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The wave function of an electron in a hydrogen atom is given by

Wavefunction

  1. Is this wave function an eigenfunction of Jz , the z-component of the electron’s total angular momentum? If yes, find the eigenvalue. (Hint: For this, you need to calculate Jz Psi21*mlms*.)
  2. If you measure the z-component of the electron’s spin angular momentum, what values will you obtain? What are the corresponding probabilities?
  3. If you measure J^2, what values will you obtain? What are the corresponding probabilities?

How can I solve this problem or with which rules can be obtains.

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1 Answer 1

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The last spin state is wrong. It should be $\frac{|\uparrow\downarrow\rangle+ |\downarrow\uparrow\rangle}{\sqrt{2}}$, to be the $S_z=0$, $S=1$ spin state.

  1. The state would then be an eigenstate of $J_z$, with eigenvalue $+1$.
  2. Measurements of $S_z$ could be $S_z=0$ ($P=1/3$) and $S_z=+1(P=2/3)$.
  3. $J^2=2*3=6$, with $P=1$.
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