QM: Clarifying Pauli Exclusion Principle, State of System

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In summary, fermions follow the Pauli Exclusion Principle and have anti-symmetric wave functions, meaning only one particle can occupy a single state. This is due to the fact that identical particles are indistinguishable and their wave function must remain unchanged under particle exchange. In the case of fermions, this results in the wave function being either totally symmetric or totally antisymmetric. This explains why two fermions cannot occupy the same one-particle state.
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Maybe_Memorie
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Just want to clarify my understanding of these things.

Fermions obey the Pauli Exclusion Principal, meaning meaning only one particle can occupy a single state. This means the wave function is anti-symmetric under particle exchange. That's the part that isn't making much sense. Is it because if we have two particles designated by quantum numbers n, l, m1 and n, l, m2 and we swap the particles the only way for the states to be different is if m1 = -m2?Also, when we speak of the state of a system...
Consider the basic problem of a spin 1/2 particle in a constant magnetic field (0, 0, B).
H = γSz , γ = geB/2mc, or something like that..

We write H|E> = E|E>
H2|E> = E2|E>
From this we determine the eigenvalues E1 and E2, and the state at some time t is given by

|ψ> = C+eaE1t|E1> + C-eaE2t|E2>What does this actually mean? Is it the energy at some later time t? The spin?
 
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The logics concerning Fermions is somewhat the other way. The point is that identical particles are indistinguishable, which means that under exchange of any two particles in an N-particle state the wave function must not change (except through a phase factor). As it turns out by some complicated arguments, if space has 3 (or more) dimensions there can only be two realizations of this principle, namely either the wave function is totally symmetric under interchange of particles or totally antisymmetric. The corresponding particles are called bosons or fermions, respectively.

For the fermions with antisymmetric wave functions, it's clear that the interchange of two particles that are in the same one-particle state doesn't change the wave function at all, but according to the rule for fermions it must get an additional minus sign. That means the wave function [itex]\psi=-\psi[/itex], but this can only be true if [itex]\psi=0[/itex]. Thus, there doesn't exist a state, where two fermions occupy the same one-particle state.
 
  • #3
vanhees71 said:
The logics concerning Fermions is somewhat the other way. The point is that identical particles are indistinguishable, which means that under exchange of any two particles in an N-particle state the wave function must not change (except through a phase factor). As it turns out by some complicated arguments, if space has 3 (or more) dimensions there can only be two realizations of this principle, namely either the wave function is totally symmetric under interchange of particles or totally antisymmetric. The corresponding particles are called bosons or fermions, respectively.

For the fermions with antisymmetric wave functions, it's clear that the interchange of two particles that are in the same one-particle state doesn't change the wave function at all, but according to the rule for fermions it must get an additional minus sign. That means the wave function [itex]\psi=-\psi[/itex], but this can only be true if [itex]\psi=0[/itex]. Thus, there doesn't exist a state, where two fermions occupy the same one-particle state.

That explanation was perfect, thanks!
 

1. What is the Pauli Exclusion Principle?

The Pauli Exclusion Principle is a fundamental principle in quantum mechanics that states that no two identical fermions can occupy the same quantum state simultaneously. This applies to particles such as electrons, protons, and neutrons, and is a key factor in determining the electronic structure of atoms.

2. How does the Pauli Exclusion Principle relate to the state of a system?

The Pauli Exclusion Principle dictates the allowed states of a quantum system by limiting the number of fermions that can occupy a given energy level. This leads to the organization of electrons in atoms into distinct energy levels and sublevels, and ultimately determines the chemical and physical properties of elements.

3. Can the Pauli Exclusion Principle be violated?

No, the Pauli Exclusion Principle is a fundamental law of quantum mechanics and has been rigorously tested and confirmed through experiments. It is considered a principle of nature and cannot be violated.

4. How does the Pauli Exclusion Principle impact the behavior of matter?

The Pauli Exclusion Principle is a key factor in determining the stability and behavior of matter. It is responsible for preventing atoms from collapsing and explains why matter takes up space and has a defined shape. It also plays a role in the formation of molecules and the properties of materials.

5. Is the Pauli Exclusion Principle related to other principles or laws in physics?

Yes, the Pauli Exclusion Principle is closely related to other fundamental principles in physics, such as the principle of superposition and the conservation of energy. It also has implications in other fields of physics, such as nuclear physics and condensed matter physics.

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