I went again through some of my undergraduate books of quantum mechanics to get a new look at it as a futur PhD (not in QM though). I got answers for some old questions that bugged me at the time but their is one I can't figure out. I left it years ago and coming back to it I still can't get the point of the exercice.

It is from the book "Quantum mechanics in simple matrix form" by Thomas F. Jordan. I liked it a lot because the author starts from scratch concerning imaginary numbers and matrix algebra.

However, in the seventh chapter, their is only one exercise and I don't see the point. The other exercices in the previous chapter were straight forward computing to get a first hand on algebra and were quite useful for me as a student. But in chapter 7 (no mention of quantum mechanics has been done so far except Heisenberg inequality, which is the guideline of the book), the exercice reads as follows:

Two quantities are represented by the matrices: $$M=\left[\begin{array}{l}3&0&-i\\0&1&0\\i&0&3\end{array}\right]\ N=\left[\begin{array}{l}3&0&2i\\0&7&0\\-2i&0&3\end{array}\right]$$ The possible values of the quantity $M$ are $1,2$ ans $4$. What are the possible values of the quantity represented by N?

The author never mentionned the link between probabilities of a given value and its matrix representation so I'm a bit confused. I don't understand how the exercice is supposed to be solvable and what is the expected solution. What is the point of this exercice? I'm sure I miss something but I can't tell what.

  • $\begingroup$ @JohnForkosh Thank you. I'll dig it if I can find it. I'm actually revisiting my old textbooks to, as I mentionned, check old questions I couldn't answer at the time. Jordan's book is quite good but the beginning is really intented to laymen with absolutely no clue about complex numbers and matrix algebra. Yet I try to answer the questions only with what has been mentionned so far and it is sometime quite tricky. $\endgroup$ – G.Clavier Mar 15 '17 at 8:40

The possible values are the eigenvalues. There is no link between a matrix and probabilities: the probabilities depend on the modulus squared of overlap between the quantum state $\vert \psi\rangle$ under investigation and the eigenvectors. As stated the question is fine as it only asks for possible outcomes.

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  • $\begingroup$ That was my initial guess. But the author never mentioned any diagonalisation method so I tried to make a guess using only what as been described in the book so far (basic matrix algebra and probabilities). So I was wondering is their was any way to make a guess about eigenvalues using only squared matrices or matrix product. It seems not from what you're saying. $\endgroup$ – G.Clavier Mar 14 '17 at 19:57
  • $\begingroup$ @G.Clavier actually your matrices are equivalent to block diagonal matrices so by inspection $1$ and $7$ are eigenvalues of the first and second matrices, respectively. Otherwise you have to slug it out. $\endgroup$ – ZeroTheHero Mar 14 '17 at 20:11
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    $\begingroup$ You can avoid the slugging by noting that N = -2M + 9I, so the eigenvalues of N are −2∗(eigenvalues of M)+9, or {7,5,1}. $\endgroup$ – Paul G Mar 17 '17 at 17:17
  • $\begingroup$ @PaulG This is exactly the link I was missing, thanks! $\endgroup$ – G.Clavier Apr 5 '17 at 14:00

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