Find Expectation Value for Particle Moving in N Steps of Length L

In summary, the problem is asking for the expectation value of z^2 for a particle moving in a sequence of steps with length L. The polar angle for each step is taken from a normalized probability density p(\theta), while the azimuthal angle is uniformly distributed. The formula for finding the expectation value is given, and the solution for one step is <z^2> = L^2/2. However, it is unclear how to treat N steps. The given probability density is p(\theta) = (2/pi)*cos^2(theta/2).
  • #1
Pacopag
197
4

Homework Statement


A particle moves in a sequence of steps of length L. The polar angle [tex]\theta[/tex] for each step is taken from the (normalized) probability density [tex]p(\theta)[/tex]. The azimuthal angle is uniformly distributed. Suppose the particle makes N steps.
My question is how do I find the expectation value (say [tex]<z^2>[/tex] for example).

Homework Equations


Usually for a probability density p(x) we have
[tex]<x^m>=\int x^m p(x) dx[/tex].

The Attempt at a Solution


I think that I can get the values for one step. eg.
[tex]<z^2>=\int_0^\pi (Lcos(\theta))^2p(\theta)d\theta={L^{2}\over 2}[/tex]
Note: the density [tex]p(\theta)[/tex] is normalized.
I just don't know how to treat N steps. Do I just multiply the one-step result by N?
 
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  • #2
What is [tex]p(\theta)[/tex]? Is it given?
 
  • #3
Oh ya. Sorry. It is
[tex]p(\theta) ={2 \over \pi}cos^2({\theta \over 2})[/tex]
 

FAQ: Find Expectation Value for Particle Moving in N Steps of Length L

1. What is an expectation value for a particle?

An expectation value for a particle is a measure of the average value of a specific physical quantity, such as position or momentum, that the particle is expected to have over a large number of measurements or observations.

2. How is the expectation value calculated for a particle moving in N steps of length L?

The expectation value for a particle moving in N steps of length L can be calculated by taking the sum of the product of the probability of the particle being at a specific position and the value of that position, over all possible positions. This can be represented mathematically as σ x*P(x), where x is the position and P(x) is the probability of the particle being at that position.

3. What is the significance of finding the expectation value for a particle's movement?

Finding the expectation value for a particle's movement allows us to make predictions about where the particle is likely to be located, based on its average position. It also gives us information about the particle's overall behavior and can be used to analyze and understand physical systems.

4. How does the expectation value change as the number of steps and length of each step vary?

The expectation value for a particle's movement will change depending on the number of steps and the length of each step. Generally, as the number of steps increases, the expectation value will approach the average of all possible positions. Similarly, increasing the length of each step will also affect the expectation value, with longer steps leading to a wider distribution of possible positions and potentially a larger expectation value.

5. Can the expectation value be used to determine the exact position of a particle at any given time?

No, the expectation value cannot be used to determine the exact position of a particle at any given time. It only provides information about the average position of the particle over a large number of measurements. To determine the exact position of a particle, other methods such as measurement or calculation based on specific conditions must be used.

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