How exactly does diagonalization work and how is it useful in qm?

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SUMMARY

Diagonalization is a crucial mathematical process used in quantum mechanics (QM) to solve the stationary Schrödinger equation, represented as H | Ψ_n ⟩ = E_n | Ψ_n ⟩. This equation allows for the determination of eigenvectors | Ψ_n ⟩ and eigenvalues E_n, which correspond to the allowed energy spectrum of the Hamiltonian operator H. The Hamiltonian, being a linear operator, can be expressed as a matrix, transforming the problem into a standard matrix diagonalization challenge. Understanding diagonalization is essential for grasping more complex concepts such as nonabelian anomalies in quantum physics.

PREREQUISITES
  • Familiarity with the stationary Schrödinger equation
  • Understanding of linear operators in quantum mechanics
  • Knowledge of matrix representation and diagonalization techniques
  • Basic concepts of eigenvalues and eigenvectors
NEXT STEPS
  • Study the stationary Schrödinger equation in detail
  • Learn about linear operators and their applications in quantum mechanics
  • Explore matrix diagonalization methods and algorithms
  • Investigate the implications of eigenvalues and eigenvectors in quantum systems
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Students and professionals in physics, particularly those focusing on quantum mechanics, as well as mathematicians interested in linear algebra applications in physics.

captain
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how exactly does it work and how is it useful in qm?
 
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All these questions you keep asking are found either on wikipedia or in any textbook on the subject.

Also, its sort of pointless to explain what nonabelian anomalies are if you don't know what diagonalization means and is used for. There's a separation of about 4 years worth of undergrad material before the latter makes sense.

So if you truly are interested in all this physics material, I highly recommend starting from the beginning and working your way through step by step. Otherwise its all going to be hopelessly opaque and serves no purpose other than to clutter the board up.
 
captain said:
how exactly does it work and how is it useful in qm?

The stationary Schroedinger equation

[tex]H | \Psi_n \rangle = E_n|\Psi_n \rangle[/tex]...(1)

is used to find eigenvectors [itex]| \Psi_n \rangle[/itex] and eigenvalues [itex]E_n[/itex] (allowed energy spectrum) of the Hamiltonian [itex]H[/itex]. Since Hamiltonian [itex]H[/itex] is a linear operator, it can be represented by a matrix in an appropriate basis. Then eq. (1) becomes the traditional matrix diagonalization problem.

Eugene.
 

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