Quantum Operators (or just operators in general)

vela

The normalization condition always turns out to be real because the squared terms always involve the multiplication of a complex number by its conjugate and the cross terms are complex conjugates of each other so when they combine, the imaginary part cancels out.

In this problem, because c is assumed to be real, a and b can be real simultaneously. You could always multiply the state by an arbitrary phase and get complex coefficients if you want, but the relative phase of a and b will remain unchanged.

Plutoniummatt

Thanks guys! have been very helpful!

kuruman

They don't have to be. You can use coefficients

[tex]ae^{i\phi_a}; \:be^{i\phi_b}[/tex]

where a and b are real. Then the normalization condition will be in terms of a, b and the (arbitrary) phase difference φ_a-φ_b. Still two equations and two unknowns.

kuruman

Finally, vela and I converged.

Plutoniummatt

ok guys, so:

a + bc = 0


aa* + ab*c + ba*c + bb* = 1
2aa* + bb* - ccbb* = 1

are my equations, where do I go from here? If a and b are in general complex, how do I solve for them? does the phase not matter like whatsoever? so I can just write aa* as a^2

vela

Use the first equation to eliminate a from the second equation. You should write bb* = |b|². The best you can do is get the relative phase between a and b, so you can assume one is real.