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.
Eigenstates of spin 1/2 are quantum states that describe the spin of a particle with a spin quantum number of 1/2. They represent the possible orientations of the particle's spin in a given direction.
The spin operator is a mathematical tool used to describe the spin of a particle. Eigenstates of spin 1/2 are the eigenvectors of the spin operator, meaning they are the states that remain unchanged when acted upon by the spin operator.
Eigenstates of spin 1/2 play a crucial role in quantum mechanics as they represent the fundamental states of a particle's spin. They are used to describe the spin of particles in various physical systems and have important implications in areas such as nuclear physics and quantum computing.
Eigenstates of spin 1/2 are unique in that they have a spin quantum number of 1/2, which is the minimum value for a particle's spin. They also have specific properties such as being orthogonal to each other and forming a complete basis for spin measurements.
No, eigenstates of spin 1/2 cannot be directly observed in experiments. However, their effects can be observed through measurements of spin properties such as spin projections and spin precession. These measurements confirm the existence of eigenstates and their importance in describing the spin of particles.