- #1
Dixanadu
- 254
- 2
Homework Statement
Hey guys, so here's the question:
The energy eigenstates of the hydrogen atom [itex]\psi_{n,l,m}[/itex] are orthonormal and labeled by three quantum numbers: the principle quantum number n and the orbital angular momentum eigenvalues l and m. Consider the state of a hydrogen atom at [itex]t=0[/itex] given by a linear combination of states:
[itex]\Psi=\frac{1}{3}(2\psi_{0,0,0}+2\psi_{2,1,0}+\psi_{3,2,2})[/itex]
(a) What is the probability to find in a measurement of energy [itex]E_{1}, E_{2}, E_{3}[/itex]?
(b) Find the expectation values of the energy [itex]\vec{\hat{L}}^{2}[/itex] and [itex]L_{z}[/itex].
(c) Does this state have definite parity? (HINT: use orthonormality of the [itex]\psi_{n,l,m}[/itex] and the known eigenvalues of [itex]\psi_{n,l,m}[/itex] with respect to [itex]\hat{H}, \vec{\hat{L}}^{2}, \hat{L}_{z}[/itex].
Homework Equations
So here's what we need I think:
Eigenvalues of [itex]\vec{\hat{L}}^{2} = \hbar^{2}l(l+1)[/itex]
Eigenvalues of [itex]\hat{L}_{z} = \hbar m[/itex]
Eigenvalues of [itex]\hat{H} = E_{n}[/itex]..right?
The Attempt at a Solution
so for part (a)...is this just really trivial, that the [itex]E_{1}=\frac{2}{3}, E_{2}=\frac{2}{3}, E_{3}=\frac{1}{3}[/itex] or am I missing something?
(b) I've got something pretty weird...like [itex]<\vec{\hat{L}}^{2}>=\frac{8}{3}\hbar^{2}[/itex] and [itex]<\hat{L}_{z}>=\frac{2}{3}\hbar[/itex] which doesn't seem right to me...
(c) I have no idea!
could you guys gimme a hand please?
thanks a lot!