Review

1. Checklist of Key Concepts¶

Section 2.1: Introduction to Statistical Mechanics¶

1. Microscopic–Macroscopic Connection

   • Macroscopic (thermodynamic) properties can be understood as statistical averages of microscopic properties.

   • “Expected value” (ensemble average) is the central idea:

   $$\langle X \rangle \;=\; \sum_i X_i\,p_i.$$

   (1)

2. Arithmetic Average vs. Expected Value

   • Arithmetic average:

   $$\bar{X} \;=\; \frac{1}{M}\sum_{i=1}^M X_i.$$

   (2)

   • Expected value:

   $$\langle X \rangle \;=\; \sum_i X_i\,p_i, \quad p_i = \text{probability of microstate } i.$$

   (3)

3. Microstates and Ensembles

   • Microcanonical ensemble $(N, V, E)$: system is isolated, fixed total energy $E$.

   • Canonical ensemble $(N, V, T)$: system in thermal contact with a reservoir at temperature $T$.

   • Grand canonical ensemble $(\mu, V, T)$: system can exchange both energy and particles with a reservoir.

4. Fundamental Postulate (Microcanonical)

   • For an isolated system, each accessible microstate is equally probable.

Section 2.2: Canonical Ensemble¶

1. Closed System

   • Exchanges energy (heat) with surroundings; no exchange of matter.

2. Boltzmann Factor and Partition Function

   • Probability of microstate $i$:

   $$p_i \;=\; \frac{e^{-\beta E_i}}{Q}, \quad \beta \;=\;\frac{1}{k_{\mathrm B}\,T}.$$

   (4)

   • Partition function $Q$:

   $$Q \;=\; \sum_{i} e^{-\beta E_i},$$

   (5)

   which normalizes probabilities.

3. Two-State System

   • A simple example with energies $E_1$ and $E_2$.

   • Partition function:

   $$Q \;=\; e^{-\beta E_1} + e^{-\beta E_2}.$$

   (6)

   • Probabilities (if $\Delta E = E_2 - E_1$):

   $$p_1 = \frac{1}{1 + e^{-\beta\,\Delta E}}, \quad p_2 = 1 - p_1.$$

   (7)

4. Interpretation of $Q$

   • $Q$ is like an “effective count” of accessible microstates.

   • At low $T$, only the lowest energy states matter; at high $T$, many states are accessible.

Section 2.3: Ensemble Averages¶

1. Internal Energy

   • $U \equiv \langle E \rangle$.

   • In the canonical ensemble:

   $$U \;=\; \frac{1}{Q} \sum_i E_i\, e^{-\beta E_i} \;=\; -\,\bigl(\tfrac{\partial \ln Q}{\partial \beta}\bigr)_{N,V}.$$

   (8)

2. Heat Capacity $(C_V)$

   • Measures how internal energy changes with temperature:

   $$C_V = \bigl(\tfrac{\partial U}{\partial T}\bigr)_{N,V}.$$

   (9)

   • Also related to energy fluctuations:

   $$\sigma_E^2 \;=\; \langle (E - \langle E\rangle)^2\rangle \;=\; k_{\mathrm B}\,T^2\,C_V.$$

   (10)

3. Pressure (brief introduction)

   • In the canonical ensemble:

   $$P \;=\; k_{\mathrm B}\,T\, \bigl(\tfrac{\partial\ln Q}{\partial V}\bigr)_{N,T}.$$

   (11)

Section 2.4: Molecular Partition Functions¶

1. Many Identical and Independent Particles

   • For $N$ independent distinguishable particles with one-particle partition function $q$, the total partition function is

   $$Q \;=\; q^N.$$

   (12)

   • For indistinguishable particles,

   $$Q \;=\; \frac{q^N}{N!}.$$

   (13)

2. Molecular Partition Function

   • Within the Born–Oppenheimer approximation, a molecule’s energy decomposes into translational, rotational, vibrational, and electronic contributions:

   $$\varepsilon = \varepsilon_{\mathrm{trans}} + \varepsilon_{\mathrm{rot}} + \varepsilon_{\mathrm{vib}} + \varepsilon_{\mathrm{elec}}.$$

   (14)

   • Thus,

   $$q \;=\; q_{\mathrm{trans}}\, q_{\mathrm{rot}}\, q_{\mathrm{vib}}\, q_{\mathrm{elec}}.$$

   (15)

Section 2.5: Particle in a Box¶

1. Quantum Levels

   • For a 1D box $(0 \le x \le L)$,

   $$E_n = \frac{h^2}{8mL^2}\,n^2,\;\; n=1,2,3,\dots$$

   (16)

   • In 3D (a rectangular box of sides $L_x, L_y, L_z$):

   $$E_{n_x,n_y,n_z} \;=\; \frac{h^2}{8m} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right).$$

   (17)

2. Partition Function (3D Box)

   • For a cubic box of volume $V = L^3$:

   $$q \;=\; \sum_{n_x=1}^\infty \sum_{n_y=1}^\infty \sum_{n_z=1}^\infty \exp\!\Bigl[ -\beta \frac{h^2}{8mL^2}\,(n_x^2 + n_y^2 + n_z^2) \Bigr].$$

   (18)

   • At high $T$ or large $L$, we approximate the sum by an integral and obtain the classical partition function:

   $$q_{\mathrm{trans}} \;=\; \frac{V}{\Lambda^3}, \quad \Lambda \;=\; \sqrt{\frac{h^2}{2\pi m k_{\mathrm B} T}}\,\, \text{(thermal de Broglie wavelength).}$$

   (19)

3. Many-Particle System (Ideal Gas)

   • For $N$ indistinguishable, non-interacting particles:

   $$Q \;=\; \frac{q^N}{N!} \;=\; \frac{1}{N!} \biggl(\frac{V}{\Lambda^3}\biggr)^{\!N}.$$

   (20)

   • Leads directly to the ideal gas law and to:

   $$U \;=\; \frac{3}{2}\,N\,k_{\mathrm B}T, \quad C_V \;=\; \frac{3}{2}\,N\,k_{\mathrm B}, \quad P = \frac{N\,k_{\mathrm B}\,T}{V}.$$

   (21)

Section 2.6: Harmonic Oscillator¶

1. Quantum Harmonic Oscillator

   • Energy levels for a 1D harmonic oscillator of frequency $\omega$:

   $$E_n \;=\; \hbar\omega\,\Bigl(n + \tfrac{1}{2}\Bigr), \quad n=0,1,2,\dots$$

   (22)

2. Partition Function

   • Summing over these energy levels:

   $$q \;=\; \sum_{n=0}^{\infty} e^{-\beta E_n} \;=\; \frac{e^{-\frac{1}{2}\,\beta\,\hbar\omega}}{1 - e^{-\beta\,\hbar\omega}}.$$

   (23)

3. Ensemble Averages

   • Internal Energy:

   $$U \;=\; \frac{\hbar\omega}{2} \;+\; \frac{\hbar\omega}{\,e^{\beta\,\hbar\omega}-1\,}.$$

   (24)

   • Heat Capacity:

   $$C_V \;=\; k_{\mathrm B} \Bigl(\frac{\hbar\omega}{k_{\mathrm B} T}\Bigr)^{2}\, \frac{\,e^{\hbar\omega/(k_{\mathrm B}T)}\,} {\bigl(e^{\hbar\omega/(k_{\mathrm B}T)} - 1\bigr)^{2}}.$$

   (25)

   • At high temperature ($k_{\mathrm B}T \gg \hbar\omega$), each harmonic oscillator recovers the classical limit $U \to k_{\mathrm B}T$ and $C_V \to k_{\mathrm B}$.

Section 2.7: Linear Rigid Rotor¶

1. Energy Levels

   • For a rigid, linear rotor of moment of inertia $I$:

   $$E_J \;=\; \frac{\hbar^2}{2I}\,J\,(J+1), \quad J=0,1,2,\dots$$

   (26)

   • Each level $E_J$ has degeneracy $g_J = 2J + 1$.

2. Rotational Partition Function

   • Exact form:

   $$q_{\mathrm{rot}} \;=\; \sum_{J=0}^\infty (2J + 1)\, e^{-\beta \,\frac{\hbar^2}{2I}\,J\,(J+1)}.$$

   (27)

   • Define the rotational temperature $\Theta_{\mathrm{rot}} = \frac{\hbar^2}{2k_{\mathrm B}I}$.

   • High-temperature limit ($k_{\mathrm B}T \gg \frac{\hbar^2}{2I}$) gives

   $$q_{\mathrm{rot}} \;\approx\; \frac{T}{\Theta_{\mathrm{rot}}} \quad (\text{for a heteronuclear diatomic}).$$

   (28)

   • For a homonuclear diatomic, include a symmetry factor $\sigma=2$, so

   $$q_{\mathrm{rot}} \;\approx\; \frac{T}{\sigma\,\Theta_{\mathrm{rot}}}.$$

   (29)

3. Ensemble Averages (High-$T$ Approximation)

   • Internal Energy ($U_{\text{rot}}$):

   $$U_{\text{rot}} \;\approx\; k_{\mathrm B}\,T.$$

   (30)

   (One linear rotor has 2 rotational degrees of freedom $\rightarrow \frac{2}{2} k_{\mathrm B}T$.)

   • Heat Capacity:

   $$C_V^{(\text{rot})} \;\approx\; k_{\mathrm B}.$$

   (31)

Section 2.8: Molecular Statistical Mechanics¶

1. Combining All Degrees of Freedom

   • For a general molecule, the total one-molecule partition function is

   $$q \;=\; q_{\mathrm{trans}}\, q_{\mathrm{rot}}\, q_{\mathrm{vib}}\, q_{\mathrm{elec}}.$$

   (32)

   • Extends to polyatomic molecules with more complex rotational constants $\Theta_{\mathrm{rot},A}, \Theta_{\mathrm{rot},B}, \Theta_{\mathrm{rot},C}$ and multiple vibrational frequencies $\Theta_{\mathrm{vib},j}$.

2. Symmetry Considerations

   • Symmetry factor $\sigma$ must be included for molecules with indistinguishable orientations (e.g., homonuclear diatomics, symmetrical polyatomics).

3. Summary Table

   • Often, we tabulate $q_{\mathrm{trans}}, q_{\mathrm{rot}}, q_{\mathrm{vib}}, q_{\mathrm{elec}}$ for different molecule types (linear, nonlinear, spherical top, symmetric top, etc.), applying high-temperature (classical) or more exact quantum results as needed.

2. Checklist of Most Important Equations¶

Below is a unified list of the major equations from Sections 2.1–2.8.

A. Expected Value of a General Variable

• Applicability: general definition in statistical mechanics/probability theory.

B. Microcanonical Ensemble (Fundamental Postulate)

• Applicability: isolated system, $(N, V, E)$.

C. Canonical Ensemble Probability

• Applicability: closed system in thermal contact at $T$.

D. Canonical Partition Function

• Applicability: $(N, V, T)$ ensemble; sum/integral over all microstates.

E. Internal Energy (Canonical)

F. Heat Capacity at Constant Volume

Also,

G. Pressure (Canonical)

H. Molecular Systems: Indistinguishability Factor

• Applicability: identical, indistinguishable particles (e.g., ideal gases).

I. Translational Partition Function (Particle in a 3D box at high $T$)

J. Harmonic Oscillator Partition Function (1D)

• Internal Energy:

K. Rotational Partition Function (Linear Rotor, High $T$)

• Internal Energy:

• Heat Capacity:

L. Polyatomic Molecules

• General form (neglecting interactions):

• For linear vs. nonlinear rotors or multiple vibrational modes, each factor is included appropriately (with possible symmetry factors).