Consider an electron, described initially by psi = 1/sqrt(10) with column vector (1 3). A measurement of the spin component is made along a certain axis, described by operator A, it has the eigenvalues +h/2 and - h/2 as possible outcomes, and the corresponding eigen states are;
Psi(1) = 1/sqrt(10) column vector (1 3) and Psi(2) = 1/sqrt(10) column vector (3 -1)
Explain why A returns the result h/2 with certainty.
for the second part you must measure spin is the Z component
Eigen equation A psi(1) = a(1) psi (1)
The Attempt at a Solution
Originally i thought the reason why it produced with certainty h/2 was because when you operate on a wavefunction, the eigenstate that is produced is the same as the initial wavefunction, thus the second eigenstate is different from that of the initial wave function and cannot be correct.
Whats confusing me is if this is true, why does psi 1 still have an amplitude associated with spin down i.e the column vector (1 3) (the 3?)
Could someone perhaps explain to me what it means to have two eigenstates corresponding to an operator?
For part b, i get the same constant as my initial wavefunction however my column vector becomes (1 -3) as the eigenstates for spin are always either +h/2 or -h/2 - this again confuses me, as i thought the eigenstate had to be the same as the initial wavefunction (from the eigen equation)