Occupation probability and examples

Where this sits. This is the final synthesis chapter of PH11001, "Occupation probability, integration, and exam preparation," which connects the course's four blocks — oscillations, waves, EM waves, and wave mechanics. The wave-mechanics block already established that bound systems such as a particle on a chain or a quantum simple harmonic oscillator have discrete, quantized energy levels. This article asks why the pendulum and spring-mass systems from Chapter 1, governed by the classical equation $\ddot x+\omega_0^2x=0$, never visibly jump between such levels.

The Boltzmann factor. For a system in thermal equilibrium with a reservoir at temperature $T$, the relative probability of occupying a state of energy $E$ is

where: $k_B = 1.38\times10^{-23}\ \text{J/K}$ is Boltzmann's constant, $E$ is the energy of the state measured from the ground state, and $T$ is the absolute temperature of the bath. Higher-energy states are exponentially less likely to be occupied than lower ones, and the ratio $E/k_BT$ sets how steep that suppression is — a large ratio crushes the higher levels toward zero probability, a small ratio leaves them almost as likely as the ground state.

Applying it to a quantized oscillator. The wave-mechanics block gives the SHO energy levels as equally spaced:

where: $n = 0,1,2,\ldots$ labels the level, $\omega$ is the oscillator's angular frequency, and $\hbar = 1.055\times10^{-34}\ \text{J}\,\cdot\,\text{s}$ is the reduced Planck constant. Because the spacing $\Delta E=\hbar\omega$ is the same between every pair of adjacent levels, the single ratio $\Delta E/k_BT$ decides how the whole ladder of levels is populated relative to its neighbours.

Two regimes. If $\hbar\omega \gg k_BT$, the ratio $\Delta E/k_BT$ is huge, $e^{-\Delta E/k_BT}$ is vanishingly small, and the system sits overwhelmingly in its ground state — quantization is sharply visible. This is the regime of atomic and molecular vibrations at room temperature. If $\hbar\omega \ll k_BT$, the ratio is tiny, $e^{-\Delta E/k_BT}\approx1$, and thousands of adjacent levels are populated with nearly equal probability — the discreteness smears into what looks like a continuous range of energy. This is exactly the regime of a macroscopic pendulum or spring, whose $\omega_0$ of a few rad/s makes $\hbar\omega_0$ fantastically smaller than $k_BT$ at ordinary temperatures, recovering classical continuous-energy behaviour.

Macroscopic spring-mass oscillator. A mass $m=0.20\ \text{kg}$ is attached to a spring of stiffness $k=20\ \text{N/m}$, oscillating at room temperature $T=300\ \text{K}$. Compare the quantum level spacing to $k_BT$ and estimate how many levels are thermally populated.

Step 1. Find the angular frequency: $\omega_0 = \sqrt{k/m} = \sqrt{20/0.20} = \sqrt{100} = 10\ \text{rad/s}$.

Step 2. Find the quantum level spacing: $\Delta E = \hbar\omega_0 = (1.055\times10^{-34}\ \text{J}\,\cdot\,\text{s})(10\ \text{rad/s}) = 1.06\times10^{-33}\ \text{J}$.

Step 3. Find the thermal energy scale: $k_BT = (1.38\times10^{-23}\ \text{J/K})(300\ \text{K}) = 4.14\times10^{-21}\ \text{J}$.

Step 4. Compare and interpret: $\dfrac{\Delta E}{k_BT} = \dfrac{1.06\times10^{-33}}{4.14\times10^{-21}} \approx 2.6\times10^{-13}$. This ratio is essentially zero, so $e^{-\Delta E/k_BT}\approx1$ and roughly $k_BT/\Delta E \approx 4\times10^{12}$ levels are thermally accessible. The spring's energy therefore looks perfectly continuous, consistent with the classical formula $E=\tfrac12kA^2$ you used for this oscillator back in Chapter 1.

Competitive stretch

Molecular vibration for contrast. A diatomic molecule vibrates at $\omega = 4\times10^{14}\ \text{rad/s}$ (typical of a chemical bond). Here $\Delta E = \hbar\omega = (1.055\times10^{-34})(4\times10^{14}) = 4.22\times10^{-20}\ \text{J}$, so $\Delta E/k_BT = 4.22\times10^{-20}/4.14\times10^{-21} \approx 10.2$. Since this is far greater than 1, $e^{-10.2}\approx4.5\times10^{-5}$: almost no thermal population above the ground vibrational state at room temperature, and the molecule's quantization stays sharply observable. Only the size of $\omega$ separates this outcome from the spring above — both obey the same $E_n=(n+\tfrac12)\hbar\omega$ law.

• Thinking the Boltzmann factor "turns off" quantization for macroscopic objects — it doesn't remove the discrete levels; it makes adjacent levels so nearly equally occupied that the system explores enormously many of them at once, which looks continuous.
• Comparing $E_n$ itself to $k_BT$ instead of comparing the level spacing $\Delta E=\hbar\omega$ to $k_BT$ — the spacing, not the absolute energy, is what governs the occupation ratio between neighbours.
• Mixing up $\hbar$ and $h$: angular frequency $\omega$ (rad/s) pairs with $\hbar$ in $\Delta E=\hbar\omega$, while ordinary frequency $f$ (Hz) pairs with $h$ via $E=hf$.
• Reaching for partition functions or Bose-Einstein/Fermi-Dirac statistics — this chapter uses only the single-particle Boltzmann factor as a qualitative illustration, nothing more.

Ask the tutor: want to work through $\Delta E/k_BT$ for a swinging pendulum or a vibrating atom in a solid, or see how this ratio connects back to the classical energy formula $E=\tfrac12m\omega_0^2A^2$ from Chapter 1?