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 also need help with part (ii), i am unure about this.
You got the part (i) correct. The probability of the door being measured as 'Open' is $1/5$.
On the part (ii), the door will stay as 'Closed' after being measured as in 'Closed' state. It will change only if the system is interacted by other operations.