Finding E, L and Lz from wavefunction

[the riddick25]

Homework Statement

We were given the wavefunction for a hydrogen atom (ignoring spin) as shown in the link below
We are asked to find the probability of obtaining E=E1, L^2=2 hbar^2 and Lz=hbar

Homework Equations

http://s633.photobucket.com/albums/uu58/john_sharkey/?action=view&current=equation.jpg

The Attempt at a Solution

i have no idea what equations i would need to find the probabilities of finding these results.
If somebody could point me in the right direction it would be much appreciated.

 

[ideasrule]

That wavefunction is a superposition of two stationary states, namely psi_100 and psi_211. Each stationary state has a definite E, L^2, and Lz: if you know that a particle is in the state psi_100, for example, you will always get E1 when you measure its energy.

This particle is not yet in any state, but when you take a measurement, it randomly collapses into one of the two stationary states. The question is essentially asking what the probability of it collapsing into the state with energy E1 is, and ditto for L^2 and Lz.

 

[the riddick25]

is the probability of obtaining each stationary state just 2/3 and 1/3 respectively?

Also thank you very much for your help

 

[ideasrule]

Yup, and you're welcome.

 

[paranormal]

I would approach this problem by first understanding the physical meaning of the wavefunction. The wavefunction represents the probability amplitude of finding an electron at a certain position in space. From this, we can derive the probability of finding the electron in a certain energy state, angular momentum state, and angular momentum projection state.

To find the probability of obtaining E=E1, we can use the probability density function, which is the square of the wavefunction. We can integrate this over all space to find the total probability of finding the electron in the energy state E1.

To find the probability of obtaining L^2=2 hbar^2, we can use the eigenvalue equation for the angular momentum operator. This will give us a set of allowed values for L^2, and we can then calculate the probability of obtaining the desired value.

Similarly, to find the probability of obtaining Lz=hbar, we can use the eigenvalue equation for the z-component of angular momentum. Again, this will give us a set of allowed values for Lz, and we can then calculate the probability of obtaining the desired value.

Overall, the key equations to use in this problem are the probability density function, the eigenvalue equations for the energy and angular momentum operators, and the normalization condition for the wavefunction. With these, we can calculate the probabilities of obtaining the desired results and solve the problem.