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Q. 2. Consider an ideal gas in the semiclassical limit, confined to a box of volume $L^3$ at temperature T. The single particle partition function for such a system can be written as $\qquad Z_1 = \sum_{n_x=1}^{\infty} e^{-\beta \epsilon_n} = \sum_{n_x=1}^{\infty} \sum_{n_y=1}^{\infty} \sum_{n_z=1}^{\infty} e^{-\frac{\beta \hbar^2 \pi^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)} = \left[ \sum_{n_x=1}^{\infty} e^{-\frac{\beta \hbar^2 \pi^2}{8mL^2} n_x^2} \right]^3 = Z_1^3$ As the number of quantum states is very large for a macroscopic system so we can convert the above sum to an integral as $\qquad Z_1 = \int_0^{\infty} e^{-\frac{\beta \hbar^2 \pi^2}{8mL^2} n_x^2} dn_x - 1$ Using the above considerations, show that the semiclassical partition function for a particle in a one dimensional box of length L can be written as $\qquad Z_1 = \frac{1}{\hbar} \int_0^L dx \int_0^{\infty} dp e^{-\beta \frac{p^2}{2m}}$ (10)

          Q. 2. Consider an ideal gas in the semiclassical limit, confined to a box of volume $L^3$ at
temperature T. The single particle partition function for such a system can be written as
$\qquad Z_1 = \sum_{n_x=1}^{\infty} e^{-\beta \epsilon_n} = \sum_{n_x=1}^{\infty} \sum_{n_y=1}^{\infty} \sum_{n_z=1}^{\infty} e^{-\frac{\beta \hbar^2 \pi^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2)} = \left[ \sum_{n_x=1}^{\infty} e^{-\frac{\beta \hbar^2 \pi^2}{8mL^2} n_x^2} \right]^3 = Z_1^3$
As the number of quantum states is very large for a macroscopic system so we can convert the
above sum to an integral as
$\qquad Z_1 = \int_0^{\infty} e^{-\frac{\beta \hbar^2 \pi^2}{8mL^2} n_x^2} dn_x - 1$
Using the above considerations, show that the semiclassical partition function for a particle in a
one dimensional box of length L can be written as
$\qquad Z_1 = \frac{1}{\hbar} \int_0^L dx \int_0^{\infty} dp e^{-\beta \frac{p^2}{2m}}$
(10)

Q. 2. Consider an ideal gas in the semiclassical limit, confined to a box of volume L^3 at
temperature T. The single particle partition function for such a system can be written as
Z1 = ∑nx=1^∞ e^-β = ∑nx=1^∞∑ny=1^∞∑nz=1^∞ e^-(βħ^2 π^2)/(8mL^2)(nx^2 + ny^2 + nz^2) = [ ∑nx=1^∞ e^-(βħ^2 π^2)/(8mL^2) nx^2]^3 = Z1^3
As the number of quantum states is very large for a macroscopic system so we can convert the
above sum to an integral as
Z1 = ∫0^∞ e^-(βħ^2 π^2)/(8mL^2) nx^2 dnx - 1
Using the above considerations, show that the semiclassical partition function for a particle in a
one dimensional box of length L can be written as
Z1 = (1)/(ħ)∫0^L dx ∫0^∞ dp e^-β(p^2)/(2m)
(10)

Added by Rhonda M.

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