Identical fermions in a box - degenerate states

In summary, the ground state for two identical fermions in a 1D box is given by a specific wave function. The exclusion principle dictates that the \psi_{22} state is not allowed. However, because the particles are identical, the order of the subscripts does not matter, making \psi_{12} and \psi_{21} essentially the same state. This results in a non-degenerate ground state, with the next possible superposition being the first excited state \psi_{13}.
  • #1
dipole
555
151
The ground state for two identical fermions in a box (in 1D) is given by:

[itex]\psi (x_{1},x_{2})_{12} = \frac{\sqrt{2}}{a}[sin(\pi x_{1}/a)sin(2\pi x_{2}/a) - sin(2\pi x_{1}/a)sin(\pi x_{2}/a)] [/itex]

The book I'm reading though says that this state is non degenerate, and that the next excited state is [itex]\psi_{13}[/itex]. My question is, why is the ground state not degenerate? Why can't you have either [itex]\psi_{12}[/itex] or [itex]\psi_{21}[/itex] which would have the same energy?

Thanks.
 
Physics news on Phys.org
  • #2
Pauli's exclusion principle. EDIT: See below.
 
Last edited:
  • #3
I don't see how that answers my question.

Pauli exclusion principle determines the form of the wave function, and it forbids the [itex] \psi_{22} [/itex] state, but that doesn't explain to me why you can't have either [itex] \psi_{12} [/itex] or [itex] \psi_{21} [/itex] and have a doubly degenerate ground state.
 
  • #4
EDIT: Misread what you were asking exactly. Changing.

Sorry, thought you were asking about if you added another fermion.

But you have to remember that the particles are identical, so the order of your subscripts doesn't matter in this case because you cannot distinguish between the two particles. Pauli's Exlusion Principle is already in play here by the fact that one particle is in the ground state (of the one particle system) and another particle is in the first excited state (of the one particle system). You can't have the [itex]\psi_{11}[/itex] and by extension neither the [itex]\psi_{22}[/itex]. So the next possible superposition would be the [itex]\psi_{13}[/itex] where one particle is in the ground state and the other is in the second excited state. And since this has a higher energy than the [itex]\psi_{12}[/itex] state, then this represents the first excited state of the two particle system.
 
Last edited:
  • #5
Ah I see, so in distinguishable particles, the [itex] \psi_{12} [/itex] and [itex] \psi_{21} [/itex] are degenerate, but for indistinguishable particles they really represent the same state.

That makes perfect sense, thanks.
 

Related to Identical fermions in a box - degenerate states

1. What is the significance of identical fermions in a box?

Identical fermions in a box refers to a system in which multiple fermions, such as electrons, are confined to a finite space. This scenario is important in understanding the behavior and properties of quantum particles, as it allows for the study of degenerate states and the effects of quantum statistics.

2. What are degenerate states in the context of identical fermions in a box?

Degenerate states in this context refer to energy states that have the same energy level. In a system of identical fermions, the Pauli exclusion principle states that no two fermions can occupy the same energy state. Therefore, as the number of fermions increases, the available energy states become more crowded and some energy levels become degenerate.

3. How does quantum statistics affect the behavior of identical fermions in a box?

Quantum statistics, specifically the Pauli exclusion principle, plays a crucial role in the behavior of identical fermions in a box. This principle states that no two fermions can occupy the same quantum state, leading to the formation of degenerate energy states and the distribution of fermions according to their energy levels.

4. Can identical fermions in a box exhibit any properties of classical particles?

No, identical fermions in a box cannot exhibit any properties of classical particles. This is because their behavior is governed by quantum mechanics, which has different principles and laws than classical physics. Additionally, the exclusion principle prevents fermions from occupying the same state, which is a behavior unique to quantum particles.

5. How does the confinement of identical fermions in a box affect their interactions?

The confinement of identical fermions in a box can affect their interactions in several ways. For example, the Pauli exclusion principle dictates that fermions cannot occupy the same energy state, leading to interactions and exchanges of energy between particles. Additionally, the finite space of the box can also affect the wave function and phase of the particles, which can influence their interactions with each other.

Similar threads

I I want to expand a Gaussian wavepacket in terms of sines
    • Quantum Physics
Replies
2
Views
594
I Do I need energy to measure position without uncertainty?
    • Quantum Physics
Replies
5
Views
2K
I Writing the wave function solutions for a particle in a 2-D box
    • Quantum Physics
Replies
1
Views
649
A Bosonization Formula and its Effects on Fermion Number
    • Quantum Physics
Replies
5
Views
1K
Spatial Wave function of two indistinguishable particles
    • Advanced Physics Homework Help
Replies
26
Views
2K
I Projection postulate and the state of a system
    • Quantum Physics
Replies
9
Views
978
I Linear combination of states with Pauli's principle
    • Quantum Physics
Replies
9
Views
1K
A Can anyone give me the details of creating a cat state in Circuit QED?
    • Quantum Physics
Replies
16
Views
1K
Quantum Well Centred at the Origin Doubled in Size at time t=0. Symmetry seems to render the question incorrect
    • Advanced Physics Homework Help
Replies
4
Views
173
A Weyl Fermion in an infinite well
    • Quantum Physics
Replies
1
Views
775

Hot Threads

Recent Insights

Back
Top