Tensor Product of Hamiltonians

[Woolyabyss]

Homework Statement

Problem statement attached as image

Relevant Equations

Schrodinger equation

$U_1 \otimes U_2 = (1- i H_1 \ dt) \otimes (1- i H_2 \ dt)$

We can write $| \phi_i(t) > \ = U_i(t) | \phi_i(0)>$ where i can be 1 or 2 depending on the subsystem. The $U$'s are unitary time evolution operators.

Writing as tensor product we get
$|\phi_1 \phi_2> = (1- i H_1 \ dt) | \phi_1(0)> \otimes \ (1- i H_2 \ dt) | \phi_2(0)>$

Since these states are normalised we may write

$1 = < \phi_1 \phi_2 | \phi_1 \phi_2 > = < \phi_1(0)| 1 + H^2_1 dt^2 | \phi_1(0)> < \phi_2(0)| 1 + H^2_2 dt^2 | \phi_2(0)>$

This is as far as I've gotten. Any help would be appreciated.

 

[TSny]

Writing as tensor product we get
$|\phi_1 \phi_2> = (1- i H_1 \ dt) | \phi_1(0)> \otimes \ (1- i H_2 \ dt) | \phi_2(0)>$

Try to express the right hand side in the form $\left( \mathbb{I}_1 \otimes \mathbb{I}_2 - i ( ?) dt \right) | \, \phi_1(0) \phi_2(0) \, \rangle$

You will need to fill in the (?) with the appropriate tensor product operator(s). You only want to keep terms up through first order in dt. Looks like you are taking $\hbar = 1$.

Since these states are normalised we may write

$1 = < \phi_1 \phi_2 | \phi_1 \phi_2 > = < \phi_1(0)| 1 + H^2_1 dt^2 | \phi_1(0)> < \phi_2(0)| 1 + H^2_2 dt^2 | \phi_2(0)>$

This is as far as I've gotten. Any help would be appreciated.

You can neglect any terms of order $dt^2$ since you are only expressing things to first order in $dt$. As a result, this will reduce to something that tells you whether or not the state remains normalized as time passes. But this does not really help with constructing the Hamiltonian for the system.