Question

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 \(n\).

Short answer

The possible microstates are: (0,0,0,3), (0,0,1,2), (0,0,2,1), (0,0,3,0), (0,1,0,2), (0,1,1,1), (0,1,2,0), (0,2,0,1), (0,2,1,0), (0,3,0,0), (1,0,0,2), (1,0,1,1), (1,0,2,0), (1,1,0,1), (1,1,1,0), (1,2,0,0), (2,0,0,1), (2,0,1,0), (2,1,0,0), (3,0,0,0). The probabilities that a given oscillator is in the \(n=0, 1, 2, 3\) states are, respectively, 0.95, 0.30, 0.15, and 0.05.

Step-by-step solution

Determine all possible microstates

The possible microstates can be determined by considering all the possible ways that three quanta can be distributed among the four oscillators. The microstates are: \((n_{a}, n_{b}, n_{c}, n_{d}) = (0,0,0,3), (0,0,1,2), (0,0,2,1), (0,0,3,0), (0,1,0,2), (0,1,1,1), (0,1,2,0), (0,2,0,1), (0,2,1,0), (0,3,0,0), (1,0,0,2), (1,0,1,1), (1,0,2,0), (1,1,0,1), (1,1,1,0), (1,2,0,0), (2,0,0,1), (2,0,1,0), (2,1,0,0), (3,0,0,0)\)

Count how many microstates have an oscillator in its \(n=0\) state

A given oscillator is in the \(n=0\) state in all microstates except (3, 0, 0, 0). Only one microstate (3, 0, 0, 0) doesn't have any oscillator in the \(n=0\) state. Therefore, the total number of microstates with a given oscillator in the \(n=0\) state is 19 out of 20. Hence, the probability that a given particle will be in its \(n=0\) state is \(19/20 = 0.95). This probability is the same for each oscillator, as they are identical and indistinguishable.

Calculate the probabilities for all other possible states

The oscillator can be in states \(n=1, 2, 3\). The number of microstates with the oscillator in the \(n=1\) state is 6, for \(n=2\) is 3, and for \(n=3\) is 1. Hence, the probabilities are, respectively, 6/20 = 0.30, 3/20 = 0.15, and 1/20 = 0.05.

Plot the probability versus \(n\)

The x-axis represents the states of the oscillator, and the y-axis represents the probability that the oscillator is in that state. A bar graph can be drawn with bars at \(n=0, 1, 2, 3\) and the heights of the bars are the probabilities calculated in step 3, which are, respectively, 0.95, 0.30, 0.15, and 0.05.

Key concepts

Microstates and Macrostates

Understanding the concepts of microstates and macrostates is foundational in the study of quantum mechanics and statistical physics. In the context of quantum harmonic oscillators, like in our exercise involving particles a, b, c, and d, a macrostate refers to the overall condition of a system—such as its total energy—without concern for how that condition is achieved. For example, our exercise defines a macrostate with total energy of 3ħω_0.

A microstate, on the other hand, details the specific arrangement of the system's components that results in the macrostate. Each microstate is a unique configuration of particles and their respective quantum states. In our exercise, one example is (n_a, n_b, n_c, n_d) = (0,0,0,3), which is one of many ways to achieve the total energy macrostate. By listing all the possible microstates, we capture all the configurations consistent with the macrostate's constraints.

The concept of microstates becomes especially powerful when combined with the principles of statistical mechanics. For systems with a large number of particles, the number of microstates associated with a particular macrostate can be astronomically large. The probability of the system being in any one microstate is typically the same for each microstate (assuming equally probable states) and understanding the distribution of microstates gives us deep insights into the system's thermodynamic behavior and properties.

Quantum Mechanics Probability

The role of probability in quantum mechanics is to indicate the likelihood of a quantum system, like our set of harmonic oscillators, being found in a particular state. In this discipline, probabilities are not just a convenient tool for handling uncertainty, they are a fundamental aspect of the physical theories themselves.

For instance, when we calculate the likelihood of an oscillator being in its n=0 state, as shown in the exercise, we are making use of quantum mechanics probability to predict the expected behavior of the system under certain conditions. According to the results of our calculation, finding a given particle in the n=0 state has a probability of 0.95, which is intuitively high given that there are 19 microstates out of 20 possible ones where at least one oscillator is in that state.

The exercise we've analyzed demonstrates how probabilities can change significantly for different states of a quantum system. This understanding is imperative for interpreting experimental results and predicting system dynamics, as every measurable quantity in quantum mechanics is ultimately associated with a probability distribution.

Energy Distribution Among Oscillators

The concept of energy distribution among oscillators relates to the way energy is allocated between different parts of a quantum system. In a quantum harmonic oscillator, energy levels are quantized, meaning oscillators can only possess specific discrete amounts of energy.

In our exercise, the task of distributing a total energy of 3ħω_0 among four distinguishable oscillators showcases this energy quantization. Each configuration reflects a different distribution of energy and is represented by a unique microstate. The probabilities calculated for each quantum state — for n=0, 1, 2, and 3 — directly depict this energy distribution. As one can see, most energy configurations favor lower energy states, such as n=0, a tendency described by the Boltzmann distribution in statistical mechanics.

The distribution tells us a lot about the system's properties: lower energy states are more likely to be occupied than higher energy states in a quantum harmonic oscillator. This knowledge is essential in many areas of physics, including understanding molecular vibrations, quantum optics, and even black body radiation. Because of the fundamental role energy plays in these systems, energy distribution is a topic that's crucial for students to grasp thoroughly.