Question 1. Correlations between two particles

(It is recommended that the complement E₁₁₁ be read in order to answer question e of this exercise.)

Consider a physical system formed by two particles (I) and (2), of the same mass m, which do not interact with each other and which are both placed in an infinite potential well of width a (cf. complement HI, § 2-c). Denote by H(1) and H(2) the Hamiltonians of each of the two particles and by |ψ_I(t)> and |ψ_q(2)> the corresponding eigenstates of the first and second particle, of energies and n²Π²h²/2ma² and n²Π²h²/2ma². In the state space of the global system, the basis chosen is composed of  the states |ψ_nψ_q> defined by:

a. What are the eigenstates and the eigenvalues of the operator H - H(I) + H(2), the total Hamiltonian of the system? Give the degree of degeneracy of the two lowest energy levels.

b. Assume that the system, at time t = 0 is in the state:

|ψ(0)> = 1/√6 |φ1φ1> + 1/√3 |φ1φ2> + 1/√6 |φ2φ1> + 1/√3 |φ2φ2>

a. What is the state of the system at time t?

β. The total energy H is measured. What results can be found, and with what probabilities?

γ. Same questions if, instead of measuring H, one measures H(1).

c. a. Show that |ψ(0)> is a tensor product state. When the system is in this state, calculate the following mean values : < H(1) >, < H(2) > and < H(1)H(2) >.

Compare < H(1) > < H(2) > with < H(1 )H(2) > ; how can this result be explained?

β. Show that the preceding results remain valid when the state of the system is the state |ψ(t)> calculated in b.

d. Now assume that the state |ψ(0) is given by:

|ψ(0> = 1/√5 |φ1φ1> + √(3/5)|φ1φ2> + 1/√5|φ2φ1>

Show that |ψ(0)> cannot be put in the form of a tensor product. Answer for this case all the questions asked in c.

e. a. Write the matrix, in the basis of the vectors |φ1φ2>, which represents the density operator p(0) corresponding to the ket |ψ(0) given in b. What is the density matrix p(t) at time t? Calculate, at the instant t = 0, the partial traces:

p(1) = Tr₂p and p(2) = T_r1p

Do the density operators ρ, ρ(1) and ρ(2) describe pure states? Compare ρ with ρ(1) x ρ(2); what is your interpretation?

β. Answer the same questions as in a, but choosing for |ψ(0)> the ket given in d.

The subject of the following exercises is the density operator: they therefore assume the concepts and results of complement E_III to be known.

Question 2. Let ρ be the density operator of an arbitrary system, where |Χ₁> and Π₁ are the eigenvectors and eigenvalues of p. Write p and p² in terms of the |x₁ and Π₁. What do the matrices representing these two operators in the { |Χ₁> } basis look like - first, in the case where p describes a pure state and then, in the case of a statistical mixture of states?

(Begin by showing that, in a pure case, ρ has only one non-zero diagonal element, equal to 1, while for a statistical mixture, ρ has several diagonal elements included between 0 and 1.) Show that ρ corresponds to a pure case if and only if the trace of ρ² is equal to 1.

Question 3. Consider a system whose density operator is ρ(t), evolving under the influence of a Hamiltonian H(t). Show that the trace of ρ² does not vary over time.

Conclusion : can the system evolve so as to be successively in a pure state and a statistical mixture of states?

Question 4. Let (1) + (2) be a global system, composed of two subsystems (1) and (2). A and B denote two operators acting in the state space ε(1) ⊗ ε(2). Show that the two partial traces Tr₁, { AB } and Tr₁, { BA } are equal when A (or B) actually acts only in the space g(1), that is, when A (or B) can be written :

A = A(1) ⊗ 1(2) [or B = B(1) ⊗ 1(2)].

Application : if the operator H, the Hamiltonian of the global system, is the sum of two operators which act, respectively, only in ε(1) and only in ε(2):

H = H(1) + H(2),

calculate the variation d/dt ρ(1) of the reduced density operator ρ(1). Give the physical interpretation of the result obtained.