Interaction Hamiltonian coupling question

In summary, the conversation discusses the composition of a system with two qubits and a bath with one qubit. The interaction Hamiltonian is given by the equation $$\sigma_1^x\otimes B_1 + \sigma_2^x\otimes B_2$$ where $$B_i$$ is a 2 by 2 matrix. The question also asks if this is equivalent to the equation $$(\sigma_1^x\otimes B_1)\otimes I_2 + I_1\otimes(\sigma_2^x\otimes B_2)$$ and if, in the case where $$B_1=B_2=B$$, it can be written as $$(
  • #1

td21

Gold Member
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8
System is composed of two qubits and the bath is one bath qubit.

The interaction Hamiltonian is:

$$\sigma_1^x\otimes B_1 + \sigma_2^x\otimes B_2$$ where $$B_i$$ is a 2 by 2 matrix.

I try to interpret and understand this, is it the same as:
$$(\sigma_1^x\otimes B_1)\otimes I_2 + I_1\otimes(\sigma_2^x\otimes B_2)~?$$If the situation is that two system qubits are coupled to the bath qubit in the same way, such that $$B_1= B_2 = B$$, may I write the the interaction hamiltonian as:

$$(\sigma_1^x\otimes B + \sigma_2^x\otimes B)\stackrel{?}{=} (\sigma_1^x\otimes B)\otimes I_2 + I_1\otimes(\sigma_2^x\otimes B)\stackrel{?}{=}(\sigma_1^x + \sigma_2^x)\otimes B$$
 
  • #3
td21 said:
is it the same as:
$$(\sigma_1^x\otimes B_1)\otimes I_2 + I_1\otimes(\sigma_2^x\otimes B_2)~?$$

Yes.

td21 said:
If the situation is that two system qubits are coupled to the bath qubit in the same way, such that $$B_1= B_2 = B$$, may I write the the interaction hamiltonian as:

$$(\sigma_1^x\otimes B + \sigma_2^x\otimes B)\stackrel{?}{=} (\sigma_1^x\otimes B)\otimes I_2 + I_1\otimes(\sigma_2^x\otimes B)\stackrel{?}{=}(\sigma_1^x + \sigma_2^x)\otimes B$$

Yes. Tensor product is distributive.
 

1. What is an interaction Hamiltonian coupling?

An interaction Hamiltonian coupling is a term used in quantum mechanics to describe the interaction between two or more quantum systems. It is a mathematical expression that represents the energy exchange between these systems, and it plays a crucial role in understanding the behavior of particles and their interactions.

2. How is the interaction Hamiltonian coupling calculated?

The calculation of the interaction Hamiltonian coupling involves finding the overlap between the wavefunctions of the systems involved. This can be done using various mathematical techniques, such as matrix multiplication or perturbation theory. The resulting value represents the strength of the interaction between the systems.

3. What factors influence the strength of the interaction Hamiltonian coupling?

The strength of the interaction Hamiltonian coupling is influenced by several factors, including the distance between the systems, the nature of the particles involved, and the energy levels of the systems. Additionally, the type of interaction (e.g., electromagnetic, nuclear, etc.) can also impact the strength of the coupling.

4. What is the significance of the interaction Hamiltonian coupling in quantum mechanics?

The interaction Hamiltonian coupling is a crucial concept in quantum mechanics as it helps us understand the behavior of particles and their interactions. It allows us to predict the outcomes of quantum processes and can aid in the development of new technologies, such as quantum computing.

5. Can the interaction Hamiltonian coupling be experimentally measured?

Yes, the interaction Hamiltonian coupling can be experimentally measured using various techniques, such as spectroscopy or quantum state tomography. These methods involve manipulating and measuring the quantum systems involved to determine the strength of their interaction. Such measurements can provide valuable insights into the behavior of particles and their interactions.

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