An Expression for a Measurement Equation

In summary, the equations k and J(t) are related to a measurement and a local coupling between an observer and an observed system. The expression \psi*(\Pi_{k}(\psi)) represents the manifestation of the two fields into a single expression. However, there may be errors in the given equations and it is suggested to treat J(t) and \psi*(\Pi_{k}(\psi)) separately.
  • #1
ManyNames
136
0

Homework Statement



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.

Homework 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]

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?
 
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  • #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]
 
  • #3
Can no one answer my question? I'd be very grateful.
 
  • #4
ManyNames said:

Homework Statement



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.

Homework 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]

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?

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?
 
  • #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]
 
  • #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]
 

FAQ: An Expression for a Measurement Equation

1. What is an expression for a measurement equation?

An expression for a measurement equation is a mathematical representation of a relationship between physical quantities. It is used to calculate the value of a measured quantity by using known values of other quantities.

2. Why is an expression for a measurement equation important?

An expression for a measurement equation is important because it allows scientists to accurately measure and quantify physical phenomena. By using mathematical equations, scientists can obtain precise and reliable measurements, which are crucial for understanding the natural world.

3. How do you create an expression for a measurement equation?

To create an expression for a measurement equation, you first need to identify the physical quantities involved in the measurement. Then, you can use mathematical operations such as addition, subtraction, multiplication, and division to express the relationship between these quantities. It is essential to use the correct units and follow the rules of mathematical operations to ensure the accuracy of the expression.

4. Can an expression for a measurement equation be simplified?

Yes, an expression for a measurement equation can be simplified by using algebraic techniques such as factoring, combining like terms, and using mathematical identities. Simplifying an expression can make it easier to understand and use in calculations.

5. What are some examples of expressions for measurement equations?

Examples of expressions for measurement equations include Ohm's Law (V=IR), the Ideal Gas Law (PV=nRT), and the Pythagorean Theorem (a²+b²=c²). These equations are used to calculate values for electrical resistance, gas pressure, and the length of the sides of a right triangle, respectively.

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