# How Do You Express the Tensor Product of Hamiltonians?

• Woolyabyss
So, in summary, the conversation discusses expressing a state as a tensor product and finding a way to express the right hand side in a specific form to determine if the state remains normalized over time. However, this does not provide much help in constructing the Hamiltonian for the system.
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.

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Woolyabyss said:
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.

Woolyabyss

## What is a Tensor Product of Hamiltonians?

A Tensor Product of Hamiltonians is a mathematical operation that combines two or more Hamiltonians to form a new Hamiltonian. It is commonly used in quantum mechanics to describe the dynamics of a composite system.

## How is the Tensor Product of Hamiltonians calculated?

The Tensor Product of Hamiltonians is calculated by taking the tensor product of the individual operators that make up the Hamiltonians. This involves multiplying the matrices that represent the operators and then taking the Kronecker product of the resulting matrices.

## What is the significance of the Tensor Product of Hamiltonians in quantum mechanics?

The Tensor Product of Hamiltonians is significant in quantum mechanics because it allows us to describe the dynamics of a composite system, where the individual subsystems are described by their own Hamiltonians. This is particularly useful in understanding entanglement and other quantum phenomena.

## Can the Tensor Product of Hamiltonians be applied to any type of Hamiltonian?

Yes, the Tensor Product of Hamiltonians can be applied to any type of Hamiltonian, as long as the operators are compatible and can be multiplied together. This includes both finite and infinite-dimensional systems.

## Are there any limitations or drawbacks to using the Tensor Product of Hamiltonians?

One limitation of the Tensor Product of Hamiltonians is that it can become computationally expensive when dealing with large systems, as the size of the resulting Hamiltonian matrix grows exponentially with the number of subsystems. Additionally, it may not always accurately describe the dynamics of highly entangled systems.

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