Question

You have six shelves, one above the other and all above the floor, and six volumes of an encyciopedia, A. B. C. D. \(E\), and \(F\). (a) L.ist all the ways you can arrange the volumes with five on the floor and one on the sixth/top shelf. One way might be \(\mid \mathrm{ABCDE},-,-,-,-,-, \mathrm{F}\\}\) (b) List all the ways you can arrange them with four on the floor and two on the third shelf. (c) Show thal there are many more ways, relative to parts \((\mathrm{a})\) and \((\mathrm{b})\), to strange the six volumes with two on the floor and two eachon the first and second shelves. (There are several ways to answer this, but even listing them all won't take forever it's fewer than \(100 .)\) (d) Suddenly, a fantastic change! All six volumes are volume \(X\) - \(\mathrm{it}\) 's impossible to tell them apar. For each of the three distributions described in parts (a), (b), and (c). how many different (distinguishable) ways are there now? (e) If the energy you expend to lift a volume from the floor is proportional to a shelf's height, how do the total energies of distributions (a), (b), and (c) compare? (I) Use these ideas to atgue that the relative probabili. ties of occupying the lowestenergy states should be higher for hosons than for classically distinguishable particles. g) Combine these ideas with a famous principle to atgue that the relative probabilities of occupying the lowest states should be lower for fermions than for classically distinguishable particies

Short answer

For (a), there are 6 ways; for (b), there are 15 ways; for (c), there are 270 ways. For (d), there is only one case for each distribution a, b, and c. For (e), the total equivalent energies for distributions a, b and c are all 6E. As for (f), the relative probabilities of bosons occupying the lowest energy states should be higher than classically distinguishable particles; for (g), the relative probabilities of fermions occupying the lowest energy states should be lower than classically distinguishable particles.

Step-by-step solution

Arranging volumes with five on the floor and one on the top shelf

Since the books are distinguishable, arranging five on the floor and one on the top shelf can be done in \(6\) ways because any one of the six books can be on the top shelf.

Arranging volumes with four on the floor and two on the third shelf

This can be done by choosing 2 out of 6 volumes in \(\binom{6}{2}\) ways, which is 15. Other volumes are placed on the floor.

Arranging volumes with two on the floor and two each on the first and second shelves

We can select two books to be placed on the floor in \(\binom{6}{2}\) ways, then select two of the remaining to be placed on the first shelf and the rest on the second shelf. This gives \(\binom{6}{2} \cdot \binom{4}{2}\) which equals to \(45\cdot6 = 270\) ways.

Arranging identical volumes

Since all the books are identical, in each of the above distributions, they are considered one and the same. So the number of distinguishable ways to arrange them as per part a, b, and c, will be only \(1\) for each case as the books are identical.

Comparing energy spent in lifting a volume

The total energy spent is proportional to the shelf's height. For distributions a, b, and c, we have:a) One book is lifted to the 6th shelf, so the energy is \(6E\).b) Two books are lifted to the 3rd shelf, so the energy is \(2*3E = 6E\).c) Four books are lifted, two on the 1st shelf and two on the 2nd, so the energy is \(2*2E + 2*1E = 6E\). Thus, the total energies are equal for a, b, and c configurations.

Arguing about relative probabilities for bosons and classical particles

Since the energy expenditure is equal for all states in each case and bosons can exist in the same state, the relative probabilities of bosons occupying the lowest energy states should be higher than classically distinguishable particles as these particles cannot occupy the same state.

Arguing about relative probabilities for fermions and classical particles

Since fermions, unlike bosons, follow the Pauli Exclusion Principle and can't occupy the same state, the relative probabilities of fermions occupying the lowest energy states should be lower than classically distinguishable particles.

Key concepts

Permutations and Combinations

When it comes to arranging objects, the concepts of permutations and combinations are essential. Permutations refer to the number of ways we can arrange a set of objects, where the order does indeed matter. For instance, arranging the volumes A, B, C, D, E, and F on a shelf would involve permutations because swapping two books would create a completely different arrangement.
In the case of combinations, the order does not matter. This is best seen in part of the original exercise, where we had to choose 2 books out of the 6 to place on a certain shelf—here, we used the combination formula \( \binom{6}{2} \).
This foundational knowledge helps solve problems where multiple entities need arranging or selecting, underpinning many complex topics in quantum physics. Here are some key aspects:

• In permutations, order matters: use when different arrangements matter, like arranging books in a sequence.
• In combinations, order does not matter: use when the arrangement does not affect the outcome, like selecting books for a group.

Understanding these concepts can greatly aid in grasping more advanced topics like quantum statistics, which involve the arrangement of particles in various energy states.

Bosons and Fermions

In the world of quantum statistics, particles are classified into two categories: bosons and fermions. Each of these particles adheres to distinct statistical rules which significantly impact their behavior in physical systems.
Bosons are particles that obey Bose-Einstein statistics. These particles can occupy the same quantum state without restrictions, making them fundamentally different from fermions. This capability allows bosons to form phenomena such as Bose-Einstein condensates, where many particles occupy the lowest energy state. Because of this peculiar trait, bosons have higher probabilities of being in the same state, which is a key concept when comparing them to classical particles in physics.
Fermions, on the other hand, strictly follow Fermi-Dirac statistics and adhere to the Pauli Exclusion Principle. This principle dictates that no two fermions can occupy the same quantum state simultaneously. As a consequence, the relative probability of fermions occupying the lowest energy state is lower compared to bosons, or even classical distinguishable particles.
Some core ideas to remember include:

• Bosons can share quantum states: leading to phenomena like lasers and superfluid helium.
• Fermions cannot share quantum states: leading to the formation of electron shells in atoms.

Each of these properties shapes the distinct behaviors and applications of these subatomic particles in the universe.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental concept in quantum physics, playing a crucial role in the behavior of fermions, such as electrons, protons, and neutrons. Formulated by Wolfgang Pauli in 1925, this principle declares that no two identical fermions can occupy the same quantum state within a quantum system simultaneously.
This rule has profound implications across numerous fields of physics and chemistry. For example, it explains electron configurations in atoms—electrons fill orbitals starting from the lowest energy level, unable to pile into the same state due to this principle. This restriction leads to many of the unique properties of elements in the periodic table.
The Pauli Exclusion Principle is not just limited to explaining atomic structures. It also provides essential insight into the stability and density of matter. Without it, matter as we understand it would not exist in its current stable form. Here are key interpretations:

• Defines electron arrangement in atoms: leading to the structure of the periodic table.
• Ensures matter's stability: prevents collapse of atomic structures.

Grasping this principle is integral to understanding the nature of matter and the foundational rules governing quantum mechanics.