 Problem Statement
 Assume that the measurement of the energy on a state A> always yields the value 'a' and the measurement of the energy on a state B> always yields the value 'b'. Now consider a quantum system in the superposition state (1+2i)A> + (1i)B>. What are the probabilities Pa and Pb to measure energy and 'a' and 'b' respectively?
 Relevant Equations

For a state Ψ = αA> + βB>;
Pa ∝ α^2
and Pb ∝ β^2
Using the fact that
Pa ∝ α^2 and Pb ∝ β^2, we get:
Pa = kα^2 and Pb = kβ^2
Since the probability of measuring the two states must add up to 1, we have Pa + Pb = 1 => k = 1/(α^2 + β^2). Substituting this in Pa and Pb, we get:
Pa = α^2/(α^2 + β^2)
and Pb = β^2/(α^2 + β^2)
And using these equations, I could get the correct answer. However, I assumed that the constant of proportionality is the same for calculating Pa and Pb and in fact, it is. But I am not sure why that is the case...why do they have to be the same?
Pa ∝ α^2 and Pb ∝ β^2, we get:
Pa = kα^2 and Pb = kβ^2
Since the probability of measuring the two states must add up to 1, we have Pa + Pb = 1 => k = 1/(α^2 + β^2). Substituting this in Pa and Pb, we get:
Pa = α^2/(α^2 + β^2)
and Pb = β^2/(α^2 + β^2)
And using these equations, I could get the correct answer. However, I assumed that the constant of proportionality is the same for calculating Pa and Pb and in fact, it is. But I am not sure why that is the case...why do they have to be the same?