Definition of action of quantum hamiltonian, two state system

In summary, the conversation discusses a two-state system with orthonormal states |1> and |2>. The Hamiltonian of the system is represented by a 2x2 matrix and the question asks for the action of H on the two states. The meaning of "action" in this context is not clear and further clarification is needed.
  • #1
theidiot
2
0

Homework Statement



Consider a two-state system. We denote the two orthonormal states by |1>and |2>. The Hamiltonian of the system is given by a 2 × 2 matrix:
[omitted in this post, has 4 entries of course, not very interesting]
Write the action of H on the states |1> and |2>.

2. The attempt at a solution

Well, I don't know what is meant by action here! No doubt it's simple but I need to know what 'action' means in this context. Thanks for assistance!
 
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  • #2
Just to clarify, 'H' is the Hamiltonian and is represented by a 2x2 (hermitian) matrix. I have not explicitly written out the matrix in these posts because there is no need. No reply yet, so just thought to clarify that the 'H' referred to is the matrix of the Hamiltonian. What is meant by its 'action' on the two orthonormal states of the two state system?
 

FAQ: Definition of action of quantum hamiltonian, two state system

What is the definition of action in quantum mechanics?

In quantum mechanics, action is defined as the integral of the Lagrangian over time, which represents the total energy of a system. It is a fundamental concept in quantum mechanics and plays a crucial role in the equations that describe the behavior of quantum systems.

What is a quantum Hamiltonian?

A quantum Hamiltonian is an operator that represents the total energy of a quantum system, including both its kinetic and potential energy. It is a key mathematical concept in quantum mechanics and is used to describe the evolution of a quantum system over time.

How does the quantum Hamiltonian act on a two state system?

In a two state system, the quantum Hamiltonian acts as a mathematical representation of the energy levels of the system. It is a matrix operator that describes the transitions between the two states and can be used to calculate the probability of a system being in a particular state at a given time.

What is the role of the quantum Hamiltonian in the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. The quantum Hamiltonian plays a central role in this equation, as it represents the total energy of the system and is used to calculate the wave function, which describes the probability of finding the system in a particular state at a given time.

How is the quantum Hamiltonian related to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that it is impossible to simultaneously know the precise position and momentum of a particle. The quantum Hamiltonian plays a role in this principle, as it is related to the particle's momentum through the Schrödinger equation. This means that changes in the quantum Hamiltonian can lead to uncertainties in the particle's position and momentum, as described by the uncertainty principle.

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