Quantum Amplitude: Find N, Probability & State |psi'|

[joker_900]

Homework Statement

The eigenstates of two commuting operators A and B are denoted |a,b> and satisfy the eigenvalue equations A|a,b>=a|a,b> and B|a,b>=b|a,b>. A system is set up in the state

|psi> = N(|1,2> + |2,2> + |1,3>)

What is the value of the normalization constant N?

A measurement of the value of A yields the result 1. What is the probability of this happening? What is the new state |psi'> of the system?

Homework Equations

None?

The Attempt at a Solution

So I did <psi|psi>=1 and got N=sqrt(1/3)

Then I thought that a measurement of A is a measurement of it's eigenvalue, so I need the probability of the system being in a state |1,b>. I think the constants in front of an eigenstate here is the amplitude of the system being in a state |a,b> (i.e. the amplitude that a measurement of A will yield a result a). So the total amplitude of measuring a=1 is 2*sqrt(1/3). However this gives a probability of 4/3!

 

[CompuChip]

I think the constants in front of an eigenstate here is the amplitude of the system being in a state |a,b> (i.e. the amplitude that a measurement of A will yield a result a). So the total amplitude of measuring a=1 is 2*sqrt(1/3). However this gives a probability of 4/3!

Up to there everything is all right. However, you are a bit off in the first sentence I quoted (apart from that, I don't think 2*sqrt(1/3) = 4/3). Remember that, in general, coefficients may be complex in quantum mechanics. So how do we relate them to probabilities (which must be real)? Also note that the sum of probabilities must be 1 (in this case, probability of finding 1 + that of finding 2 for the measurement of A).