QM Two-State System: Determine Eigenvalues & Eigenvectors of A

Finally, use the eigenvectors and eigenvalues to write the general state |phi> and solve for the coefficients C1 and C2 using the normalization condition. In summary, the states of a system in QM can be described by linear superpositions of two linearly independent state vectors, psi1 and psi2, which are normalized but not orthogonal to each other. To find the eigenvalues and eigenvectors of a hermitian operator A acting on these states, one must first determine the eigenvalues and eigenvectors in the psi1, psi2 basis, then normalize the eigenvectors using the normalization condition. This can then be used to write the general state |phi> and solve for the coefficients C1 and C2
  • #1

Homework Statement


QM states of a system are described by linear super positions of two linearly independent state vectors psi1 and psi2. These two states are normalized but are NOT orthogonal to each other. A hermitian operator A actes on the two states in the following way.

A|psi1> = 5|psi1>+3|psi2>
A|psi2> = -3|psi1> - 5|psi2>

Determine the eigenvalues and properly normalized eigenvectors of A.


The Attempt at a Solution



I attempted to introduce a general state |phi> = C1|psi1> + C2|psi2>, act A on |phi> and apply the normalization condition on |phi> to determine C1 and C2 but have gotten nowhere. I have found the value of <psi2|psi1>. Thanks for your help.
 
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  • #2
First find the eigenvalues and eigenvectors of A in the psi1, psi2 basis. The eigenvectors must be orthogonal since the operator is supposed to be hermitian. Now using that psi1 and psi2 are normalized, you should be able to normalize the eigenvectors.
 

1. What is a QM Two-State System?

A QM Two-State System refers to a quantum mechanical system that can exist in two distinct states. It is a simplified model used to study the behavior of particles in quantum mechanics.

2. How do you determine the eigenvalues of A in a QM Two-State System?

To determine the eigenvalues of A, you need to solve the characteristic equation det(A-λI)=0, where A is the matrix representing the system and λ is the eigenvalue. The solutions to this equation will give you the eigenvalues of A.

3. What are eigenvectors in a QM Two-State System?

Eigenvectors in a QM Two-State System are the vectors that do not change direction when multiplied by the matrix representing the system. They are associated with the eigenvalues and can be used to describe the state of the system.

4. How do you determine the eigenvectors of A in a QM Two-State System?

To determine the eigenvectors of A, you need to substitute the eigenvalues obtained from the characteristic equation into the equation (A-λI)x=0 and solve for x. The solutions to this equation will give you the eigenvectors of A.

5. Why are the eigenvalues and eigenvectors important in a QM Two-State System?

The eigenvalues and eigenvectors of A are important because they provide information about the energy levels and the state of the system. They also help in solving mathematical equations and understanding the behavior of particles in quantum mechanics.

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