The energy of a 2-dimensional oscillator is E =p^2x/2m+p^2y/2m+ax^2/2+by^2/2+ε, where px and py are its momentumcomponents, x and y are the Cartesian coordinates, m is the mass, aand b are constants, and ε is its internal energy which does notdepend on x, y, px and py. From Gibbsâs distribution finddistributions dwx and dwy of its coordinates x and y at temperatureT. Find average potential energy U. Sketch the distributionfunction f(x) for large and small a/kT.
[Additional problem of choice (you solve it only if you want to !):derive the probability distribution dwr of the distance r from theorigin = the probability that the oscillator is within the distancerange from r to r+dr (clue: change from (x, y) to the polarcoordinates (r, Ï))].
The energy of a 2-dimensional oscillator is E =p^2x/2m+p^2y/2m+ax^2/2+by^2/2+ε, where px and py are its momentumcomponents, x and y are the Cartesian coordinates, m is the mass, aand b are constants, and ε is its internal energy which does notdepend on x, y, px and py. From Gibbsâs distribution finddistributions dwx and dwy of its coordinates x and y at temperatureT. Find average potential energy U. Sketch the distributionfunction f(x) for large and small a/kT.
[Additional problem of choice (you solve it only if you want to !):derive the probability distribution dwr of the distance r from theorigin = the probability that the oscillator is within the distancerange from r to r+dr (clue: change from (x, y) to the polarcoordinates (r, Ï))].
