Four distinguishable harmonic oscillators a, b, c, and d may...

Four distinguishable harmonic oscillators $a, b, c,$ and $d$ may exchange energy. The energies allowed particle $a$ are $E_{a}=n_{d} h \omega_{0} ;$ those allowed particle $b$ are $E_{b}=n_{b} h \omega_{0}$ and so $\mathrm{cm}$. Consider an overall state (macrustate) in which the total energy is $3 \hbar \omega_{0}$. One possible microstate would have particles $\alpha$ b. and $c$ ' in their $n=0$ states and particle $d$ in its $n=3$ state: that is, $\left(n_{u}, n_{b}, n_{c}, n_{d}\right)=(0,0,0,3)$ (a) List all possible microstates. (b) What is the probability that a given particle will be in its $n=0$ state? (c) Answer par (b) for all other possible values of $n$. (d) Plot the probability versus $n$.

Four distinguishable harmonic oscillators $a, b, c,$ and $d$ may exchange energy. The energies allowed particle $a$ are $E_{a}=n_{d} h \omega_{0} ;$ those allowed particle $b$ are $E_{b}=n_{b} h \omega_{0}$ and so $\mathrm{cm}$. Consider an overall state (macrustate) in which the total energy is $3 \hbar \omega_{0}$. One possible microstate would have particles $\alpha$ b. and $c$ ' in their $n=0$ states and particle $d$ in its $n=3$ state: that is, $\left(n_{u}, n_{b}, n_{c}, n_{d}\right)=(0,0,0,3)$
(a) List all possible microstates.
(b) What is the probability that a given particle will be in its $n=0$ state? (c) Answer par (b) for all other possible values of $n$.
(d) Plot the probability versus $n$.

Key Concepts

Probability Distribution

When all microstates are assumed to have equal probability, the probability that a given oscillator occupies a specific energy level is determined by counting the number of microstates in which the oscillator has that energy relative to the total number of microstates. This forms a probability distribution over the allowed energy levels.

Microstates

A microstate is a specific detailed configuration in which the system can exist at the microscopic level, defining the quantum numbers of each individual oscillator such that the overall constraint is satisfied. In statistical mechanics, the enumeration of microstates is essential because the probability of any macrostate is determined by the number of corresponding microstates.

Macrostates

A macrostate defines the overall, coarse-grained state of the system in terms of macroscopic quantities like total energy. In systems with quantized energy levels, many different microstates (specific assignments of energies to particles) can correspond to the same macrostate, and understanding this relationship is key to linking microscopic behavior to macroscopic properties.

Quantization of Energy Levels

Harmonic oscillators in quantum mechanics have discrete energy levels, often expressed as multiples of a fundamental energy quantum (such as ???). This quantization implies that only specific energy values are allowed, and thus the state of each oscillator is characterized by a quantum number that determines its energy level.

Distinguishable Particles

When particles (or oscillators) are distinguishable, each configuration of energy assignments is considered unique, even if the same set of energy levels is occupied. This distinction affects the counting of microstates, as every permutation of energy assignments among different particles needs to be taken into account.

Combinatorial Enumeration

Counting the number of microstates for a given total energy often involves combinatorial methods such as the stars and bars technique. This method is used to find all possible combinations of quantum numbers (subject to a total energy constraint) by allocating discrete units of energy among the oscillators.

Data Visualization

Plotting the probability versus quantum number (n) helps to visualize how energy is statistically distributed among the oscillators. Such plots provide insight into the most and least probable energy states, reflecting the underlying combinatorial structure and the constraints imposed by the total energy of the system.

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