# Eigenvalues and Probabilities

1. Nov 13, 2009

### bluebandit26

1. The problem statement, all variables and given/known data

Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A1>, |A2>, and |A3> such that A|Ak> = ak|Ak>. The system is at state:

|psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3>.

Provide the possible measured values of a and corresponding probabilities.

2. Relevant equations

(A)(psi) = sum[anCnPsin]

3. The attempt at a solution
It would seem that A1 = (3)^(-1/2), A2 = 2((3)^(-1/2), and A3 = (5/3)^(1/2), but the state is not normalized so the probabilities don't add up to one... so I am confused about how to handle this.

2. Nov 13, 2009

### MathematicalPhysicist

So normalise |psi>.
BTW, the probabilities of recieving the eigenvalue a_k is the square of the coefficient of |A_k> in |psi>.

3. Nov 13, 2009

### MathematicalPhysicist

In general, if $$|\psi>=\sum c|n>$$ the the probability to find |psi> in the state |n> (and measuring its eigenvalue) is |c|^2=cc* where c* is the complex conjugate of c.