notes 1

ξ1= (~r1, σ1)ξ2= (~r2, σ2)
ψ(ξ1, ξ2) = φa(ξ1)φb(ξ2)
φa,b
P12
P12ψ(ξ1, ξ2)def =ψ(ξ2, ξ1) = φa(ξ2)φb(ξ1)6=ψ(ξ1, ξ2)
ψ(ξ1, ξ2)ψ(ξ2, ξ1)
ψ(ξ1, ξ2) = eıθψ(ξ2, ξ1)
P2
12ψ(ξ1, ξ2) = P12ψ(ξ2, ξ1) = ψ(ξ1, ξ2)
±1
P12ψ(ξ1, ξ2) = ψ(ξ2, ξ1)
P12ψ(ξ1, ξ2) = −ψ(ξ2, ξ1)

ψ+(ξ1, ξ2) = N+[φa(ξ1)φb(ξ2) + φa(ξ2)φb(ξ1)]
N+
ψ−(ξ1, ξ2) = N−[φa(ξ1)φb(ξ2)−φa(ξ2)φb(ξ1)]
ψ±
H=H1+H2
=p2
1
2m+V(x1)
| {z }
H1
+p2
2
2m+V(x2)
| {z }
H2
φn(x)H1H2
ψn1n2(x1, x2) = φn1(x1)φn2(x2)
H
Hψn1n2(x1, x2)=[H1+H2]φn1(x1)φn2(x2)
= [H1φn1(x1)] φn2(x2)+[H2φn2(x2)] φn1(x1)
=En1φn1(x1)φn2(x2) + En2φn1(x1)φn2(x2)
= [En1+En2]φn1(x1)φn2(x2)
= [En1+En2]ψn1n2(x1, x2)
φaφbN+= 1/√2

Hixi
ψn1n2(x1, x2)
R2
ψn1n2(x1, x2)
ψ+
n1n2(x1, x2) = ([φn1(x1)φn2(x2) + φn1(x2)φn2(x1)] /√2n16=n2
φn1(x1)φn1(x2)n1=n2
n H s
Sz
def
=~σz/2
Szχs=sχs.
χss
±~/2
χ↑=|↑i =1
0
χ↓=|↓i =0
1
n s
φs
n(x) = φn(x)χs
n1=n2=n s1=s2=s
s16=s2n16=n2
ψ−
n1n2s1s2(x1, x2) = 1
√2[φn1(x1)χs1φn2(x2)χs2−φn1(x2)χs2φn2(x1)χs1]

V(x)
H=H1+H2
=p2
1
2m+1 2mω2x2
1
| {z }
H1
+p2
2
2m+1 2mω2x2
2
| {z }
H2
φn(x)
φ0(x) = s1
σ√πexp −x2
2σ2
φ1(x) = s2
σ3√πxexp −x2
2σ2
σ2=~/mω
ψ00(x1, x2) = 1
σ√πexp −x2
1+x2 2
2σ2
ψ10(x1, x2) = φ1(x1)φ0(x2) = √2
σ2√πx1exp −x2
1+x2 2
2σ2
ψ01(x1, x2) = φ0(x1)φ1(x2) = √2
σ2√πx2exp −x2
1+x2 2
2σ2
ψ+
00(x1, x2) = ψ00(x1, x2)
=1
σ√πexp −x2
1+x2 2
2σ2
ψ+
10(x1, x2) = 1
√2[ψ10(x1, x2) + ψ01(x1, x2)]
=1
σ2√π(x1+x2) exp −x2
1+x2 2
2σ2

s1=−s2
ψ−
00↑↓(x1, x2) = ψ00(x1, x2)1
√2|↑i|↓i − |↓i|↑i
=1
σ√2πexp −x2
1+x2 2
2σ2|↑i|↓i − |↓i|↑i
φ0(x)φ1(x)
ψ−
10↑↑(x1, x2) = 1
√2[ψ10(x1, x2)−ψ01(x1, x2)] |↑i|↑i
=1
σ2√π(x1−x2) exp −x2
1+x2 2
2σ2|↑i|↑i
ψ−
10↓↓(x1, x2) = 1
√2[ψ10(x1, x2)−ψ01(x1, x2)] |↓i|↓i
=1
σ2√π(x1−x2) exp −x2
1+x2 2
2σ2|↓i|↓i
ψ−
10↑↓(x1, x2) = 1
√2[ψ10(x1, x2)|↑i|↓i − ψ01(x1, x2)|↓i|↑i]
=1
σ2√π(x1|↑i|↓i − x2|↓i|↑i) exp −x2
1+x2 2
2σ2
ψ−
10↓↑(x1, x2) = 1
√2[ψ10(x1, x2)|↓i|↑i − ψ01(x1, x2)|↑i|↓i]
=1
σ2√π(x1|↓i|↑i − x2|↑i|↓i) exp −x2
1+x2 2
2σ2
x1x2
X y
Xdef =x1+x2
2
Ydef =x1−x2