Probability that a measurement of Sx will yield hbar for this state

[sabre729]

Homework Statement

Alright, so there is a problem in the Townsend quantum mechanics book that reads: What is the probability that a measurement of Sx will yield the value hbar for this state:


The state is given by:
|ψ> = 1/sqrt(14) * (1,2,3i)
(But the vector is represented as a column vector in state ψ)

It's in the Sz basis, and it is a spin 1 particle.

Homework Equations

The Attempt at a Solution

In another problem I determined the eigenstates of Sx for a spin 1 particle in terms of the eigenstates for |1,1>, |1,0>, |1,-1>. So I figured it would be easy to take that vector for hbar and just do the probability amplitude on it with the given ψ. That vector I have is:

(1, sqrt(2), 1)

Again as a column vector. Or row vector, either way really. But when I did that I got a value bigger than 1. So I'm obviously missing something here, and I was hoping someone could tell me what.

Also I have a huge issue when bases switch, and yes, it's just a lin alg thing and I should review it, but as I do, does anyone have a quick trick for these kinds of problems? It's really appreciated.

 

[vela]

Hint: Check the normalization of the L_x eigenstate you used.