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In Heisenberg picture, my textbook says that the following matrix

$A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity.

the book says this is because $\frac{5}{3}\Sigma_1$ and $\frac{4}{3}\Sigma_2$ cannot have definite values.

At here, $\Sigma_1$ and $\Sigma_2$ represent Pauli matrices.

I am curious why this is; how can $\frac{5}{3}\Sigma_1$ and $\frac{4}{3}\Sigma_2$ cannot have definite values?

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any comment....? –  Dotzero Sep 25 '12 at 6:54
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2 Answers 2

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I) Recall that the observables in quantum mechanics are given by self-adjoint operators (or matrices).

The matrix

$$A ~=~ \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$$

is not self-adjoint

$$A^{\dagger} ~=~ \frac{5}{3}\Sigma_1 - i\frac{4}{3}\Sigma_2,$$

so $A$ is not an observable in the traditional sense.

II) However, there is a caveat to the above. If we e.g. have two mutually commuting self-adjoint operators $B\!=\!B^{\dagger}$ and $C\!=\!C^{\dagger}$, we can construct a so-called "complex observable"

$$A~:=~B+iC.$$

Then we could find a complete set of simultaneous eigenstates for $B$ and $C$. And then $A$ would have complex-valued eigenvalues on these eigenstates. (Technically $A$ is then known as a normal operator. This is the quantum version of what we know from classical geometry, that we can equivalently represent a point in the 2D plane as two real coordinates $(x,y)\in\mathbb{R}^2$, or as a single complex number $z=x+iy$.)

In our case we have $B=\frac{5}{3}\Sigma_1$ and $C=\frac{4}{3}\Sigma_2$. But the commutator $[B,C]\neq 0$ is not zero, so we can not interpret $A$ as a "complex observable".

III) Conclusion: $A$ is neither a "real" (i.e. self-adjoint) nor a "complex" (i.e. normal) physical observable.

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The book wants to say that the two Pauli matrices $\sigma_1,\sigma_2$ don't commute with each other, so they can't have definite values simultaneously. This is just another example of the Heisenberg uncertainty principle. They're making the same comment about the Pauli matrices as the usual comment about the position and momentum that can't be determined at the same time.

For $A$ to be well-defined, you would need the state to be an eigenstate of both Pauli matrices because $A$ is a complex combination of them. If the relative coefficient were real, then $A$ would be just another, slanted component of the Pauli matrix vector, and it could have eigenvalues even though they would be eigenvalues of neither $\sigma_1$ nor $\sigma_2$.

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Question: But the book says that because $\frac{5}{3}\sigma_1$ and $\frac{4}{3}\sigma_2$ represents real quantities that cannot have definite values together, matrix $A$ cannot be a physical quantity.. I do get that these two matrices cannot have definite values simultaneously, but then, following the book's explanation, how can any combination of (Pauli or maybe non-Pauli) matrices be used to represent physical quantity? Am I seriously mistaken? –  Dotzero Sep 25 '12 at 12:52
    
Dear Dotzero, a real combination of these two operators would simply be a component of the spin along another axis, one that is neither x nor y but has some angle in between both. –  Luboš Motl Sep 25 '12 at 15:03
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