Question

Determine the mean position of the particle at temperature T when a particle is restricted to the region x>=0 by a potential.

Answer 1

Answer:

The mean position of the particle at temperature T when a particle is restricted to the region x>=0 by a potential is kbT/uo

Explanation:

Let u(x) =uox is the potential which is increased linearnly.

The particle function Z is defined as Z= 1/h∫(e^(-p^2/2mkbT)dp ∫e^(-uox/kbT) dx

(x) = xp(x)dxdp

(x) =∫∫(xe^(-p^2/2mkbT)dp ∫e^(-uox/kbT)) dx/∫∫e^(-p^2/2mkbT)dp ∫e^(-uox/kbT)) dx

=∫(xe^(-uox/kbT) dx)/∫e^(-uox/kbT) dx)

=(kbT/uo) ^2/(kbT/uo) ∫(te^-t)dt/∫(e^-t)dt

=kbT/uo