# Quantum mechanics , probability of quantum door measurement

Suppose that the Hilbert space of a quantum-mechanical system - which we will call the quantum door - is generated by two states, |open> and |closed>, forming an orthonormal basis. Suppose also that the system is prepared in the state

$|\psi(x)> = \frac{1}{\sqrt{5}}(|OPEN> + 2|CLOSED>)$ We are given a device that measures whether the quantum door is open or closed.

(i)If we perform a measurement, which probability do we have to find the quantum door open?

(ii) Suppose the measurement returns that the quantum door is closed, and assume that the quantum Hamiltonian is identically 0 for this system at any future times. Does the door stay closed forever?

For part (i) I get

$P_{Open} = ( <open|\frac{1}{\sqrt{5}}(|OPEN> + 2|CLOSED>)^{2}$ = 1/5 ?

• I'm no expert, but this seems underspecified: what if open and closed are both eigenfunctions of $H$ with the same eigenvalue? – Ian Jan 3 '17 at 21:46
You got the part (i) correct. The probability of the door being measured as 'Open' is $1/5$.