# Three particles case, finding ground energy state

Here I came up with three particles in a box problem.

(Assumption: Here I do not consider the interaction between particles and spin for simplicity.)

What I want to do is express the ground state's energy and its wave function. Then the $n^\mathrm{th}$ energy level and wavefunction are denoted by $E_n$ and $\psi_n$.

1. three electrons case

2. three pions case

3. two electron and proton case

Here I came up with my own solutions for 1 and 2.

1. three electrons --> three indistinguishable fermions. Due to Pauli's principle $n=1,2,3$:

$E_n = E_1 + E_2 + E_3$

The wave function is determined by the Slater determinant: \begin{align} \psi(x_1, x_2, x_3) = \frac{1}{\sqrt{3!}} \left( \psi_1(x_1) \psi_2(x_2) \psi_3(x_3) - \psi_1(x_1) \psi_3(x_2) \psi_2 (x_3) - \psi_2(x_1) \psi_1(x_2) \psi_3(x_3) + \psi_2(x_1) \psi_3(x_2) \psi_1(x_3) - \psi_3(x_1) \psi_2(x_2) \psi_1(x_3) + \psi_3(x_1) \psi_1(x_2) \psi_2(x_3) \right) \end{align}

1. three pions --> three indistinguishable bosons: $E_n = 3E_1$.

The wavefunction is \begin{align} \psi(x_1, x_2, x_3) = \psi_1(x_1) \psi_1(x_2) \psi_1(x_3) \end{align}

What I am confused is the third case. I know the proton is a fermion, thus the third case is a problem with three fermions.

I have problems with interpreting these three fermions. Three indistinguishable fermions? two indistinguishable fermion and one different fermion.

Is the ground state energy is the same as case 1? How about the wavefunction?

Also for distinguishable fermions, does the same Pauli's exclusion principle work? (I guess it works). Thus my answer for the third case, the ground state energy is $E_n= E_1 + E_2 + E_3$. But I am confused with determining the wavefunction.

• Don't you need to account for the spin states of the electrons as well? Two electrons can have the same spatial wavefunction if their spin states are already antisymmetric. – Michael Seifert Mar 24 '16 at 13:08
• Check out the Fadeev equation: en.wikipedia.org/wiki/Faddeev_equations – Lewis Miller Mar 24 '16 at 13:19
• Because the electron and proton are not identical, Pauli's exclusion principle doesn't apply, even though they are both fermions as you say. The ground state would have the electrons in levels 1 and 2 (neglecting spin), and the proton in level 1. Note however that the energy of the proton in level 1 may be different from the energy of the electron in level 1, as $E_n$ may depend on the mass of the particle. As for the wavefunction, it is given by the product of the proton wavefunction and the electrons' wavefunction, where the electron wavefunction is appropriately antisymmetrised. – gj255 Mar 24 '16 at 13:56
• Just to be clear, these are non-interacting 'electrons'? – Emilio Pisanty Mar 24 '16 at 14:30
• @EmilioPisanty, Here i did not consider the interaction between particles. Thus yes, i treated them as non-interacting electrons. – phy_math Mar 24 '16 at 14:38