How Do You Calculate the Probabilities of Measured Values for a Quantum State?

[bluebandit26]

Homework Statement

Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A₁>, |A₂>, and |A₃> such that A|A_k> = a_k|A_k>. 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.

Homework Equations

(A)(psi) = sum[a_nC_nPsi_n]

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.

 

[MathematicalPhysicist]

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

 

[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.