- #1
noamriemer
- 50
- 0
Hello!
My exam is tomorrow... Help will be much appreciated...
Say l=0 is given. My hemiltonian is [itex] \bar {H}=\frac {1} {\hbar} \Omega (\vec {n} \cdot \vec {L})^2 [/itex]
And we use "base states" that are [itex] L_z [/itex] eigenstates, |1 m>
n is a general direction unit vector.
Now, I am looking for all the possible energies I can measure.
What I thought was the answer is - sincs [itex] L^2=l(l+1)\hbar^2 |lm> [/itex]
There is only one option- since l=1. But the solution shows a different thing:
They refer to the possible m's: since l=1, m=-1,0,1
So they actually use [itex] L_z^2 [/itex] instead of [itex] L^2 [/itex]
Why is that true?
In the next section of the question, the vector [itex] \vec {n}= \frac {1} {\sqrt 2} (1,1,0) [/itex] is given.
Now, we put the system(for t=0) in such a way so that [itex] |\psi>= |1,-1> [/itex] and we are looking for the wave function for any time t.
So now is what I don't understand: in the first section, they used [itex] L_z^2 [/itex] in the hemiltonian, instead of [itex] L^2 [/itex] .
But now, using the given n, in the hemiltonian, since n does not have a z component, the Hemiltinian should be zero!
What an I not getting right?
Thank you so much!
My exam is tomorrow... Help will be much appreciated...
Say l=0 is given. My hemiltonian is [itex] \bar {H}=\frac {1} {\hbar} \Omega (\vec {n} \cdot \vec {L})^2 [/itex]
And we use "base states" that are [itex] L_z [/itex] eigenstates, |1 m>
n is a general direction unit vector.
Now, I am looking for all the possible energies I can measure.
What I thought was the answer is - sincs [itex] L^2=l(l+1)\hbar^2 |lm> [/itex]
There is only one option- since l=1. But the solution shows a different thing:
They refer to the possible m's: since l=1, m=-1,0,1
So they actually use [itex] L_z^2 [/itex] instead of [itex] L^2 [/itex]
Why is that true?
In the next section of the question, the vector [itex] \vec {n}= \frac {1} {\sqrt 2} (1,1,0) [/itex] is given.
Now, we put the system(for t=0) in such a way so that [itex] |\psi>= |1,-1> [/itex] and we are looking for the wave function for any time t.
So now is what I don't understand: in the first section, they used [itex] L_z^2 [/itex] in the hemiltonian, instead of [itex] L^2 [/itex] .
But now, using the given n, in the hemiltonian, since n does not have a z component, the Hemiltinian should be zero!
What an I not getting right?
Thank you so much!