40-4 Exclusion Principle and Multiple Electrons in a Trap#
Prompts
State the Pauli exclusion principle. Why would atoms collapse if it did not hold?
For a 1D infinite well, how many quantum numbers does each electron have? How many electrons can occupy the lowest energy level? Explain using the exclusion principle.
In a 2D rectangular corral, energy levels can be degenerate (e.g., \(E_{2,1} = E_{1,2}\)). How many distinct quantum states share that energy? How many electrons can occupy that level?
Describe the procedure for building the ground-state configuration of \(N\) electrons in a trap: add electrons one by one, fill lowest levels first, obey the exclusion principle. What do empty, partially occupied, and fully occupied mean?
For multiple electrons in a trap (neglecting electron–electron interaction), how do you find the total energy? How do you identify the first excited state and the energy required to reach it?
Lecture Notes#
Overview#
The Pauli exclusion principle forbids two electrons in the same trap from having the same set of quantum numbers — a fundamental rule that explains atomic structure and stability.
Electrons are placed in traps one by one, filling the lowest available states; each state can hold at most one electron (or two if they differ only by spin).
Levels are empty, partially occupied, or fully occupied; the electron configuration is the listing of occupied states.
For multiple electrons (neglecting interaction), the total energy is the sum of single-electron energies; the first excited state is the lowest-energy configuration that differs from the ground state by one electron jump.
The Pauli Exclusion Principle#
No two electrons confined to the same trap can have the same set of values for their quantum numbers.
For atoms (Section 40-5), this means no two electrons can share the same \((n, \ell, m_\ell, m_s)\). All electrons have \(s = \tfrac{1}{2}\), so any two must differ in at least one of \(n\), \(\ell\), \(m_\ell\), or \(m_s\).
Why it matters
Without the exclusion principle, all electrons would collapse into the lowest energy level. Atoms would not have the structure they do, and matter as we know it could not exist.
Multiple Electrons in Rectangular Traps#
To prepare for atoms, consider electrons in the traps of Chapter 39, now including spin (\(m_s = \pm\tfrac{1}{2}\)).
Trap |
Quantum numbers |
Max electrons per level |
|---|---|---|
1D (\(L\)) |
\(n\), \(m_s\) |
2 (opposite spins) |
2D corral (\(L_x \times L_y\)) |
\(n_x\), \(n_y\), \(m_s\) |
2 per distinct \((n_x,n_y)\); degenerate levels hold more |
3D box (\(L_x \times L_y \times L_z\)) |
\(n_x\), \(n_y\), \(n_z\), \(m_s\) |
2 per distinct \((n_x,n_y,n_z)\) |
Procedure: Add electrons one by one. Each goes into the lowest energy level that still has an available state (i.e., a unique set of quantum numbers not yet used).
Occupancy Terminology#
Empty (unoccupied): No electrons in that level.
Partially occupied: Some states in that level are filled, others are available.
Fully occupied: No more electrons can be added to that level without violating the exclusion principle.
The electron configuration is a listing (or diagram) of which levels the electrons occupy, or of the quantum numbers of each electron.
Finding the Total Energy#
We assume electrons do not electrically interact with each other. Then:
Use the one-electron energy-level diagram (from Chapter 39).
Place electrons according to the exclusion principle.
Total energy = sum of the energies of the individual electrons.
The ground state of the system is the configuration with the lowest total energy. The first excited state is the next-lowest configuration, reached by moving one electron to a higher level (the jump must be to a level that is not fully occupied).
Example: Seven Electrons in a Square Corral#
For a square corral (\(L_x = L_y = L\)), energies are \(E_{n_x,n_y} = (n_x^2 + n_y^2)h^2/(8mL^2)\). The lowest levels (in units of \(h^2/8mL^2\)) are:
\(E_{1,1} = 2\) — 1 distinct state, 2 electrons max (opposite spins)
\(E_{2,1} = E_{1,2} = 5\) — 2 distinct states, 4 electrons max
\(E_{2,2} = 8\) — 1 distinct state, 2 electrons max
\(E_{3,1} = E_{1,3} = 10\) — 2 distinct states, 4 electrons max
Ground state for 7 electrons:
2 in \(E_{1,1}\), 4 in \(E_{2,1}/E_{1,2}\), 1 in \(E_{2,2}\)
\(E_{\text{gr}} = 2(2) + 4(5) + 1(8) = 32\,h^2/8mL^2\)
First excited state
Question: What is the smallest energy input needed to reach the first excited state?
Answer: An electron must jump to a level that is not fully occupied. The three smallest possible jumps are:
\(E_{1,1} \to E_{2,2}\): \(\Delta E = 6\)
\(E_{2,1}/E_{1,2} \to E_{2,2}\): \(\Delta E = 3\)
\(E_{2,2} \to E_{3,1}/E_{1,3}\): \(\Delta E = 2\)
The minimum is \(\Delta E = 2\,h^2/8mL^2\). The electron in \(E_{2,2}\) jumps to the empty \(E_{3,1}/E_{1,3}\) level. The first excited state has energy \(E_{\text{fe}} = E_{\text{gr}} + \Delta E = 34\,h^2/8mL^2\).
Energy-Level Diagrams for the System#
One can draw either:
One-electron diagram: Show each electron on the single-particle energy levels (arrows for spin).
System diagram: Show the total energies \(E_{\text{gr}}\), \(E_{\text{fe}}\), \(E_{\text{se}}\), … of the system as a whole.
The ground state is the lowest total energy; the first excited state is the next; and so on.
Summary#
The Pauli exclusion principle forbids two electrons from sharing the same quantum numbers; it is essential for atomic structure.
Electrons fill traps one by one into the lowest available states; levels can be empty, partially occupied, or fully occupied.
With no electron–electron interaction, total energy is the sum of single-electron energies.
The first excited state is reached by the smallest-energy single-electron jump to a non-full level.