I am studying a Quantum Mechanics course and I have come across something that I am a little stuck on, mathematically. Physically it seems to make sense but I'm not sure which equations to use to justify the behaviour of the system.
Here is the question:
Here is an excerpt of our notes:
The equation given (80) to find the probability of a quantum state seems to be for a one-dimensional system with time t and position x on the real number line. My problem has two dimensions: open and closed.
I'm unsure what approach to take, but I gave it a shot: I can only post two links, the imgur code is PFnqyTY.jpg
I expect that the probability of the door to be open is 1/5 but I'm not sure how to get to the answer without just jumping to it. I'd like to do it completely and respect the vectors and the notation involved. I'm slowly getting the hang of quantum systems and understanding all of the notation and their physical meanings, but I'm stuck on this one.
Any help with this would be greatly appreciated. Thank you!
Your equation (80) does not mean that $x$ is a real number, $x$ is the state. For particular systems, it may correspond to a position, but that is clearly meaningless in this question. Also, your state does not depend on time, so you can just omit that. The probability of the door being open is thus
$$P_\text{open}(t) = |\langle \text{open} | \psi \rangle |^2$$ which is thus constant in time. We expect this since nothing depends on time here.
You were told that $|\text{open} \rangle$ and $|\text{closed} \rangle$ are orthonormal, so $$ \langle \text{open} | \text{closed} \rangle = 0, \qquad \langle\text{open}|\text{open} \rangle = \langle\text{closed}|\text{closed} \rangle = 1.$$