Is the Hydrogen Atom Ground State Wave Function Normalized?

[jg370]

Homework Statement

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.

Homework Equations

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

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

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.

Thanks for your kind assistance

jg370

 

[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

 

[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

 

[jg370]

Tks. I wil try again using spherical coordinates