Oakey1w

2022-06-27

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
$|\psi \left(x\right)>=\frac{1}{\sqrt{5}}\left(|OPEN>+2|CLOSED>\right)$ 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
${P}_{Open}=\left(+2|CLOSED>{\right)}^{2}$ = 1/5 ?
I also need help with part (ii), i am unure about this.

tennispopj8

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.

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Recalculate according to your conditions!