- #1

space-time

- 218

- 4

1. I know that if you apply a spin operator σ (which is a matrix) to an eigenvector <a> (I would type it in the form of a ket vector, but I don't see that option in the latex), then you will get an eigenvalue λ multiplied by the same ket vector <a>. In the case of spin, those eigenvalues can be either +1 or -1. Furthermore, a spin can be measured as either up or down. My question to you guys is:

Does +1 correlate to measuring spin up and -1 to spin down?

I am asking this because I know that particles can actually be said to be in a linear superposition of states. For example: Spin up (which is represented by <1, 0 > ) can be written as 1 <up> + 0 <d>.

Now this one is obviously spin up, however:

Spin when measured against a certain axis has an eigenvectors < 1/squrt(2) , 1/squrt(2) > which can be written as:

1/squrt(2) <up> + 1/squrt(2) <down>

In this case, the 1/squrt(2) terms are ,probability amplitudes. When you square the magnitude of these probability amplitudes, you get the probability that the particle will be measured to have that probability amplitude's respective state. This means that in the case of the equation above, the particle would have a 50% chance of being measured as spin up and a 50% chance of being measured as spin down.

If you apply the spin operator that goes with this particular state vector to said vector however, you get an eigenvalue of +1. Proof:

σ = a 2 by 2 matrix. In this case the matrix is all 0 on the diagonal and it has the number 1 as the other two entries. When you multiply this matrix by the vector < 1/squrt(2) , 1/squrt(2) > , your result is just that same vector (which means that the eigenvalue is +1).

Now if +1 implies up and - 1 implies down, then getting this eigenvalue of +1 would seem to guarantee that you would measure the particle to be spin up if you have the eigenvector < 1/squrt(2) , 1/squrt(2) > . This guarantee however, is a direct contradiction to the notion that you'd have a 50% chance of measuring up and a 50% chance of measuring down.

That is why I came here to verify whether or not +1 and -1 implied up and down respectively, or if these eigenvalues represent something else. If it is something else, what exactly do these eigenvalues tell you? If I was right in thinking that +1 was up and -1 was down, then what becomes of the notion of the linear superposition of states?

2. Leonard Susskind used 3 different 2 by 2 σ matrices. σ

_{1}had 0's on the diagonal and 1's as the other entries. σ

_{2}had 0's on the diagonal, - i as the top right entry and i as the bottom left entry. σ

_{3}had 1 as the top left entry, -1 as the bottom right entry and 0's as the other entries. Now this may seem like a rather obvious question, but I just want to verify this:

The 3 different matrices simply correspond to the 3 different Cartesian axes correct? In other words,

σ

_{1}corresponds to if you're measuring spin against the x- axis while σ

_{2}corresponds to the y - axis and σ

_{3}is the z - axis correct?