Eigenvectors of the Hamiltonian

  • Thread starter Ajihood
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Hey guys (this is not a HW problem, just general discussion about the solution that is not required for the assignment),

So I am doing this problem where I had to find the eigenvalues and eigenvectors of the Hamiltonian:

H = A*S[itex]_{x}[/itex][itex]^{2}[/itex] + B*S[itex]_{y}[/itex][itex]^{2}[/itex] + C*S[itex]_{z}[/itex][itex]^{2}[/itex].

Easy enough, just basic linear algebra.

However, I want to interpret what the results I get. So I understand the eigenvalues of H represents the energy levels of the system but what physical interpretation should i take to the eigenvectors?

So by finding the eigenvectors, I find a basis that spans the space I am working in. Why is this important to know? I dont want to lose the forest for the trees and just trying to grapple with why the eigenvectors are important or what they physically mean? (eg energy levels have a physical meaning to me, so the eigenvalues make sense but the eigenvectors seem abstract to me). I know I should follow the math in QM but I like to understand the world, not just apply math tricks... Thanks!
 

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  • #2
181
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Hey guys (this is not a HW problem, just general discussion about the solution that is not required for the assignment),

So I am doing this problem where I had to find the eigenvalues and eigenvectors of the Hamiltonian:

H = A*S[itex]_{x}[/itex][itex]^{2}[/itex] + B*S[itex]_{y}[/itex][itex]^{2}[/itex] + C*S[itex]_{z}[/itex][itex]^{2}[/itex].

Easy enough, just basic linear algebra.

However, I want to interpret what the results I get. So I understand the eigenvalues of H represents the energy levels of the system but what physical interpretation should i take to the eigenvectors?

So by finding the eigenvectors, I find a basis that spans the space I am working in. Why is this important to know? I dont want to lose the forest for the trees and just trying to grapple with why the eigenvectors are important or what they physically mean? (eg energy levels have a physical meaning to me, so the eigenvalues make sense but the eigenvectors seem abstract to me). I know I should follow the math in QM but I like to understand the world, not just apply math tricks... Thanks!
Eigenvectors are state of definite eigenvalue. In the case of Hamiltonian, eigenvectors are states with definite energy. Now, quantum states evolves by the factor [itex]e^{i E t}[/itex] from the Schrodinger's equation. So a state with definite energy evolves by only multiplying a phase factor in front, i.e. A state with definite energy remains a state with definite energy. They do not become "mixed" with other states of other energy (let's suppose states are non-degenerate) as they evolve in time.

That's why eigenvector of H is important and they are called "stationary states" for this reason.
 
  • #3
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Hi, Ajihood.

Please let me know what kind of system are you dealing with this Hamiltonian.

Thank you in advance.
 

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