A one-dimensional quantum harmonic oscillator (whose ground...

A one-dimensional quantum harmonic oscillator (whose ground state energy is $\hbar \omega / 2$ ) is in thermal equilibrium with a heat bath at temperature $T$. (a) What is the mean value of the oscillator's energy, $\langle E\rangle$, as a function of $T$ ? (b) What is the value of $\Delta E$, the root-mean-square fluctuation in energy about $\langle E\rangle$ ? (c) How do $\langle E\rangle$ and $\Delta E$ behave in the limits $k T \ll \hbar \omega$ and $k T \gg \hbar \omega$

A one-dimensional quantum harmonic oscillator (whose ground state energy is $\hbar \omega / 2$ ) is in thermal equilibrium with a heat bath at temperature $T$.
(a) What is the mean value of the oscillator's energy, $\langle E\rangle$, as a function of $T$ ?
(b) What is the value of $\Delta E$, the root-mean-square fluctuation in energy about $\langle E\rangle$ ?
(c) How do $\langle E\rangle$ and $\Delta E$ behave in the limits $k T \ll \hbar \omega$ and $k T \gg \hbar \omega$

Key Concepts

Boltzmann Factor

The Boltzmann factor, given by exp(-E/kT), is a key concept in statistical mechanics that determines the probability of a system occupying a particular energy state in thermal equilibrium. It underpins the derivation of the partition function and is essential for understanding how temperature influences the statistical distribution of states.

High and Low Temperature Limits

Evaluating the behavior of physical systems in the high and low temperature limits helps to bridge quantum and classical descriptions. At low temperatures, quantum effects such as zero-point energy become significant, while at high temperatures classical principles like the equipartition theorem dominate, simplifying the statistical behavior of the system.

Energy Fluctuations

Energy fluctuations refer to the statistical spread of the energy values around their mean in a thermally equilibrated system. These fluctuations, often quantified by the variance or the root-mean-square deviation, provide insight into the stability of the system and are an important aspect of the canonical ensemble.

Quantum Harmonic Oscillator

A fundamental model in quantum mechanics, the quantum harmonic oscillator represents a particle subject to a restoring force that is proportional to its displacement. This model exhibits quantized energy levels and serves as an important paradigm for understanding more complex quantum systems, molecular vibrations, and many-body physics.

Partition Function

In statistical mechanics, the partition function is a central quantity that encapsulates all thermodynamic information about a system in thermal equilibrium. It is computed as the sum over all states of the Boltzmann factors and is used to derive observables such as mean energy and fluctuations.

Mean Energy in the Canonical Ensemble

The mean energy of a system in a thermal bath is calculated as the ensemble average over all possible energy states, each weighted by its Boltzmann factor. This concept is crucial for linking microscopic quantum states to macroscopic thermodynamic quantities and understanding how energy is distributed in equilibrium.

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