Download Microscopic states of ideal quantum gases [tln62]

Microscopic states of ideal quantum gases
Hamiltonian: ĤN =
N
X
[tln62]
ĥ` .
`=1
1-particle eigenvalue equation: ĥ` |k` i = ` |k` i.
N -particle eigenvalue equation: ĤN |k1 , . . . , kN i = EN |k1 , . . . , kN i.
Energy: EN =
N
X
` ,
` =
`=1
~2 k2`
.
2m
N -particle product eigenstates: |k1 , . . . , kN i = |k1 i . . . |kN i.
Symmetrized states for bosons: |k1 , . . . , kN i(S) .
1
• N = 2: |k1 , k2 i(S) = √ (|k1 i|k2 i + |k2 i|k1 i).
2
Antisymmetrized states for fermions: |k1 , . . . , kN i(A) .
1
• N = 2: |k1 , k2 i(A) = √ (|k1 i|k2 i − |k2 i|k1 i).
2
Occupation number representation: |k1 , . . . , kN i ≡ |n1 , n2 , . . .i.
Here k1 represents the wave vector of the first particle, whereas n1 refers to
the number of particles in the first 1-particle state.
• energy: Ĥ|n1 , n2 , . . .i = E|n1 , n2 , . . .i,
E=
∞
X
n k k .
k=1
• number of particles: N̂ |n1 , n2 , . . .i = N |n1 , n2 , . . .i,
N=
∞
X
k=1
` : energy of particle `.
k : energy of 1-particle state k.
Allowed occupation numbers:
• bosons: nk = 0, 1, 2, . . .
• fermions: nk = 0, 1.
nk .