How Do You Calculate the Expectation Value <x²> for a Particle in a Box?

[CollectiveRocker]

If the expectation value <x> of a particle trapped in a box L wide is L/2, which means its average position in the middle of the box. Find the expectation value <x squared>. How do I go about doing this? I am really confused.

 

[ehild]

If the expectation value <x> of a particle trapped in a box L wide is L/2, which means its average position in the middle of the box. Find the expectation value <x squared>. How do I go about doing this? I am really confused.

The expectation value of x^2 can be calculated if you know the probability density function of x. If it is f(x)

$$<x^2>=\int_0^L{x^2f(x)dx}$$

For a classical particle f(x) = 1/L as it has equal probability of being anywhere in the box.

If it is a quantum particle, find out how the wavefunction of that particle, trapped in a box extending from x=0 to x=L, looks like. The probability density function is the square of the absolute value of the wavefunction.

$$f(x)= \phi \phi^* \rightarrow <x^2>=\int_0^L{x^2\phi \phi^*dx}$$




ehild

 

[blashimov]

I have the same problem as one part of a larger question, and was hoping to check my answer. Did you get a²(1/3 + 3/(2*pi²*n²) for the nth state?