Tensor Product of Two States

Let F₁(x₁) and F₂(x₂) be solutions to the Schrodinger equation for the Hamiltonians H₁(x₁) and H₂(x₂)  respectively, corresponding to two separate independent parts of system respectively. I.e.

Define the following

Take the derivative of Y with respect to time and multiply by

Multiply H by Y to get

Find the difference between Eq. (5) and Eq. (6)

which reduces to

Therefore Y = F₁(x₁)F₂(x₂) is a solution of Schrodinger's equation corresponding to the Hamiltonian H =  H₁(x₁) + H₂(x₂).