Pauli spin matrices, operating on |+> with Sx

[L-x]

Homework Statement

What is the result of operating on the state |+> with the operator Sx?

here, |+> denotes the eigenstate of Sz with eigenvalue 1/2. I am working in units where h-bar is 1 (for simplicity, and because I don't know how to type it)

Homework Equations

$$S_i = \frac{1}{2} σ_i$$

The Attempt at a Solution

My understanding of the physics of the problem is that after measurement the system will be in state where the x component of its spin is certain, as [Sx,Sz] != 0 this means it will be in some superposition of the states |+> and |-> My intuition tells me that the probability for the particle to be found in either state should be equal.

However, operating with the matrix representation of Sx :$$\frac{1}{2} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$$
on |+> just gives (1/2)|->

What am i doing wrong?

 

[L-x]

I think I understand where I've gone wrong now. The problem is in my intuition, not in my appilcation of the pauli matrices.

as $$|+> = \frac{1}{\sqrt{2}}(|+_x> + |-_x>)$$
(where $$|+_x> and |-_x>$$ are the eigenvectors of S_x with eigenvalues +/- 1/2)

when we operate with Sx we obtain $$(2^{-\frac{1}{2}})(\frac{1}{2}|+_x> - \frac{1}{2}|-_x>)$$ which is equal to $$\frac{1}{2}|->$$. The confusion I had was due to a faliure to distinguish between operation with a linear operator and measurement of an ovservable.

 

[G01]

Yes, your reasoning is correct, and you have the correct result.