Ground State Energy Levels of He with Two Identical or Distinguishable Electrons

[empirekhoo]

Homework Statement

Discuss the ground state energy levels of He, if

i) two electrons are identical bosons
ii) two electrons are distinguishable particles (but with same mass and same charge)

Homework Equations

err...

The Attempt at a Solution

well...

i) no change in energy. we notice both electron(fermions) have same energy. even if we change it to boson where the quantum state for both electron are equal (ground state), there'll be no changes?

ii) errr... actually i have no idea! some hint please?

 

[Feldoh]

i.) What are the rules for identical particles? I.e. what does the Pauli exclusion say about bosons and fermions? What's different for the two?

ii.) What are the rules for distinguishable particles?

Basically you should start by thinking about the wave functions for these cases.

 

[empirekhoo]

i.) What are the rules for identical particles? I.e. what does the Pauli exclusion say about bosons and fermions? What's different for the two?

ii.) What are the rules for distinguishable particles?

Basically you should start by thinking about the wave functions for these cases.

Hm.. I was thinking of:
1. write down both formula for distinguishable, fermions & bosons

$$\psi \left( r_{{1}},r_{{2}} \right) =\psi_{{a}} \left( r_{{1}} \right) \psi_{{b}} \left( r_{{2}} \right)$$

and

$$\psi \left( r_{{1}},r_{{2}} \right) =1/ \sqrt {2} \left( \psi_{{a}} \left( r_{{1}} \right) \psi_{{b}} \left( r_{{2}} \right) +\psi_{{a}} \left( r_{{2}} \right) \psi_{{b}} \left( r_{{1}} \right) \right)$$

2. Then I use $$H\psi=E\psi$$ to get the energies?

However I notice it's quite impossible to compare directly (ie without substituting the wavefunction for He). Am I wrong, or it's a must to substitute wavefunction for each case to compare?

 

[Feldoh]

You don't need to know the exact states, what you need to know are what particles can be in any given eigenvalue of the energy, and how many can be in the same eigenvalue.

 

[paranormal]

I would like to address the ground state energy levels of He in the two given scenarios. In the first case, where two electrons are identical bosons, there will be no change in the ground state energy levels. This is because identical bosons, like fermions, follow the Pauli exclusion principle which states that no two particles can occupy the same quantum state. Therefore, the two electrons will have the same energy level, and there will be no difference from the ground state energy levels of He with two identical fermions.

In the second case, where the two electrons are distinguishable particles with the same mass and charge, the ground state energy levels of He will be affected. This is because distinguishable particles do not follow the Pauli exclusion principle, and thus, can occupy the same quantum state. In this case, the two electrons can occupy different energy levels, resulting in a different ground state energy level for He. This can lead to a slightly higher energy level compared to the case of identical particles.

In summary, the ground state energy levels of He are affected by the nature of the particles, whether they are identical bosons or distinguishable particles. Identical bosons have no effect on the energy levels, while distinguishable particles can result in slightly higher energy levels due to their ability to occupy the same quantum state. Further research and experiments can provide more insights and a deeper understanding of the ground state energy levels of He in these two scenarios.