# Tensor Product of Hamiltonians

#### Woolyabyss

Problem 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.

#### Attachments

• 36.8 KB Views: 77
Related Advanced Physics Homework News on Phys.org

#### TSny

Homework Helper
Gold Member
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.

"Tensor Product of Hamiltonians"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving