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Homework Help: An Expression for a Measurement Equation

  1. Jan 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Knowing the given equations, [tex]k[/tex] is equal to a measurement where [tex]J(t)[/tex] implies some local coupling between the observer and observed system. If a field is considered to collapse [tex]J(t)[/tex], how would the two manifest into a single expression.

    2. Relevant equations

    [tex]k=\frac{(t<t_0)-(t>t_1)}{\int_{t_0}^{t_1} dt J(t)}[/tex]

    [tex]J(t)=\int_{\Omega}\Pi |\psi|^2[/tex]

    3. The attempt at a solution

    [tex]\psi*(\Pi_{(k)}(\psi))[/tex]

    Do you think this expression helps imply the two fields [tex]\psi[/tex] with the given information?
     
  2. jcsd
  3. Jan 28, 2009 #2
    The first equation is meant to look like

    [tex]k=\frac{(t<t_0)-(t>t_1)}{\int_{t_0}^{t_1} dt J(t)}[/tex]
     
  4. Jan 29, 2009 #3
    Can no one answer my question? I'd be very grateful.
     
  5. Jan 30, 2009 #4
    Also, it seems that [tex]J(t)[/tex] should be treated as independant from the field describing the collapse in the expression. In the expression, there could be another two quantum wave fields;

    [tex]\psi* (\Pi_{(k)}(\psi))=|\psi|^2 (\Pi_{(k)})[/tex]

    Does this seem reasonable?
     
  6. Jan 31, 2009 #5
    Am i allowed to deduct the following?

    [tex]\frac{d\Lamba \psi \rightarrow d\Pi(|\psi|^2)}{\int_{t_0}^{t_1}\Pi_{k}^n}=\sum^{\Pi}_{n} \xi^n(t) |\psi|^2[/tex]

    Where [tex]\xi^n(t)[/tex] is the probability of global changes, where it must vanish totally upon the square of the density. Then back to the original assumptions, we can treat [tex] J(t)[/tex] as it is and [tex]\psi*(\Pi_{k}(\psi))[/tex] as:

    [tex]\psi_i*(\Pi_{k}(\psi_i))[/tex]

    So thus implying a series with a linear function. It would be fair to analyse the convergent monotonic series idea with both fields in J(t) and contained in [tex]\psi_i*(\Pi_{k}(\psi_i))[/tex] as:

    [tex]J(t)_i=\begin{pmatrix} i=2k \\ i=2k+1 \end{pmatrix}[/tex]

    [tex]\psi_i*=\begin{pmatrix} i*=2k \\ i*=2k+1 \end{pmatrix}[/tex]

    If they converge synomynously, then it is fair to say they are asympototically-equivalant in respect to time:

    [tex]\sum^{\Pi}_{n} \xi^{n}(t) |\psi|^2 \approx J(t)[/tex]
     
  7. Jan 31, 2009 #6
    That was riddled with errors. Hopefully this is more clear

    Am i allowed to deduct the following?

    [tex]\frac{d\Lamba \psi \rightarrow d\Pi(|\psi|^2)}{\int_{t_0}^{t_1}\Pi_{k}^n}=\sum^{\Pi}_{n} \xi^n(t) |\psi|^2[/tex]

    Where [tex]\xi^n(t)[/tex] is the probability of global changes, where it must vanish totally upon the square of the density. Then back to the original assumptions, we can treat [tex] J(t)[/tex] as it is and [tex]\psi*(\Pi_{k}(\psi))[/tex] as:

    [tex]\psi_i*(\Pi_{k}(\psi_i))[/tex]

    So thus implying a series with a linear function. It would be fair to analyse the convergent monotonic series idea with both fields in J(t) and contained in [tex]\psi_i*(\Pi_{k}(\psi_i))[/tex] as:

    [tex]\Pi J(t)_i=\begin{pmatrix} i=2k \\ i=2k+1 \end{pmatrix}[/tex]

    [tex]\Lambda \psi_i*=\begin{pmatrix} i*=2k \\ i*=2k+1 \end{pmatrix}[/tex]

    If they converge synomynously, then it is fair to say they are asympototically-equivalant in respect to time:

    [tex]\sum^{\Pi}_{n} \xi^{n}(t) |\psi|^2 \sim J(t)[/tex]
     
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