Proving the Existence of a Hydrogen Electron in Space via Wavefunction Integral

[bigevil]

Homework Statement

A wavefunction for a hydrogen electron is given by $$\Psi = - <br /> <br /> \sqrt{\frac{3}{8 \pi}} sin\theta e^{i \phi} (\frac{1}{2a^3})^{3/2} <br /> <br /> \frac{re^{-r/2a}}{a \sqrt{3}}$$

Prove that the electron exists in space, ie, $$\int {\Psi}^2= 1$$

2. Homework Equations & attempt at solution

Apologise in advance for the shortcuts, these equations are terrible to type

out.

Expressed in spherical polar coordinates, $$dV = r^2 sin \theta dr <br /> <br /> d\theta d\phi$$

The squared wavefunction,
$${\Phi}^2 = \frac{1}{64\pi a^5} r^2 {sin}^2 \theta e^{2i\phi}$$

With respect to r, $$\int^{\infty}{\0} r^4 e^{r/a} = 24 a^5$$

This is a pain to do due to iterated application of integration by parts, but

by inspection,

$$<br /> \int^{\infty}{\0} r^4 e^{r/a} = 4a \int^{\infty}{\0} r^3 e^{r/a} = 4.3a^2 <br /> <br /> \int^{\infty}{\0} r^2 e^{r/a}... = 24a^5<br />$$


With respect to $$\theta$$,

$$\int^{\pi}{\0} {sin}^3 \theta d\theta = \frac{4}{3}$$

This gives us,

$$\int {\Phi}^2 dV = \frac{1}{2\pi} \int^{2\pi}{\0} e^{2i\phi} d\phi$$

I'm stuck at this point. How do I proceed? Was my earlier working correct?

If the earlier integration was right, then the last integral must be equal to 2pi.


Exploration
From using traditional methods the answer I actually get is 0. How does the pi term come about.

 

[HallsofIvy]

Shouldn't that be
$$\int |\Psi|^2 dV$$?

In other words, you want the absolute value squared, not the function. And
[tex]|e^{i\phi}|= 1[/itex][/tex]

 

[bigevil]

Thanks, HallsofIvy, I think that explains that one. (Sorry, I mixed up psi/phi.) The rest of my solution is ok, right?

My text explicitly gave the condition required as

$$\int {\Psi}^2 dV = 1$$

Which leads me to another question. What is the physical significance of the modulus? (I'm not studying a text on quantum physics at the moment, I'm working on a mathematical physics text.)

 

[Ygggdrasil]

$$|\Psi |^2$$ is interpreted to be a probability density function describing the probability of finding your particle in a specific state.