Hydrogen atom and Vibrational SPec

[dcm18]

I was wondering if anyone could point me in the right direction for any of these questions...

 

Show that the wavefunction for the 1s electron orbital for the hydrogen atom is a solution of the 3-D Schrodinger equation. What is teh total energy eigenvalue for this wavefunction?

 

The bond vibrational energy levels for the 1H35Cl molecule can be described by Morse Potential with De=7.41E-19 J. The force constant k=516.3 N/m and frequency=8.97E13 1/s

 

a) Calculate the lowest four energy levels using the Morse potential.

 

b) Calculate the fundamental frequency associated with teh n=0 --> n=1 transition and the frequencies of the first three overtone vibrations.

 

 

....any guidance would be much appreciated.

 

Thanks

[Testo]

Essentially you need to plug the 1s wavefunction (radial and angular parts) into the 3D time independant schrodinger equation and show that both sides are equal. Theres some tricky partial differenciation involved here but this should also allow you to find the eigenvalue for this eigenfunction.

 

For the vibrational levels you will need the anharmonic equation

G = w(v+1/2)-xw(v+1/2)^2 where w = (1/2pi)x(k/reduced mass)^1/2 and x = w/4D(e) then let v = 0,1,2,3 (four lowest energy levels)

 

The last part is simply subtraction of these levels