Question

Consider a system with just two linearly independent states |1? = ?1? and |2? = ?0? ?0? ?1? and the Hamiltonian matrix given by ? = ? u v ?. ? v u ? Solve the time-independent Schrödinger equation ?|?? = E|?? by showing that the eigenvalues are E± = u ± v (using the characteristic equation) and the normalized eigenvectors of ? are |?+? = 1/?2 ?1? and |?-? = 1/?2 ? 1 ?. ?1? ?-1?

          Consider a system with just two linearly independent states

|1? = ?1? and |2? = ?0?
      ?0?           ?1?

and the Hamiltonian matrix given by

? = ? u v ?.
    ? v u ?

Solve the time-independent Schrödinger equation ?|?? = E|?? by showing that the eigenvalues are E± = u ± v (using the characteristic equation) and the normalized eigenvectors of ? are

|?+? = 1/?2 ?1? and |?-? = 1/?2 ? 1 ?.
              ?1?               ?-1?

Consider a system with just two linearly independent states

|1? = ?1? and |2? = ?0?
?0? ?1?

and the Hamiltonian matrix given by

? = ? u v ?.
? v u ?

Solve the time-independent Schrödinger equation ?|?? = E|?? by showing that the eigenvalues are E± = u ± v (using the characteristic equation) and the normalized eigenvectors of ? are

|?+? = 1/?2 ?1? and |?-? = 1/?2 ? 1 ?.
?1? ?-1?

Added by William J.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Madhur L

Step 1

First, we need to find the characteristic equation of the Hamiltonian matrix H. The characteristic equation is given by det(H - EI) = 0, where E is the eigenvalue and I is the identity matrix. det(H - EI) = det([1-E, 2; 4, 3-E]) = (1-E)(3-E) - 8 = E^2 - 4E - 5 = Show more…

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Consider a system with just two linearly independent states (|1) and |2) and the Hamiltonian matrix given by: H = [41) /2] Solve the time-independent Schrodinger equation H|ψ⟩ = E|ψ⟩ by showing that the eigenvalues are E1 = 4 and E2 = 1 (using the characteristic equation) and the normalized eigenvectors of H are: |ψ+⟩ = [1/√2 1/√2] and |ψ-⟩ = [-1/√2 1/√2]