Normalizing etc. Wavefunction of hydrogen atom

[Kentaxel]

Homework Statement

Find the constant A such that the equation

\psi(r,\theta,\varphi)=\sqrt{6\pi}A\sqrt{r}e^{-r/a}

Wich describes one electron in a hydrogenatom, is normalized

The Attempt at a Solution

I figured this equation is seperable in the form

\psi(r,\theta,\varphi)=R(r)Y_{l,m}(\theta,\varphi)

Such that Y_{l,m} is the first spherical harmonic

Y_{0,0}=\frac{1}{\sqrt{4\pi}}

Enabling me to write ψ in the form

\sqrt{24}{\pi}A\sqrt{r}e^{-r/a}Y_{0,0}(\theta\varphi)

and since the spherical harmonics are all normalized it is suficient to normalize this acording to

\int\left|\sqrt{24}{\pi}A\sqrt{r}e^{-r/a}\right|^2 r^2 dr =1

Which gives me the result

A=\frac{1}{3\pi^2 a^4}the problem is just that there is no solution available and I am not exactly 100% that this is correct so i would appreciate some input.

 

[anjelin]

Your solving way is right. But why you choose l=0, m=o. I think you have to write in general form i.e.,

I attached file.

 

[Jorriss]

Homework Statement

Find the constant A such that the equation

\psi(r,\theta,\varphi)=\sqrt{6\pi}A\sqrt{r}e^{-r/a}

Wich describes one electron in a hydrogenatom, is normalized

The Attempt at a Solution

I figured this equation is seperable in the form

\psi(r,\theta,\varphi)=R(r)Y_{l,m}(\theta,\varphi)

Such that Y_{l,m} is the first spherical harmonic

Y_{0,0}=\frac{1}{\sqrt{4\pi}}

Enabling me to write ψ in the form

\sqrt{24}{\pi}A\sqrt{r}e^{-r/a}Y_{0,0}(\theta\varphi)

and since the spherical harmonics are all normalized it is suficient to normalize this acording to

\int\left|\sqrt{24}{\pi}A\sqrt{r}e^{-r/a}\right|^2 r^2 dr =1

Which gives me the result

A=\frac{1}{3\pi^2 a^4}


the problem is just that there is no solution available and I am not exactly 100% that this is correct so i would appreciate some input.

Method seems fine. Viewing it as a spherical harmonic though is unnecessary. All you needed to do was integrate the given wave function over all space and the angular contributions would give 4pi (full solid angle).

 

[Kentaxel]

The reason i choose to separate it was because of the following questions which are

Show that ψ is not an energy eigenfunction.

and

Compute the probability to find the system in the groundstate of the hamilton operator when measuring the energy.

Where regarding the first one i thought that in writing the given wavefunction in that form it was obvious that it is not one of the solutions listed for an electron in a hydrogenic atom. This however feels allot like a guess on my part.

For the second one i used the fact that i knew l and m was 0 and tried to solve it by taking \left|c_{1,0,0}\right| ^2 = \left|<\psi_{1,0,0}|\psi > \right| ^2

Otherwise how would i know what groundstate psi to use?

Edit: actually since the ground state is non degenerate i suppose that would not matter?