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Introduction

I'm reading the book "Quantum Mechanics in Simple Matrix Form" by T. Jordan. I try to solve the problem sets. With problem 7-1 i have a solution, i think, but i'm not sure if the writer had this solution in mind.

The Problem 7-1

Two quantities are represented by the matrices:

$$ M = \left( \begin{array}{ccc} 3 & 0 & -i \\ 0 & 1 & 0 \\ i & 0 & 3 \end{array} \right) $$

$$ N = \left( \begin{array}{ccc} 3 & 0 & 2i \\ 0 & 7 & 0 \\ -2i & 0 & 3 \end{array} \right) $$

The possible values of the quantity represented by $M$ are 1,2 and 4. What are the possbile values of the quantity represented by $N$? Explain how you know that.

My answer:

The eigenvalues of $N$ are 5,1 and 7. So that is my answer.

My Question for help:

But this is probably to advanced. In the book eigenvalues are not described at all and should not be used, i think. Is there an other (better) way to find the possible values?

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  • $\begingroup$ What do the book mean by "possible values". It can be different types of things for a matrix, eigenvalues being just one of the examples? $\endgroup$ – Secret Aug 28 '16 at 9:44
  • $\begingroup$ From PREFACE of the book : "The book is ambitious in making basic quantum mechanics accessible with minimum mathematics. It avoids the mathematics of Hilbert space, Hermitian and unitary matrices, eigenvalues and eigenvectors, and the like. There are no state vectors or wave functions at all. There are no differential equations. The book does not even use calculus or trigonometry. It assumes only basic algebra." How is this possible? $\endgroup$ – Frobenius Aug 28 '16 at 13:20
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    $\begingroup$ See my comment on essentially the same question: link $\endgroup$ – Paul G Jun 22 '17 at 4:39
  • $\begingroup$ duplicate of physics.stackexchange.com/questions/318777/… $\endgroup$ – ZeroTheHero Dec 9 '18 at 4:36
  • $\begingroup$ Possible duplicate of How is simple matrix representation related to quantum probabilities? $\endgroup$ – Buzz Dec 10 '18 at 2:32
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The goal of the problem is to identify the relationship between the two representations of the quantities, and then determine the values of the quantity represented by the matrix N in terms of the values of the quantity represented by the matrix M.

In the context of the textbook chapter, the matrix representation of a quantity can be expressed in the form $$ N = zM + w1 $$ where z is a number and w is a number while 1 is the identity matrix. By inspection, $$ z = -2 $$ $$ w = 9 $$ in order to relate the representation M to the representation N. Given the possible values of the quantity represented by M, we may now evaluate the possible values of the quantity represented by the matrix N using this relationship.

For the value 1, we find the value of the quantity represented by N must be -2*1 + 9 = 7. Similarly, we find the possible values 5 and 1 for the quantity represented by the matrix N given the values 2 and 4 for the quantity represented by M, respectively.

The possible values of the quantity represented by N found here are in agreement with the eigenvalues of N.

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