Two fermions systems – Wave function (1)

Two fermions in an infinite square wel. The wave function.

Consider two fermions in an infinite square well of width L. On is in the n=1 state and one has been excited to the n=4 state.

a) Write down the full state $|ψ_{12}>$, including the spatial wave function and spin states for the single spin state and the triplet state.

a)

The spatial total wave function is a linear combination of the individual spatial wave functions of the two given electrons. To obtain the full state wave function, we will multiply this spatial function with the two electron total spin wave function.

The individual spatial wave functions for particles on 1 and 4 energy levels are

$φ_1 (x)=\sqrt{2/L}*\sin(π/L*x)$ and $φ_2 (x)=\sqrt{2/L}*\sin⁡(4π/L*x)$

The spatial part of the combined wave function can be either

$ψ_{12}^S (x_1,x_2 )=|ψ_12^S>=1/sqrt{2} *[φ_1 (x_1 ) φ_2 (x_2 )+φ_1 (x_2 ) φ_2 (x_1 ) ]$ symmetric

$ψ_{12}^A (x_1,x_2 )=|ψ_12^A>=1/√2 [φ_1 (x_1 ) φ_2 (x_2 )-φ_1 (x_2 ) φ_2 (x_1 )]$ antisymmetric

For Fermions there are possible two distinctive states: the singlet state (anti-symmetric) |0,0> and the triplet state (symmetric): |1,M> where M=-1,0 or 1. These states (singlet or triplet) can be written either in the uncoupled basis ($m_{s1,2}=±1/2$)

$|m_{s1},m_{s2}>$ that is $|++>,|+->,|-+>,|- ->$

or in the coupled basis ($S=S_1+S_2$ and $m_s=m_{s1}+m_{s2}$)

$|S,m_s>$ that is |$11>, |10>, |1,-1>, |00>$

The correspondence between the states written in the two above bases is

$\left\{\begin{matrix} |11>=|++>\\ |10>=(1/\sqrt{2})*(|+->+|-+>) \\ |11>=|–> \end{matrix}\right.$

for Symmetric triplet |1M> and

$|00>=(1/\sqrt{2})*(|+->-|-+>)$  for Antisymmetric singlet |00>

For Fermions the total combined wave function is always anti-symmetric so that it can be either

$|ψ_{12}^{SA}>=1/\sqrt{2}*(2/L)*[\sin⁡((π/L)*x_1 )*\sin⁡(4(π/L)*x_2 )+$

$+\sin⁡((π/L)*x_2 )*\sin⁡((4π/L)*x_1 ) ]*|0,0>$

$|ψ_{12}^{AS}>=(1/\sqrt{2})*(2/L)*[sin⁡(π/L*x_1 )*sin⁡(4π/L*x_2 )-$

$-sin⁡(π/L*x_2 )*sin⁡(4π/L*x_1 ) ]*|1,M>$

with M=1,0 or-1. The above first wave function corresponds to singlet states existence in the quantum well and second wave function corresponds to triplet states existence in the quantum well.