Excited States of Helium

The question asks: The two electrons of Helium are found in the antisymmetrized version of $$\Psi_{He} = \chi(r_{1})\Theta(r_{2})$$ where $$\chi(r) = (\frac{1}{\sqrt5} \psi_{100} + \frac{2}{\sqrt5}(1+i) \psi_{211})|\uparrow>$$ and $$\Theta(r)$$ = $$\psi_{100}|\uparrow>$$.

a) Write the normalized antisymmetric form of the above wavefunction

b)Using the above wavefunction calculate the following expectation values: E = $$<\Psi|\hat{H}|\Psi>$$,$$<\Psi|\hat{L^{2}}|\Psi>$$, and $$<\Psi|\hat{L_{x}}|\Psi>$$.

The Attempt

a) I just antisymmetrized the wave function and said that $$\Psi_{He} = \frac{1}{\sqrt2}(\chi(r_1)\Theta(r_2) -\chi(r_2)\Theta(r_1))$$

I think i did this right, but it's possible that I made a mistake... I'm more concerned about part b)

b) For B I started with trying to solve for E by doing:

E = $$\frac{1}{2}(<\chi(1)|<\Theta(2)| - <\chi(2)|<\Theta(1)|)(H_1+H_2)(|\chi(1)>|\Theta(2)>-|\chi(2)>|\Theta(1)>)$$

Which I reduced eventually to : $$\frac{1}{2}(<\chi(1)|(H_1)|\chi(1)> + <\Theta(2)|H_2|\Theta(2)> - <\chi(1)|H_1|\Theta(1)> - <\Theta(1)|(H1)|\chi(1)>$$

$$-<\Theta(2)|H_2|\chi_2> - <\chi_2|H_2|\Theta_2>+<\chi(2)|(H_1)|\chi(2)> + <\Theta(1)|H_2|\Theta(1)>$$

but somehow i keep getting imaginary values in my energy which doesn't qualitatively make sense to me? Any advice?

• I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. – Ben Crowell Jan 24 at 19:22
• didn't know that was a thing! Thanks Ben! – AWiltzer Jan 24 at 19:28
• Could you detail the last equation until you get the imaginary values as you mentioned? – rnels12 Jan 29 at 14:29