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--- un:bose-einstein-statistics [2025/11/30 07:33] – [Bose–Einstein occupation curves] asad
+++ un:bose-einstein-statistics [2025/11/30 07:37] (current) – asad
@@ Line 15: / Line 15: @@
 $$
-Here the factor of 2 accounts for two photon polarization states. For massive bosons, this factor is replaced by the appropriate spin degeneracy. This function expresses a key difference between classical and quantum statistics: the denominator can become small, allowing **large occupation of low-energy states**.
+Here the factor of 2 accounts for two photon polarization states. For massive bosons this factor is replaced by the appropriate spin degeneracy \( g \). This function expresses a key difference between classical and quantum statistics: the denominator can become small, allowing large occupation of low-energy states.
-For photons, the variable \( \nu \) labels frequency, and this same distribution underlies the Planck spectrum of [[Blackbody radiation]]. For massive bosons, the energy is instead \( \epsilon = p^{2}/2m \), but the statistical form is identical:
+For photons, the variable \( \nu \) labels frequency, and this same distribution underlies the Planck spectrum of [[Blackbody radiation]]. For massive bosons, the energy is instead \( \epsilon = p^{2}/(2m) \), but the statistical form is identical:
 $$
-f = \frac{g}{h^{3}} \frac{1}{e^{(\epsilon-\mu)/(kT)} - 1},
+f = \frac{g}{h^{3}} \, \frac{1}{e^{(\epsilon-\mu)/(kT)} - 1}.
 $$
-with \( g \) the spin degeneracy. The form of \( f \) therefore unifies the physics of blackbody photons and ideal Bose gases.
+The form of \( f \) therefore unifies the physics of blackbody photons and ideal Bose gases.
 ===== Mean occupation of quantum states =====
-The phase-space distribution function determines the **mean number of bosons occupying each single-particle quantum state**. Integrating \(f\) over a cell associated with the state \(i\) leads directly to the Bose–Einstein occupation number:
+The phase-space distribution function determines the mean number of bosons occupying each single-particle quantum state. Integrating \( f \) over a phase-space cell associated with state \( i \) leads to the Bose–Einstein occupation number
 $$
-\bar{n}_i = \frac{1}{\exp\!\left(\frac{\epsilon_i - \mu}{kT}\right) - 1}.
+\bar{n}_i = \frac{1}{e^{(\epsilon_i - \mu)/(kT)} - 1}.
 $$
-This quantity plays the same role for massive bosons that the photon distribution function plays for radiation. Summing over all states,
+This quantity plays the same role for massive bosons that the photon distribution function plays for radiation. Summing over all states gives
 $$
@@ Line 41: / Line 41: @@
 $$
-In the continuum limit with density of states \(g(\epsilon)\),
+In the continuum limit with density of states \( g(\epsilon) \),
 $$
 n = \frac{N}{V}
-  = \int_0^\infty \frac{g(\epsilon)}{\exp[(\epsilon-\mu)/(kT)] - 1} \, d\epsilon.
+  = \int_0^\infty \frac{g(\epsilon)}{e^{(\epsilon-\mu)/(kT)} - 1} \, d\epsilon.
 $$
@@ Line 58: / Line 58: @@
 $$
-the Bose–Einstein distribution reduces to the Maxwell–Boltzmann form,
+the Bose–Einstein distribution reduces to the Maxwell–Boltzmann form
 $$
-\bar{n}_i \approx \exp\!\left(-\frac{\epsilon_i-\mu}{kT}\right),
+\bar{n}_i \approx e^{-(\epsilon_i-\mu)/(kT)},
 $$
@@ Line 67: / Line 67: @@
 $$
-f \approx \frac{g}{h^{3}} e^{-(\epsilon-\mu)/(kT)}.
+f \approx \frac{g}{h^{3}} \, e^{-(\epsilon-\mu)/(kT)}.
 $$
-Quantum statistical effects appear when the thermal de Broglie wavelength,
+Quantum statistical effects appear when the thermal de Broglie wavelength
 $$
 \lambda_{\mathrm{th}}
-= \sqrt{\frac{2\pi\hbar^{2}}{m kT}},
+= \sqrt{\frac{2\pi\hbar^{2}}{m kT}}
 $$
@@ Line 80: / Line 80: @@
 $$
-n\lambda_{\mathrm{th}}^{3} \gtrsim 1.
+n \lambda_{\mathrm{th}}^{3} \gtrsim 1.
 $$
@@ Line 87: / Line 87: @@
 ===== Bose–Einstein condensation =====
-For a uniform ideal Bose gas, the chemical potential satisfies \( \mu \le \epsilon_{0} \). As the gas is cooled at fixed density, \( \mu \rightarrow \epsilon_{0} \), and the excited states are no longer capable of holding all particles. The excess particles accumulate in the ground state, producing macroscopic occupation:
+For a uniform ideal Bose gas, the chemical potential satisfies \( \mu \le \epsilon_{0} \). As the gas is cooled at fixed density, \( \mu \to \epsilon_{0} \), and the excited states are no longer capable of holding all particles. The excess particles accumulate in the ground state, producing macroscopic occupation,
 $$
-\bar{n}_{0} \rightarrow \text{large}.
+\bar{n}_{0} \gg 1.
 $$
-This is **Bose–Einstein condensation**, a striking manifestation of the Bose–Einstein distribution function. The condensation phenomenon is therefore the direct consequence of the form of \(f\) and the associated occupation numbers \(\bar{n}_i\): large phase-space density implies large ground-state occupation.
+This is Bose–Einstein condensation, a striking manifestation of the Bose–Einstein distribution function. The condensation phenomenon is therefore the direct consequence of the form of \( f \) and the associated occupation numbers \( \bar{n}_i \): large phase-space density implies large ground-state occupation.
 In contrast, for photons the chemical potential is fixed to zero, and instead of condensing, photons redistribute themselves to form the blackbody spectrum.
@@ Line 99: / Line 99: @@
 ===== Bose–Einstein occupation curves =====
-To analyze how the Bose–Einstein distribution behaves, it is useful to define dimensionless variables,
+To analyze how the Bose–Einstein distribution behaves, it is useful to define dimensionless variables
 $$
@@ Line 112: / Line 112: @@
 $$
-Small \(a\) (i.e., large \( \mu \)) enhances the low-energy peak of the distribution. The special case \(a=0\) corresponds exactly to the photon distribution function above.
+Small \( a \) (that is, large \( \mu \)) enhances the low-energy peak of the distribution. The special case \( a = 0 \) corresponds exactly to the photon distribution function in blackbody radiation.
 <html>
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-// Axis labels with NO MathJax
+// Axis labels with no MathJax
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   margin: {l: 95, r: 20, t: 10, b: 60},
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   * Large low-energy occupation arises because the denominator may become small as \( \mu \to \epsilon_{0} \).
   * Classical statistics emerges only when \( e^{(\epsilon-\mu)/(kT)} \gg 1 \), suppressing large occupation.
-  * Quantum degeneracy appears when \( n\lambda_{\mathrm{th}}^{3} \gtrsim 1 \), forcing the use of Bose–Einstein statistics.
+  * Quantum degeneracy appears when \( n \lambda_{\mathrm{th}}^{3} \gtrsim 1 \), forcing the use of Bose–Einstein statistics.
   * Bose–Einstein condensation results directly from the inability of excited states to hold all particles as \( \mu \to \epsilon_0 \).
   * For photons, the condition \( \mu = 0 \) transforms the Bose–Einstein distribution into the Planck spectrum of blackbody radiation.
@@ Line 222: / Line 222: @@
 ===== Inquiries =====
   * Derive the phase-space distribution \( f = g h^{-3} [e^{(\epsilon-\mu)/(kT)} - 1]^{-1} \) from the grand canonical ensemble.
-  * Show how the occupation number \( \bar{n}_i \) emerges from integrating \(f\) over a state’s phase-space cell.
+  * Show how the occupation number \( \bar{n}_i \) emerges from integrating \( f \) over a state’s phase-space cell.
   * Demonstrate explicitly how the Maxwell–Boltzmann limit arises from the Bose–Einstein distribution.
-  * Using \( n\lambda_{\mathrm{th}}^{3} \sim 1 \), estimate the temperature at which a dilute Bose gas becomes quantum degenerate.
+  * Using \( n \lambda_{\mathrm{th}}^{3} \sim 1 \), estimate the temperature at which a dilute Bose gas becomes quantum degenerate.
-  * Explain how varying \(a=-\mu/(kT)\) alters the shape of the occupation curves in the interactive figure.
+  * Explain how varying \( a = -\mu/(kT) \) alters the shape of the occupation curves in the interactive figure.