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Commutation

Commutation. The most important property of operators. operators. |> and <| are vectors and represent STATES Operators represent VARIABLES and OBSERVABLES (measurables) there are operators for Energy momentum position, etc.

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Commutation

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  1. Commutation The most important property of operators

  2. operators |> and <| are vectors and represent STATES Operators represent VARIABLES and OBSERVABLES (measurables) there are operators for Energy momentum position, etc In CM, observables commute  it doesn’t matter which one we measure first (the order of the measurement is not important) IN QM, observables NOT always commute it matters which one we measure first (the order of the measurement is FUNDAMENTAL)

  3. Adjoint operators Adjoint = Complex Conjugate

  4. Self-Adjoint operators Self-Adjoint operators are also called HERMITIAN.

  5. The eigenvalue-eigenvector equation For some kets |>, when an operator is applied to the |>, the same ket is obtained, multiplied by a number  eigenvalue-eigenvector eq.

  6. Theorems of Hermitian operators Eigenvalues of Hermitian operators are real numbers

  7. 2 Two eigenvectors of an Hermitian operator with two  eigenvalues are orthogonal

  8. More properties

  9. Expansion of a vector Complete set: any vector obeying the same boundary conditions can be expressed by using this set of vectors

  10. Expectation value If we know the eigenket of an operator, it is easy to know what will be the outcome of a measurement, but what happens if the state (ket) is not and eigenstate of that operator?

  11. cont Lets first do it for a superposition of 2 states

  12. cont We can now do the generalization

  13. Superposition of states Any superposition of independent eigenvectors having the same eigenvalue is also an eigenvector of the operator with the same eigenvalue

  14. Eigenfunctions of commuting operators If there is a complete set {Gi} which are eigenvectors of two operators, then the two operators commute

  15. cont If two HERMITIAN operators commute, we can select a common complete set of eigenvectors for them We will not prove the degenerate case

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