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

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

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.