Integration/ boundary conditions

[Abigale]

Hi guys,

I regard a particle in an Potential.

I have callculated the partition function and the probability density function $F_{1}$.

$$
H= \frac{p^{2}_{x}}{2m}
+ \frac{p^{2}_{z}}{2m}+ \frac{p^{2}_{\phi}}{2I}+ mgz
$$

For callculating an average value I do:
$$
<mgz>=\int \limits_{\color{Brown}?}^{\color{Brown}?}dx\int \limits_{\color{Brown}?}^{\color{blue}+ \color{blue}\infty}dz\int \limits_{\color{Brown}?}^{\color{Brown}?}d\phi~~~\int \limits_{-\infty}^{+\infty}dp_{x}\int \limits_{-\infty}^{+\infty}dp_{z}\int \limits_{-\infty}^{+\infty}dp_{\phi}
~
~~~~F_{1} ~mgz
$$

The boundary conditions are:
$$
0 \le x \le L \\
0 \le z \le {\color{blue} + \color{blue}\infty}\\
0\le \phi \le 2\pi \\
$$

Do I have to integrate to $+/- \infty$ or to the boundary conditions?

Thanks a lot
Abby

 

[eljose79]

Hi Abby,

It looks like you are on the right track with your calculations and understanding of the partition function and probability density function. In order to calculate the average value of $mgz$, you will need to integrate over all possible values of the variables $x$, $z$, and $\phi$ as well as the momentum variables $p_x$, $p_z$, and $p_{\phi}$. This means you will need to integrate from $-\infty$ to $+\infty$ for the momentum variables, and from the boundary conditions given for the position variables. For example, for $z$, you will integrate from $0$ to $+\infty$ according to the given boundary condition. This will give you the average value of $mgz$ for the system described by the potential function you provided.

I hope this helps clarify things for you. Keep up the good work!