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In a book I have been reading recently a question as follows came up as a problem and I am unsure how to solve it:

Two quantities are represented by the matrices

$$ \text{M = } \left[\begin{array}{rrr} 3 & 0 & -i \\ 0 & 1 & 0\\ i & 0 & 3 \end{array}\right] $$

$$ \text{N = } \left[\begin{array}{rrr} 3 & 0 & 2i \\ 0 & 7 & 0\\ -2i & 0 & 3 \end{array}\right] $$

The possible values of the quantity represented by M are 1, 2 and 4.

What are the possible values of the quantity represented by N?

Explain how you know that.

Any help would be greatly appreciated!

André Caldas
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Joshua
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1 Answers1

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Note that the matrices $M$ and $N$ are self-adjoint. Given that you're reading a book on quantum mechanics, it makes sense to look at their spectrum (set of eigenvalues).

The spectrum of $M$ is $\{1,2,4\}$ and the spectrum of $N$ is $\{1,5,7\}$.

t.b.
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  • See also: http://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29 – t.b. Apr 17 '12 at 18:45
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    This is a good suggestion; you can read a substantial portion of the book on Amazon's preview, including the problem the OP asks (on pg 59). However, in my opinion, the book is just awful. It introduces matrices assuming the reader has not seen them, and introduces other very elementary ideas (including $i$ as $\sqrt{-1}$) so it's a huge leap to assume the reader would know about eigenvalues, if that was the author's intent. – Fixee Apr 17 '12 at 18:49
  • This sounds like the correct answer, thank you very much! I would agree with Fixee that for the book to spring Eigenvalues on me when it has not even mentioned it yet is a huge leap to expect from readers especially as it is only supposed to assume a basic knowledge of algebra. Thank you all! – Joshua Apr 17 '12 at 18:52
  • Also, out of interest is there a good web page describing how to calculate the Eigenvalues of 3x3 matrices? From what I've read I'm guessing it comes down to a cubic equation which has to be solved which gives the 3 possible values. – Joshua Apr 17 '12 at 18:57
  • @Joshua: unfortunately, I don't know one that I can vouch for: try the Wikipedia page and maybe this lecture from Khan Academy helps. In this case I just "saw" the eigenvalues and eigenvectors: the eigenvectors are $\begin{pmatrix} 0 \ 1 \ 0\end{pmatrix}$ and $\begin{pmatrix} 1 \ 0 \ \pm i\end{pmatrix}$. I would recommend looking at a good linear algebra text, as e.g. one of those suggested here – t.b. Apr 17 '12 at 19:06
  • Here's a (less instructive) way to get the eigenvalues: http://www.wolframalpha.com/input/?i=eigenvalues+%5B%7B3%2C0%2C-i%7D%2C+%7B0%2C1%2C0%7D%2C+%7Bi%2C0%2C+3%7D%5D – Fixee Apr 17 '12 at 22:07
  • Thanks for all the input, @Fixee :) – t.b. Apr 17 '12 at 22:29
  • @t.b. Thanks for the suggestion, the wikipedia page was of help and I also found another Khan Academy lecture which focusses specifically on getting the Eigenvalues of a 3x3 matrix which was extremely helpful. – Joshua Apr 18 '12 at 05:52
  • @Fixee I'll certainly use that method when I don't have the energy to do it myself! :) – Joshua Apr 18 '12 at 05:52