Physical significance of eigenvalues?

[James MC]

Standard Pauli spin matrices are:
S_x:
$$\left(\begin{array}{cc}0&1/2\\1/2&0\end{array}\right)$$
S_z:
$$\left(\begin{array}{cc}1/2&0\\0&-1/2\end{array}\right)$$
The S_z eigenvectors are Z₊ = (x=1,y=0) and Z_- = (x=0,y=1). These yield eigenvalues 1/2 and -1/2 respectively. Similarly, eigenvectors of S_x are defined to guarantee eigenvalues 1/2 and -1/2 (e.g. by making X₊ = (1/2(x=sqrt(2),y=sqrt(2))).

My question is this:
Do they need to be those values? What is stopping me from instead defining the operators as follows:
S_x:
$$\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$
S_z:
$$\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$$
Retain the S_z eigenvectors. That gives eigenvalues +1 and -1. Make the S_x eigenvectors such that eigenvalues are +1 and -1 (e.g. by making X₊ = (x=1/sqrt(2),y=1/sqrt(2)).

Is there some reason why the spin matrices don't take this form instead?

A more general question from which this spawns: Hermitian operators are invoked so as to sure that eigenvalues are always real numbers. But why should they be real numbers? What's wrong with complex eigenvalues?

Interested in your thoughts!

 

[muppet]

In QM, operators, like the Pauli matrices (multiplied by hbar) are associated to observable quantities, and their eigenvalues are the results of a possible measurement of that quantity. (e.g. you have a hamiltonian that describes the potential felt by an electron due to the proton; that hamiltonian has eigenvalues; those eigenvalues are the possible results of a measurement of the energy of the electron.). That's why they have to be real - because they can be measured.

If you rescaled the Pauli matrices, they'd no longer describe the physics of spin-1/2 particles. They'd just be random 2x2 matrices.

 

[James MC]

Thanks for your response, but I'm not sure this really gets to the heart of the problem...

In QM, operators, like the Pauli matrices (multiplied by hbar) are associated to observable quantities, and their eigenvalues are the results of a possible measurement of that quantity. [...] That's why they have to be real - because they can be measured.

No the eigenvalues are not results of measurements, eigenvalues *refer* to the results of measurements. For e.g. an x-spin measurement might result in spin-up; and we happen to refer to spin-up with the numeral "+1/2". My questions are (i) why not refer to it with "+1" instead and (ii) why not set things up so the eigenvalue is complex?

If you rescaled the Pauli matrices, they'd no longer describe the physics of spin-1/2 particles. They'd just be random 2x2 matrices.

Yes this must have something to do with it. Can you elaborate? What exactly about the physics would one miss?

 

[Ravi Mohan]

A more general question from which this spawns: Hermitian operators are invoked so as to sure that eigenvalues are always real numbers. But why should they be real numbers? What's wrong with complex eigenvalues?

One demands self-adjointness of operators (corresponding to observables) so that their eigenvectors can form base set for the hilbert space of the system. Now why should the eigen vector form basis (preferably complete orthonormal set)? Well (according to copenhegen interpretation) we expect the system to exist either in one eigen state or the superposition of eigen states of an observable. Thus the linear combination of eigen states should be able to give us any arbitrary state in which system can exist. This can happen only if we demand eigen vectors to form basis.
The "reality" of eigen values is a welcome gift which one gets for demanding self-adjointness.

 

[The_Duck]

Let's talk specifically about the spin angular momentum operators for spin-1/2 particles. These are $\hbar$ times the Pauli matrices. The normalization of the angular momentum operators $J_x, J_y, J_z$ is fixed by the requirement that they obey the commutation relations

$[J_i, J_j] = i \hbar \epsilon_{ijk} J_k$.

If you have some $J$ operators that obey this commutation relation, and then rescale them, the rescaled operators will not obey this commutation relation. So the commutation relation fixes their normalization.

eigenvalues are not results of measurements, eigenvalues *refer* to the results of measurements. For e.g. an x-spin measurement might result in spin-up; and we happen to refer to spin-up with the numeral "+1/2".

Eigenvalues really are the results of measurements. If I measure an electron to have spin-up along the z axis, what I really mean is that its z component of angular momentum is $\hbar/2$. The eigenvalue of the angular momentum operator is numerically equal to the measured angular momentum. That's the whole point of writing down the operator in the first place.

However I think you are right that there is a little bit of arbitrariness. We get the commutation relation $[J_i, J_j] = i \hbar \epsilon_{ijk} J_k$ from the representation theory of the group of spatial rotations. From this commutation relation we can derive the fact that particle spins must be integer or half-integer multiples of $\hbar$. But in this derivation, $\hbar$ is just some undetermined constant to make the units work out. By fixing the value of $\hbar$, we are just defining what units we are using for angular momentum.

In the end, it's often simplest to choose "natural" units in which $\hbar$ is dimensionless and equal to 1. Then the angular momentum commutation relation is $[J_i, J_j] = i \epsilon_{ijk} J_k$. With this convention, angular momentum becomes dimensionless and comes in integer or half-integer units. But in principle we could change the convention and scale all our measurements of angular momenta by some constant factor.

 

[muppet]

However: If you rescale the Pauli matrices, they'll no longer satisfy the angular momentum commutation relations! Suppose you rescale $J_i \rightarrow cJ_i$ for all i. Then the commutator gets rescaled by a factor $c^2$, whilst the lone generator on the RHS only gets rescaled by a factor c.

 

[kith]

The Duck talked about rescaling spin by rescaling hbar. This corresponds to a re-labeling of your measurement apparatus. Such a rescaling implies new scales for other observables as well but is of course always possible.