QuantumPhysics.dvi

and insert the productI 1 ⊗I 2 ⊗···⊗INin betweenEi− 1 andEifor alli. The problem now

becomes one of 2×2 matrix multiplication. To see this, we work out the matrix elements of

Eithat enter here,

Tσ′,σ=〈σ′|Ei|σ〉= exp

{

βJσ′σ+

1

2

βB(σ+σ′)

}

(8.77)

The partition function is then given by

Z=

∑

{σi=± 1 }

Tσ 1 ,σ 2 Tσ 2 ,σ 3 ···TσN,σ 1 = trTN (8.78)

Written out explicitly, the matrixTis given by

T=

(

eβJ+βB e−βJ

e−βJ eβJ−βB

)

(8.79)

Its eigenvaluesλ±satisfy the equation,

λ^2 ±− 2 λ±eβJch(βB) + 2 sh(2βJ) = 0 (8.80)

which is solved by

λ±=eβJch(βB)±

√

e^2 βJsh^2 (βB) +e−^2 βJ (8.81)

Therefore, the partition function is given by

Z=λN++λN− (8.82)

for all values ofN,β,J, andB.

In statistical mechanics and thermodynamics, we are mostly interested in taking the

thermodynamics limit of the system, which here corresponds to takingN→∞. The number

of sitesN plays the role of volume, and it is then more appropriate to consider the large

volume limit of intensive quantities, such as the free energy per unit volume etc. Thus, we

shall be interested in the limit,

f= lim

N→∞

F

N

=−

1

β

lim