Calculating Normalization Constant for Wavefunction

[sarabellum02]

How do I calculate the normalization constant for a wavefunction of the form (r/a)e^(-r/2a) sin(theta)e^(i*phi)?

How would I write the explict harmonic oscillator wavefunction for quantum number 8(in terms on pi, alpha, and y)

thanx

 

[masudr]

Remember that the probability of the particle existing somewhere in all space is certain. So we have

$$\int_{-\infty}^{\infty}\psi\left(x\right)\psi^*\left(x\right)dx=1$$.

For the case of the wavefunction you have been given, an exact anti-derivative exists with these particular limits.

EDIT: Now correct for the 1D case. See jtbell's post for the correct answer.

 

[jtbell]

No, this is a three-dimensional wave function in spherical coordinates, so the integral looks like this:

$$\int_0^{2 \pi} {\int_0^{\pi} {\int_0^{\infty}{\psi^*(r, \theta, \phi) \psi(r, \theta, \phi)} r^2 \sin \theta \ dr} \ d\theta} \ d\phi} = 1$$

 

[masudr]

Yes, of course, jtbell is correct. Sorry. What I wrote was wrong even in the 1D case.

 

[dextercioby]

It was correct in the ID case,those wave functions are scalars (bosonic variables) and can be switched places inside the integral.


Daniel.

 

[dextercioby]

How would I write the explict harmonic oscillator wavefunction for quantum number 8(in terms on pi, alpha, and y)

thanx

How many dimensions does this oscillator have...?It's essential to know this fact.As for the variables you posted,they couldn't ring a bell,because notation conventions are not unique...

Daniel.