- #1
PerUlven
- 7
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The given problem:
The permitted energy values for a massless (or ultrarelativistic) particle (kinetic energy much larger than rest energy) in a 3-dimensional cubic box of volume V = L^3, can be expressed in terms of quantum numbers n_{x}, n_{y} and n_{z}:
\epsilon = \frac{hc\sqrt{n_x^2 + n_y^2 + n_z^2}}{2L},
where n_{x}, n_{y} and n_{z} must be positive integers.
a) What are the lowest two energy levels for this system and their degeneracy?
b) Write down an expression for the canonical partition function Z_1[\itex] for 1 particle at low temperature<br /> <br /> c) Determine the energy U and heat capacity C_V[\itex] in the limit of low T.<br /> <br /> <b>Relevant equations</b><br /> b) This is the equation I&#039;ve been trying to use for Z:<br /> Z = \sum_i e^{-\frac{\epsilon_i}{kT}}<br /> <br /> c) The &quot;shortcut formula&quot; U = -\frac{\partial}{\partial \beta}\ln Z, where \beta = 1/kT, <br /> and C_V = \left ( \frac{\partial U}{\partial T} \right )_{N,V}<br /> <br /> <b>Attempt at a solution</b><br /> a) \epsilon_1 = \frac{hc\sqrt{3}}{2L}, not degenerate (d = 1)<br /> \epsilon_2 = \frac{hc\sqrt{6}}{2L}, thrice degenerate d = 3<br /> <br /> b) Z_1 = e^{-\frac{\epsilon_1}{kT}} = e^{-\frac{\sqrt{3}hc}{2LkT}}<br /> <br /> Could anyone tell me if is correct? b) doesn&#039;t seem correct to me, since they&#039;re asking us to sketch U(T) and Cv(T) at low T and comment on the temperature dependence later in the task, seing as neither of them depend on T...
The permitted energy values for a massless (or ultrarelativistic) particle (kinetic energy much larger than rest energy) in a 3-dimensional cubic box of volume V = L^3, can be expressed in terms of quantum numbers n_{x}, n_{y} and n_{z}:
\epsilon = \frac{hc\sqrt{n_x^2 + n_y^2 + n_z^2}}{2L},
where n_{x}, n_{y} and n_{z} must be positive integers.
a) What are the lowest two energy levels for this system and their degeneracy?
b) Write down an expression for the canonical partition function Z_1[\itex] for 1 particle at low temperature<br /> <br /> c) Determine the energy U and heat capacity C_V[\itex] in the limit of low T.<br /> <br /> <b>Relevant equations</b><br /> b) This is the equation I&#039;ve been trying to use for Z:<br /> Z = \sum_i e^{-\frac{\epsilon_i}{kT}}<br /> <br /> c) The &quot;shortcut formula&quot; U = -\frac{\partial}{\partial \beta}\ln Z, where \beta = 1/kT, <br /> and C_V = \left ( \frac{\partial U}{\partial T} \right )_{N,V}<br /> <br /> <b>Attempt at a solution</b><br /> a) \epsilon_1 = \frac{hc\sqrt{3}}{2L}, not degenerate (d = 1)<br /> \epsilon_2 = \frac{hc\sqrt{6}}{2L}, thrice degenerate d = 3<br /> <br /> b) Z_1 = e^{-\frac{\epsilon_1}{kT}} = e^{-\frac{\sqrt{3}hc}{2LkT}}<br /> <br /> Could anyone tell me if is correct? b) doesn&#039;t seem correct to me, since they&#039;re asking us to sketch U(T) and Cv(T) at low T and comment on the temperature dependence later in the task, seing as neither of them depend on T...