Symmetric and Anti-symmetric Wave functions
Introduction to Symmetric and Asymmetric Wave Functions
In this section, the concept of symmetric and asymmetric wave functions is introduced, along with their implications for the classification of elementary particles.
Understanding a System of Two Particles (Helium Atom)
- A helium atom consists of two electrons surrounding a nucleus.
- Each electron has a position (r1 and r2) and occupies a quantum state (sy a and sy b).
- The eigenstates determine the energy levels in which the electrons are present, independent of their positions.
Identical Nature of Electrons
- Electrons, like most elementary particles, are identical and cannot be distinguished from one another.
- Swapping the positions or eigenstates of two electrons does not result in any observable change.
- Probability distributions for both systems remain the same under an exchange of electrons.
Wave Function Solutions after Exchange
- After an exchange of electrons, the combined wave function can be written as SCI(r2,r1) = psy_a(r2)cye_b(r1).
- The second electron now has position r2 and eigenstate ia, while the first electron has position r1 and eigenstate ib.
Classification of Elementary Particles
- Under an exchange of particles, there is a plus-minus sign involved in the wave function.
- Symmetric case: sy_r1_r2 = +sy_r2_r1. This behavior is observed in bosons.
- Asymmetric case: Sai_r1_r2 = -Phi_r2_r1. This behavior is observed in fermions.
Fundamental Classification Based on Particle Behavior
This section explores how the probability distribution under particle exchange leads to a fundamental classification of elementary particles into bosons and fermions.
Symmetric Case (Bosons)
- In the symmetric case, the probability distribution for electron 1 to occupy position r1 and electron 2 to occupy position r2 is the same as when electron 1 occupies r2 and electron 2 occupies r1.
- This leads to the wave function equation: sigh(R1,R2) = +/-sy(R2,R1).
Asymmetric Case (Fermions)
- In the asymmetric case, the wave function undergoes a change in sign under an exchange of particles.
- The wave function equation becomes: Sai(R1,R2) = -Phi(R2,R1).
Classification of Elementary Particles
- Bosons: Particles that do not undergo a change in sign under particle exchange.
- Fermions: Particles that undergo a change in sign under particle exchange.
Conclusion
The concept of symmetric and asymmetric wave functions provides a fundamental classification for elementary particles. Bosons do not change their wave functions under particle exchange, while fermions exhibit a change in sign. This distinction plays a crucial role in understanding the behavior of elementary particles.
New Section
This section discusses the general solution for the linear combination of wave function solutions, distinguishing between cases with positive and negative signs. It introduces the concepts of boson and fermion particles based on the sign of the wave function.
General Solution for Linear Combination of Wave Function Solutions
- The solution involves a linear combination of two possible wave function solutions.
- Cases with a positive sign indicate no change in the wave function solution.
- Cases with a negative sign indicate a change in the wave function solution upon particle exchange.
- The 1 by root 2 term is a normalization constant.
Boson and Fermion Particles
- Cases with a plus sign involve boson particles.
- Cases with a negative sign involve fermion particles.
New Section
This section explores a specific case where two electrons exist in the same quantum state. It examines their combined wave function and how it changes under particle exchange.
Two Electrons in Same Quantum State
- Consider two electrons existing in the same quantum state, denoted as "psi."
- The combined wave function is asymmetric: psi_a(r1, r2) = psi_a(r1) * psi_a(r2).
- The wave function under exchange becomes symmetric: psi_s(r2, r1) = psi_a(r2) * psi_a(r1).
Symmetric and Asymmetric Wave Functions
- For symmetric wave functions, there is no change upon exchange (psi_s(r1, r2) = psi_s(r2, r1)).
- For asymmetric wave functions, there is a change upon exchange (psi_a(r1, r2) ≠ psi_a(r2, r1)).
Pauli's Exclusion Principle
- The symmetric wave function does not vanish due to summation involved.
- The asymmetric wave function vanishes due to the negative sign.
- The vanishing of the wave function implies that two particles cannot exist in the same quantum state.
- This leads to Pauli's exclusion principle, which states that fermions cannot occupy the same quantum state.
New Section
This section discusses the Pauli exclusion principle and its implications for bosons and fermions. It explains how bosons can occupy multiple energy levels, while fermions are limited to occupying energy levels in pairs.
Pauli Exclusion Principle
- The Pauli exclusion principle applies to fermions, which follow anti-symmetric wave functions.
- Fermions cannot exist in the same quantum state within a given system.
Distinction between Bosons and Fermions
- Bosons have symmetric wave functions under particle exchange.
- Fermions have anti-symmetric wave functions under particle exchange.
Implications for Bosons and Fermions
- Bosons do not follow the exclusion principle and can occupy energy levels with multiple particles.
- Fermions follow the exclusion principle and can only occupy energy levels with two particles at most.
- Same-spin particles cannot exist in the same quantum state for fermions.
New Section
This section summarizes the distinctions between bosons and fermions based on their wave functions, behavior under particle exchange, and occupation of energy levels.
Distinctions between Bosons and Fermions
- Bosons have symmetric wave functions under particle exchange.
- Fermions have anti-symmetric wave functions under particle exchange.
- Bosons do not follow the exclusion principle, while fermions do.
- Bosons follow Bose-Einstein statistics, while fermions follow Fermi-Dirac statistics.
Occupation of Energy Levels
- Bosons can occupy any number of particles in an energy level due to their symmetric wave function behavior.
- Fermions can only occupy energy levels with two particles at most due to their anti-symmetric wave function behavior.
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