Suppose that the Hilbert space of a quantum-mechanical system - which we will call the quantum door - is generated by two states, |open&gt; and |closed&gt;, forming an orthonormal basis. Suppose also that the system is prepared in the state | ψ ( x ) &gt;= 1 5 ( | O P E N &gt; + 2 | C L O S E D &gt; ) 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 O p e n = ( &lt; o p e n | 1 5 ( | O P E N &gt; + 2 | C L O S E D &gt; ) 2 = 1/5 ?I also need help with part (ii), i am unure about this.

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Suppose that the Hilbert space of a quantum-mechanical system - which we will call the quantum door - is generated by two states, |open&amp;gt; and |closed&amp;gt;, forming an orthonormal basis. Suppose also that the system is prepared in the state      |    ψ  (  x  )  &amp;gt;=      1          5        (      |    O  P  E  N  &amp;gt;  +  2      |    C  L  O  S  E  D  &amp;gt;  ) 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          O      p      e      n        =  (  &amp;lt;  o  p  e  n      |        1          5        (      |    O  P  E  N  &amp;gt;  +  2      |    C  L  O  S  E  D  &amp;gt;      )          2       = 1/5 ?I also need help with part (ii), i am unure about this.

tennispopj8 · Accepted Answer

You got the part (i) correct. The probability of the door being measured as &#039;Open&#039; is 1/5.On the part (ii), the door will stay as &#039;Closed&#039; after being measured as in &#039;Closed&#039; state. It will change only if the system is interacted by other operations.

Suppose that the Hilbert space of a quantum-mechanical system - which we will call the quantum door

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