10 Questions
Which of the following best describes the exchange interaction?
It arises from the indistinguishability of identical particles, such as electrons.
In the context of exchange interaction, what happens when two electrons have opposite spins?
The exchange interaction is attractive, leading to a lower total energy for the system.
How does the wave function of multi-electron atoms with exchange symmetry behave?
It becomes symmetric with respect to particle exchange.
What type of particles have a positive wave function and are symmetric with respect to particle exchange?
Bosons
How does exchanging labels of a pair of electrons affect the wavefunction?
It has no effect on the wavefunction.
What is the reason for the wavefunction of a multielectron atom to be anti-symmetric with respect to particle exchange?
To satisfy the Pauli exclusion principle for fermions
In the ground state of helium, what is the configuration of the two electrons?
1s2
What is the total wavefunction for an excited helium atom with one electron in the higher shell?
$\Psi = \frac{1}{\sqrt{2}}(\psi_{\alpha}(r_1)\psi_{\beta}(r_2) + \psi_{\beta}(r_1)\psi_{\alpha}(r_2))$
What does it mean when the total wavefunction for a two-electron system is anti-symmetrical?
The spin of one electron is up and the other is down
What is the value of the total spin ($S$) for a two-electron system if $S = 0$?
$S = 0$
Study Notes
Exchange Interaction
- Exchange interaction occurs when two electrons have opposite spins, resulting in a decrease in energy.
- When two electrons have opposite spins, the Coulomb repulsion between them is reduced, leading to an attraction between the electrons.
Wave Function of Multi-Electron Atoms
- The wave function of multi-electron atoms with exchange symmetry behaves according to the Pauli Exclusion Principle.
- The wave function of a multi-electron atom is anti-symmetric with respect to particle exchange.
Particle Exchange
- Bosons are particles that have a positive wave function and are symmetric with respect to particle exchange.
- Exchanging labels of a pair of electrons affects the wavefunction, making it anti-symmetric.
Multi-Electron Atoms
- The reason for the wavefunction of a multielectron atom to be anti-symmetric with respect to particle exchange is to satisfy the Pauli Exclusion Principle.
- In the ground state of helium, the two electrons have opposite spins and occupy the same orbital.
Excited Helium Atom
- The total wavefunction for an excited helium atom with one electron in the higher shell is a linear combination of the wavefunctions of the individual electrons.
Anti-Symmetrical Wavefunction
- When the total wavefunction for a two-electron system is anti-symmetrical, it means that exchanging the labels of the electrons changes the sign of the wavefunction.
- The value of the total spin (S) for a two-electron system is 0 when S = 0, indicating that the electrons have opposite spins.
Test your knowledge about exchange and Coulomb interactions, which are types of interatomic interactions that occur between atoms or ions. This quiz covers the interaction between charged particles due to their electric fields, as well as the exchange interaction related to the Pauli exclusion principle.
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