By: Maria Christine L. Lugo, MS Physics I, MSU-IIT   Show that   a.)   b.)    Solution: a.) We can expand by its components and S_{2}^{2},   But we know that, (See link https://www.quantumsciencephilippines.com/?p=5698 )   Therefore,     b.) Using the eigenvalue equation of and substituting the basis   We have,       […]

Orthogonality of Two Eigenvectors by: Mariel A. Escobal, MS Physics I, MSU-IIT Prove that two eigenvectors of a Hermitian operator corresponding to two different eigenvalues are orthogonal. Consider two eigenvectors |ψ⟩ and |φ⟩ of the Hermitian operator Â, Â|ψ⟩ = λ|ψ⟩     (1) Â|φ⟩ = μ|φ⟩     (2) Since Â is Hermitian, we can write (2) as ⟨φ| = μ⟨φ|     (3) Multiplying […]

Prove that the eigenvalues of a Hermitian operator are real. by: Mariel A. Escobal, MS Physics I, MSU-IIT Let Ĥ be a Hermitian operator on an inner product space V over the field of complex numbers . That is Ĥ = Ĥ+. Then, for an eigenvector  |φ⟩ ∈ V,  |φ⟩ ≠ |0⟩, and eigenvalue λ ∈ . Ĥ |φ⟩ = λ |φ⟩ We know for a general operator Â on […]

Properties of Operators by: Mariel A. Escobal, MS Physics I, MSU-IIT Show the following where A & B are operators: a. (A+)+= A b. (λA)+=λ*A+ c. (A + B)+= A+ + B+ d. (AB)+= B+ A+ Solutions: a. (A+)+= A   The adjoint of an operator is defined as: ⟨ψ|Aφ⟩=⟨A+ψ|φ⟩ Now, taking the complex conjugate of […]

  by Jomari L. Tanghal, MS Physics I, MSU-IIT Derive Bhor’s quantization condition from De Broglie’s relation. Solution: Assume that an integral number of wavelengths must fit in the circumference of an orbit. The circumference of the circular orbit must be an integral of wavelength given by,     from De Broglie’s wavelength, . This […]