Finding a matrix representation of a Hamiltonian.

In summary, a matrix representation of a Hamiltonian is a mathematical representation of the Hamiltonian operator in matrix form. It is important because it allows us to solve the Schrödinger equation and determine the energy levels of a physical system. To find a matrix representation of a Hamiltonian, one can use the commutation relations between position and momentum operators or the ladder operator method. The applications of this representation include calculating energy levels and wave functions, studying quantum systems, and developing quantum computing algorithms.
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Homework Statement


The Hamiltonian H for a certain physical quantum mechanical system has three eigenvectors {|v1>, |v2>, |v3>} satisfying:
H|vj> = (2-j)a|vj>

Write down the matrix representing H in the representation {|v1>, |v2>, |v3>} .

Homework Equations

The Attempt at a Solution


I though the Hamiltonian would just be the eigenstates along the diagonal, however this seems too simple.
a 0 0
0 0 0
0 0 -a​
 
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That's correct.
 
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1. What is a matrix representation of a Hamiltonian?

A matrix representation of a Hamiltonian is a mathematical representation of the Hamiltonian operator in matrix form. The Hamiltonian operator is a mathematical operator used in quantum mechanics to describe the total energy of a physical system.

2. Why is it important to find a matrix representation of a Hamiltonian?

Finding a matrix representation of a Hamiltonian is important because it allows us to solve the Schrödinger equation and determine the energy levels of a physical system. This is crucial in understanding the behavior and properties of quantum systems.

3. How do you find a matrix representation of a Hamiltonian?

To find a matrix representation of a Hamiltonian, you first need to express the Hamiltonian operator in terms of the position and momentum operators. Then, you can use the commutation relations between these operators to transform the Hamiltonian into matrix form.

4. Are there different methods for finding a matrix representation of a Hamiltonian?

Yes, there are different methods for finding a matrix representation of a Hamiltonian. One approach is to use the ladder operator method, which involves creating a set of raising and lowering operators to transform the Hamiltonian into matrix form. Another method is to use the matrix representation of position and momentum operators and apply the commutation relations directly.

5. What are some applications of finding a matrix representation of a Hamiltonian?

The matrix representation of a Hamiltonian has many applications in quantum mechanics. It is used to calculate the energy levels and wave functions of a system, which can provide insight into the behavior of atoms, molecules, and other quantum systems. It is also used in the development of quantum computing algorithms and in the study of quantum field theory.

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