- #1

dreamspy

- 41

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We have that [tex]\epsilon \subset H[/tex] is a subspace with an orthonormal basis [tex]\{\phi_j\}[/tex] (here H is Hilbert space). We define an operator [tex]\Pi\rightarrow \epsilon[/tex] like this:

[tex]\Pi\psi=\sum_j<\phi_j,\psi>\phi_j[/tex]

Let's say that [tex]A[/tex] is a measurable property of a particle, and [tex]\hat A[/tex] is the corresponding operator. Let's make [tex]a[/tex] to be one of the eigenvalues of [tex]\hat A[/tex] and [tex]\epsilon_a[/tex] is the corrisponding eigenspace. That is: [tex]\epsilon_a [/tex] is the subspace in H that is spanned by all eigenfunctions of [tex]\hat A[/tex] with eigenvalue [tex]a[/tex]. If the particle has the wavefunction [tex]\psi_t[/tex] then the probability to measurement of [tex]A[/tex] will give the result [tex]a[/tex] is:

[tex]P(A=a)=\parallel\Pi_a\psi_t\parallel^2[/tex]

Now there is given an example to explain this. We are given wave function

[tex]\Psi=(A+Bx)e^{-x^2/a^2}[/tex]

And the parity operator [tex]\hat p[/tex] which functions like this:

[tex]\hat p \Psi(x) = \Psi(-x)[/tex]

Now it's easy to show that the eigenvalues of the parity operator are [tex]\pm 1[/tex] and the corresponding eigenfunctions are:

all even functions for eigenvalue +1

all odd functions for eigenvalue -1

Now the probability of getting [tex]p = +1[/tex] is given by:

[tex]P(p=+1) = \parallel \hat\Pi_{+1}\Psi\parallel^2 = \parallel Ae^{-x^2/a^2}\parallel^2[/tex]

Now what I don't get is how [tex]\hat \Pi_{+1}\Psi[/tex] was calculated. Anyone care to shed a light on this for me?