An Expression for a Measurement Equation

[ManyNames]

Homework Statement

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

Homework Equations

$$k=\frac{(t<t_0)-(t>t_1)}{\int_{t_0}^{t_1} dt J(t)}$$

$$J(t)=\int_{\Omega}\Pi |\psi|^2$$

The Attempt at a Solution

$$\psi*(\Pi_{(k)}(\psi))$$

Do you think this expression helps imply the two fields $$\psi$$ with the given information?

 

[ManyNames]

The first equation is meant to look like

$$k=\frac{(t<t_0)-(t>t_1)}{\int_{t_0}^{t_1} dt J(t)}$$

 

[ManyNames]

Can no one answer my question? I'd be very grateful.

 

[ManyNames]

Homework Statement

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

Homework Equations

$$k=\frac{(t<t_0)-(t>t_1)}{\int_{t_0}^{t_1} dt J(t)}$$

$$J(t)=\int_{\Omega}\Pi |\psi|^2$$

The Attempt at a Solution

$$\psi*(\Pi_{(k)}(\psi))$$

Do you think this expression helps imply the two fields $$\psi$$ with the given information?

Also, it seems that $$J(t)$$ 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;

$$\psi* (\Pi_{(k)}(\psi))=|\psi|^2 (\Pi_{(k)})$$

Does this seem reasonable?

 

[ManyNames]

Am i allowed to deduct the following?

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

Where $$\xi^n(t)$$ 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 $$J(t)$$ as it is and $$\psi*(\Pi_{k}(\psi))$$ as:

$$\psi_i*(\Pi_{k}(\psi_i))$$

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 $$\psi_i*(\Pi_{k}(\psi_i))$$ as:

$$J(t)_i=\begin{pmatrix} i=2k \\ i=2k+1 \end{pmatrix}$$

$$\psi_i*=\begin{pmatrix} i*=2k \\ i*=2k+1 \end{pmatrix}$$

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

$$\sum^{\Pi}_{n} \xi^{n}(t) |\psi|^2 \approx J(t)$$

 

[ManyNames]

That was riddled with errors. Hopefully this is more clear

Am i allowed to deduct the following?

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

Where $$\xi^n(t)$$ 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 $$J(t)$$ as it is and $$\psi*(\Pi_{k}(\psi))$$ as:

$$\psi_i*(\Pi_{k}(\psi_i))$$

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 $$\psi_i*(\Pi_{k}(\psi_i))$$ as:

$$\Pi J(t)_i=\begin{pmatrix} i=2k \\ i=2k+1 \end{pmatrix}$$

$$\Lambda \psi_i*=\begin{pmatrix} i*=2k \\ i*=2k+1 \end{pmatrix}$$

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

$$\sum^{\Pi}_{n} \xi^{n}(t) |\psi|^2 \sim J(t)$$