Computational Physics

Exercises 369

the last equation can be written in the more compact form:

ρS|uα〉=

∑

α′

〈uα|ρS|uα′〉|uα′〉.

Show that the last equation can be satisfied by requiring that|uα〉is a normalised

eigenvector of the operatorρS.

Also show that the above error is minimised by taking the eigenvectors

corresponding to the largest eigenvalues of the density operator.

(d) The previous analysis was a bit sloppy as the minimisation is subject to the

constraint that

∑

αnDαn|uαn〉be normalised, and that the|uα〉are orthonormal.

These constraints should be taken into account through Lagrange parameters.

Show that this analysis leads to the conclusion that the|uα〉form an invariant set

under the action ofρS, very similar to what we have seen in the derivation of the

Hartree–Fock formalism (see Section 4.5.2). Similar to the analysis there, we

see that a set of eigenvalues ofρScan therefore be chosen as a basis.

11.5 We now approach the same problem as in the previous exercise from another point
of view. We consider again the Hilbert spaces of the system S, the environment E
and the universe U which is the system and environment together. For simplicity, we
take all coefficients and basis functions to be real.
Suppose U is in a pure state|ψ〉. To this state there corresponds a density matrix
ρUwhich can be reduced to a density matrixρSof S or a density matrixρEof E. We
callλαthe eigenvalues ofρSandλβthose of E. The corresponding eigenstates are
denoted|uα〉and|vβ〉respectively.
Obviously we can write the states|ψ〉in terms of the eigenstates|uαvβ〉:
|ψ〉=

∑

αβ

Cαβ|uαvβ〉.

(a) Derive the following equations:

λα=

∑

β

C^2 αβ

and

λβ=

∑

α

C^2 αβ.

By consideringCαβas a matrix, show that the set of numbersλαmust be equal

to the setλβ. In the following, we shall denote both byλα.

(b) Show that|ψ〉can be expanded as

|ψ〉=

∑

α

λα|uαvα〉.

and that the eigenvalues ofρUarewα=λ^2 α. Show that the error in the

representation of|ψ〉on S by using a truncated set of eigenstates as a basis is

given as the sum over the discarded weightswα.