Gaussian State SpreadA measurement device which can be represented by a 1D quantum system (with canonical observables X and P) 'is prepared in a Gaussian state with spread s' | ψ ⟩ = 1 ( π 2 s 2 ) 1 / 4 ∫ exp ⁡ [ − x 2 2 s 2 ] d x | x ⟩Can somebody tell me what it means that its canonical variables are X and P?

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Gaussian State SpreadA measurement device which can be represented by a 1D quantum system (with canonical observables   X and   P) &#039;is prepared in a Gaussian state with spread   s&#039;  |  ψ  ⟩  =      1          (              π        2                    s        2                    )                  1                      /                    4                      ∫  exp  ⁡      [    −                  x        2                    2                  s          2                      ]              d    x  |  x  ⟩Can somebody tell me what it means that its canonical variables are   X and   P?

pheniankang · Accepted Answer

To have canonical observables             x      ^       and             p      ^       means that the eigenvalues of these operators are what you measure (denotes   x  ,  p), and the operators satisfy the &quot;canonical commutation relation&quot;  [            x      ^        ,            p      ^        ]  ≡            x      ^                  p      ^        −            p      ^                  x      ^        =  i  ℏTo prepare a system in an initial state       |    ψ  ⟩ means smily that this is the state of the system at   t  =  0; it is usually denoted as       |    ψ  (  0  )  ⟩Now, your initial state is given by  ∣  ψ  (  0  )  ⟩  =  (      π    2        s    2        )          −      1              /            4        ∫  d  x         e          −                        x          2                          2                      s            2                                ∣  x  ⟩  .Aside: A probability distribution given by   f  (  x  )  =  (      π    2        s    2        )          −      1              /            4            e          −                        x          2                          2                      σ            2                               is said to be a Gaussian with spread   σ, where   σ is the usual standard deviation from the center of the bell curve.End AsideTherefore, the meaning of  ∣  ψ  (  0  )  ⟩  =  (      π    2        s    2        )          −      1              /            4        ∫  d  x         e          −                        x          2                          2                      s            2                                ∣  x  ⟩  .is that your system at   t  =  0 is in a superposition of &quot;position eigenstates&quot;       |    x  ⟩ (i.e. eigenvectors of             x      ^      ), weighted by a gaussian distribution. That is, your system isn&#039;t just in any old superposition of position eigenstates, but that the ones near   x  =  0 are most likely and as you move away from the origin the probability of the system being in that state decreases like       e          −                        x          2                          2                      s            2                              If you are familiar with wave functions, recall that the definition of   ⟨  x      |    Ψ  ⟩  ≡  Ψ  (  x  ). Then you can get something a bit more useful. Namely, that  ⟨  x      |    ψ  (  0  )  ⟩  =  (      π    2        s    2        )          −      1              /            4        ∫  d      x    ′           e          −                        x                      ′            2                                    2                      s            2                                ⟨  x      |        x    ′    ⟩  =  (      π    2        s    2        )          −      1              /            4        ∫  d      x    ′           e          −                        x                      ′            2                                    2                      s            2                                δ  (  x  −      x    ′    )  =  (      π    2        s    2        )          −      1              /            4            e          −                        x          2                          2                      s            2                              where I changed the variable of integration to       x    ′   to make things clearer.At any rate, what this means is that your initial wave function is given by  Ψ  (  x  ,  t  =  0  )  =  (      π    2        s    2        )          −      1              /            4            e          −                        x          2                          2                      s            2                                .Where to go from here with whatever you&#039;re doing should be familiar at this point (i.e. after finding the initial wave function).

Gaussian State Spread A measurement device which can be represented by a 1D quantum system (with ca

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