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- Homework Statement
- We now focus on the Hydrogen atom wavefunctions ##\psi_{nlms}##, where n is the principal quantum number, l and m are the usual quantum numbers associated with orbital angular momentum

(##L^2## and ##L_z## respectively) and s (represented by an up or down-arrow) is the quantum

number associated with ##S_z## . Define J = L + S as the total angular momentum.

1.3. Take the three states :

$$\psi_a = \psi_{321\downarrow}$$

$$\psi_b = \frac{1}{\sqrt2}(\psi_{321↓} + \psi_{321↑}) $$

$$\psi_c = \sqrt{\frac{1}{3}}\psi_{211↓} + \sqrt{\frac{2}{3}}\psi_{210↑}$$

For each of them, and for each of the operators $$L_z , J_z$$ and $$S_x$$ (note: x-component!), say whether the state is an eigenstate or not, giving the eigenvalue if it is and calculating the expectation value otherwise. Express your results in the form of a table.

- Relevant Equations
- $$L_z*f_m^l = \hbar mf_m^l$$

I'm really not sure what the question expects me to do here but here is what I do know. If the state is an eigenstate it should satisfy the eigenvalue equation for example;

$$\hat{H} f_m^l = \lambda f_m^l$$

but is the question asking me to use each operator on each state? How do I know if its an eigenstate without using an operator? IS it just me or is the question not so clear? Also I can't get the LaTex code to parse, what are the correct delimiters?

$$\hat{H} f_m^l = \lambda f_m^l$$

but is the question asking me to use each operator on each state? How do I know if its an eigenstate without using an operator? IS it just me or is the question not so clear? Also I can't get the LaTex code to parse, what are the correct delimiters?

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