Tensor product space

Assume you have a system consisting of two spinless particles.  Its wave function is Ψ(r1,r2).  A  special case of such a wave function is a product function of the form Ψ(r1,r2) = Φ(r1)ζ(r2) . The state vector corresponding to such a product is denoted by
|Ψ> = |Φ(1)> ⊗ |ζ(2)> = |Φ(1)>|ζ(2)>.

|Ψ> is a vector in the two-particle space E, while |Φ(1)> and |ζ(2)> are vectors in the spaces E1 and E2 of particle 1 and particle 2, respectively. 
We say that E is the tensor product of E1 and E2 and write E = E1 ⊗ E2
If {|ui>} and {|vj>} are bases for E1 and E2 respectively, then the set of all tensor product vectors {|ui> ⊗ |vj>} is a basis for E.  Every vector in E is a sum of product vectors.  This, however, does not mean that every vector in E is a product vector, only that it can be written as a linear combination of product vectors.  Not every Ψ(r1,r2) is a product function.

The inner product in E satisfies
(<Φ'(1)| ⊗ <ζ'(2)|)(|Φ(1)> ⊗ |ζ(2)>) = <Φ'(1)Φ(1)> <ζ'(2)|ζ(2)>) .
Tensor products occur whenever a system has two or more independent degrees of freedom.


We can regard E = E1 ⊗ E2 as a factoring of E, and it is an important fact that a given space can be factored in many different ways.

Consider the state space of two spinless particles.  We may rewrite Ψ(r1,r2) as a function of
cm = (m1r1 + m2r2)/(m1 + m2) and r = r1 - r2.
Ψ(rcm,r) is a vector in the vector space E =  Er1 ⊗ Er2 = Ecm ⊗ Erel spanned by product functions Φ(r1)ζ(r2) or Φ(rcm)ζ(r).

Certain operators in E are product operators.  If A(1) and B(2) are operators acting in E1 and E2, respectively, we define the product operator A(1) ⊗ B(2) through the relation
(A(1) ⊗ B(2))(|Φ(1)> ⊗ |ζ(2)>) = A(1)|Φ(1)> ⊗ B(2)|ζ(2)>.
If there can be no confusion we write A(1) ⊗ B(2) = A(1)B(2).

Examples of product operators:


Consider a system of two distinguishable particles with mass m which do not interact and which are both placed in an infinite potential well of width a.  Denote by H(1) and H(2) the Hamiltonians of each of the two particles.  Let {|Φn(1)>} and {|Φq(2)>} be an eigenbasis of H(1) and H(2) respectively.  A basis for the global system is  {|Φn(1)> ⊗ |Φq(2)> = |ΦnΦq>}.
(a)  What are the eigenvectors and eigenvalues of the total Hamiltonian?   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>.
What is the state of the system at time t?  If H is measured at time t, what results can be found and with what probabilities? 
If H(1) is measured at time t, what results can be found and with what probabilities?

(c) Show that |Ψ(0)> is a tensor product state. 
Calculate at t = 0  <H1>, <H2>, and <H1H2>. 
Compare <H1><H2> with <H1H2>.  Show that this remains valid at later times.

(d)  Let |Ψ(0)> = (1/√5)|Φ1Φ1> + √(3/5)|Φ1Φ2> + (1/√5)|Φ2Φ1>.
Show that this is not a tensor product state.  Calculate at t = 0  <H1>, <H2>, and <H1H2>.  Compare <H1><H2> with <H1H2>.


In one dimension, consider two particles of mass m, coordinates x1 and x2, momenta p1 and p2, and potential energy
U(x1,x2) = mω2x12 + mω2x22 + gmω2(x1 - x2)2.
Find the eigenvalues and eigenfunctions of the Hamiltonian H of the system.