The discussion revolves around the relationship between the components (u, v) and the state alpha+ in the context of spin-1/2 eigenstates. Participants are examining the mathematical expressions and transformations involved in this quantum mechanics topic.
The conversation is actively exploring the implications of the relationships between u and v, with some participants noting the arbitrary choices made in their values. There is recognition of the normalization condition and its relevance to the probabilities involved, but no consensus has been reached on the underlying assumptions.
Participants are considering the implications of choosing u and v to be real numbers and the significance of the arbitrary sign in the context of probability calculations. The discussion reflects a focus on the mathematical properties rather than definitive conclusions.
MathematicalPhysicist said:Well, for alpha+, if you multiply (scalar multiplcation) the equation for 1/sqrt(2) (1,1), with the vector 1/sqrt(2) (exp(-i phi), exp(i phi)), the part that is multiplied by alpha minus is vanished, and you get the equality.
For u and v, you get u=v so obviously 2u^2=1, and the sign is arbitrary.
MathematicalPhysicist said:So?
That means v=u.
MathematicalPhysicist said:Well, they arbitrarily chose u and v to be real, and from u=v you plug back to get 2u^2=1.
As I said the sign is arbitrary, and because we only interested in probabilities (eventually), the sign is irrelevant.