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knowing that energy is given by $E_{n}=\frac{n^{2}\pi^{2}\hbar^{2}}{2ma^{2}}$

and that $$|\psi(t=0)\rangle=\frac{1}{\sqrt{6}}|\phi_{1}\rangle+\frac{1+i}{\sqrt{12}}|\phi_{2}\rangle+\frac{1-i}{\sqrt{4}}|\phi_{3}\rangle+\frac{i}{\sqrt{6}}|\phi_{4}\rangle$$

I want to calculate the probability of finding the value $E_{1}$ when the energy is measured

The first thing I did was to find the norm:

$$\langle\psi|\psi\rangle=\frac{1}{6}+\frac{(1+i)^{2}}{12}+\frac{(1-i)^{2}}{4}+\frac{i^{2}}{6}=-\frac{i}{3}\\ E_{1}=\frac{\pi^{2}\hbar^{2}}{2ma^{2}}$$

\begin{align} P(E_{1})&=\frac{|\langle\phi_{1}|\psi\rangle |^{2}}{\langle\psi|\psi\rangle}\\&=|\frac{1}{\sqrt{6}}\langle\phi_{1}|\phi_{1}\rangle|^{2}(-\frac{3}{i})\\&=-\frac{1}{2i}\\&=\frac{i}{2} \end{align}

Since energy was measured, shouldn't the probability of finding this value be a real number instead of complex? Or is it ok to find an imaginary probability?

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  • $\begingroup$ How come the norm of you state is not unity? $\endgroup$ – my2cts May 26 at 15:34
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The problem is not non-unitarity or normalization, the problem is that you forgot to complex-conjugate the coefficients in the bra $\langle\psi|$ when computing the norm. When you recompute this with complex conjugation you'll find the correct result, the procedure was right.

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    $\begingroup$ @BioPhysicist Is it the current policy to downvote answers to question which should be closed? Why not just wait for them to be closed? $\endgroup$ – Giorgio Comitini May 26 at 16:41
  • $\begingroup$ Because closing questions doesn't remove answers. If you agree that a question should be closed then you shouldn't post an answer to it. I down vote answers that I do not think are useful to the site. That includes answers to questions that should be closed because it gives something for other users to point to to justify posting questions that should be closed. $\endgroup$ – BioPhysicist May 26 at 16:50
  • $\begingroup$ Got it, seems fair. If it gets closed I'll remove the answer. $\endgroup$ – Giorgio Comitini May 26 at 16:53
  • $\begingroup$ @BioPhysicist Ah, accepted answers cannot be deleted! I had no idea, sorry. $\endgroup$ – Giorgio Comitini May 27 at 15:26
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Probability should never be complex. If it comes out complex, you made a mistake somewhere.

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  • $\begingroup$ The OP puts the normalization in when they calculate the probability. They just calculated it incorrectly. $\endgroup$ – BioPhysicist May 26 at 15:56
  • $\begingroup$ Yes, that was exactly where I wanted him to find his mistake, instead of giving it to him directly. (It's not very clear to me why the downvote) $\endgroup$ – Davide Morgante May 26 at 15:58
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    $\begingroup$ No, I'm saying they did what you say to do already. They just didn't calculate $\langle\psi|\psi\rangle$ correctly. I down voted because we shouldn't be answering check my work questions. $\endgroup$ – BioPhysicist May 26 at 16:00
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    $\begingroup$ I see now, it's the first time I do this type of calculation with complex coeficients, so I just did the coeficients squared rather than multiplied by them conjugated. the function is already normalised $\endgroup$ – random name May 26 at 16:07
  • $\begingroup$ the probability will then be 1/6 $\endgroup$ – random name May 26 at 16:09

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