = [-\frac{a_0^2}{2} e^{-\frac{2r}{a_0^2}]_0^{\infty}=\frac{a_0^2}{2}$

which gives $\frac{1}{2 \pi a_0}=1$

3. Jan 27, 2009

### CPL.Luke

the a_0 in your exponent shouldn't be squared, also the normalization condition says you should integrate over the bounds of your function. if r is the radial coordinate then how can it have a value of negative infinity? similarly your no longer normalizing in 1-d but 3D I recommend using spherical coordinates in which case your integral inherits a r^2 sin theta

4. Jan 28, 2009

### jg370

Tks. I wil try again using spherical coordinates