Consider a particle that lives in an isolated one-dimensional world and is tethe

Question

Consider a particle that lives in an isolated one-dimensional world and is tethered

to the origin by a harmonic spring. The total energy of the particle at any point in

time is due to its kinetic plus potential energy,

E =K + U = p^2/2m + kx^2/2

where p is the momentum of the particle and x is its position, and k is the force

constant of the spring.We define the phase space of a systemas themultidimensional

space of all of its microscopic degrees of freedom. Here the particle has two degrees

of freedom, p and x, and the phase space is a two-dimensional space.

(a) For this isolated system, show that the particle is confined to an ellipse in

phase space. Draw a graph illustrating the shape of the ellipse with principal

radii indicated in terms of E, k, and m as appropriate.

(b) Consider all points in phase space for which the particle can have energy

between E and E + E, where E is a small number. Show that this is

equivalent to an area between two ellipses in phase space. Find an expression

for this area in terms of E, E, k, and m.

(c) What if the particle lives not in a one-dimensional world, but a d-dimensional

one, still tethered to the origin by a harmonic spring? For example, d =2 for a

particle confined to a plane, or d = 3 for one in three-dimensional space. By

analogy with part (b), give an expression for the 2d-dimensional volume in

phase space for energies between E and E + E. Expand your answer and omit

terms in E with powers higher than one. Note that the volume of a 2ddimensional

ellipse is given by the product of all radii with C(2d), a d-dependent

constant.

(d) You now want to count the number of possible microscopic configurations

(microstates) that lie inside this volume so that you can compute the entropy.

In general, a microstate is defined as a unique set of positions and momenta.

You will need to discretize or “pixelate” phase space in order to perform this

counting. Discretize positions into blocks of size x and momenta into blocks

of size p. Find an expression for the entropy.

(e) Your expression should contain the term (p x), which is essentially a discrete

area in phase space. To what physical constant might this be related?

(f) Using the entropy, predict the temperature dependence of the total energy,

E(T). Do the values of p, x, or E matter?

Explanation / Answer

a. Form the given data

E = p^2/2m + kx^2/2

from these equations, E is constant, and so are 2m and k/2

so this is an equaiton of the ellipse

1 = x^2/a^2 + y^2/b^2

where a^2 = 2mE

b^2 = 2E/k

b. E = p^2/2m + kx^2/2

dE = 2p*dp/2m + 2x*k*dx/2 = p*dp/m + kx*dx

now area of ellipse with Equation E = p^2/2m + kx^2/2 is A = pi*sqroot(2mE)*sqroot(2E/k) = 2*pi*E*sqroot(m/k)

area of ellipse with equation E + dE = p^2/2m + kx^2/2 is A' = pi*sqroot(2m(E + dE))*sqroot(2(E + dE)/k) = 2*pi*(E + dE)sqroot(m/k)

now since area of the ellipse in phase plane is the number of particles which can have energy E, so differnece in area of these two ellispses will give un number of particles that can have energy between E and E + dE

so, number of particles that can have energy between E and dE = 2*pi*(E + dE)sqroot(m/k) - 2*pi*E*sqroot(m/k) = 2*pi*dE*sqroot(m/k)

c. volume of 2D ellipse = pi*a*b*C [ C depends on d, the number of dimensins]

so from previous question

number of particles that can have energy between E and E + dE = 2*pi*dE*sqroot(m/k)*C [ where C will be the d dependent constant]

d. after discretizing space in dx and momjentum into dp

dp*dx*C will be the volume of this phase space

and from the postulalate of statistical mechanis

entropy S = kb*ln(dp*dx*C)

where Kb is boltzmann's constant

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