Composite Systems - QM Homework: Equations & Solution

In summary, the conversation is discussing the appropriate form of the interaction hamiltonian for two oscillating particles connected by a weak spring. The question also asks about the commutativity of HA and Hint, and the physical significance of this. The conversation concludes that if HA and Hint commute, common eigenstates can be found, otherwise the complete hamiltonian must be solved. The correctness of the expression for Hint is confirmed to be incorrect.
  • #1
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Homework Statement



Please see attached problem


Homework Equations





The Attempt at a Solution



Ok so I am just stuck on the bit that asks us to write down the appropriate form of the interaction hamiltonian H int for two oscillating particles connected by a weak spring. Is it 1/2 k (xa - xb) ^2?

It asks whether HA commutes with Hint. Not sure about this>? What would be the physical significance?
 

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  • #2
H_A contains a derivative. H_int contains a polynomial.
Do you think they can commute? You can work it out.

If H_A and H_int commute, and that the total hamiltonian is their sum, you can find common eigenstates between H_A and H_int.
Else, no, you'll have to solve the complete hamiltonian.
 
  • #3
Is my expression for H_int correct?

I guess they don't commute then?
 
  • #4
yes.

yes.
 
  • #5



Hello there,

Thank you for reaching out for help with this problem. It looks like you are working on a composite system in quantum mechanics, which can be a complex topic. Let's break down the problem and see if we can understand it better.

First, the problem asks you to write down the appropriate form of the interaction Hamiltonian, denoted as Hint. This Hamiltonian describes the interaction between two oscillating particles connected by a weak spring. The equation you have written, 1/2 k (xa - xb)^2, is the correct form for this interaction Hamiltonian. This represents the potential energy of the spring, where k is the spring constant and xa and xb are the position operators for the two particles.

Next, the problem asks whether the Hamiltonian for the entire system, denoted as HA, commutes with Hint. This means whether the two operators can be measured simultaneously without affecting each other's measurements. In this case, the physical significance is that the two particles are not interacting strongly enough for their measurements to affect each other. This is an important concept in quantum mechanics and can have implications for how we understand and describe complex systems.

I hope this helps clarify the problem for you. If you have any further questions or need more assistance, please don't hesitate to ask. Remember, in science, it's important to take the time to fully understand a problem before moving on to the solution. Keep up the good work!
 

What is a composite system in quantum mechanics?

A composite system in quantum mechanics refers to a system made up of two or more individual systems, each of which may have its own quantum state. The composite system has its own state that is a combination of the individual states of the subsystems.

How is the state of a composite system described mathematically?

The state of a composite system is described mathematically using the tensor product of the individual states of the subsystems. This is represented by the symbol ⊗ (tensor product symbol).

What is the difference between a pure state and a mixed state in composite systems?

A pure state in a composite system refers to a state where the individual systems are in a definite state and the overall state is a superposition of these individual states. A mixed state, on the other hand, refers to a state where the individual systems are in a mixture of states and the overall state cannot be described by a single wavefunction.

What is entanglement in composite systems?

Entanglement in composite systems refers to a phenomenon where the quantum state of one subsystem depends on the state of the other subsystem, even if they are separated by a large distance. This correlation between the subsystems is a unique feature of quantum mechanics and cannot be explained by classical physics.

How are measurements performed on composite systems?

Measurements on composite systems are performed by measuring the properties of each individual subsystem separately. The results of these measurements can then be used to determine the state of the composite system as a whole.

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