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Archive for the 'Quantum Physics' Category

The Recursion Relation for larger n

Wednesday, May 22nd, 2019

By Kim S. Ponce, MS Physics I, MSU-IIT   Problem   Given the recursion relation: . (a) Find    for large . (b) Compare result from (a) to .   Solution   Starting with the recursion relation,       When , the expression above reduces to     Now, we compare this to  which […]

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Matrix representation of the square of the spin angular momentum

Tuesday, May 21st, 2019

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,       […]

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Orthogonality of Two Eigenvectors

Tuesday, May 21st, 2019

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 […]

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Eigenvalues of Hermitian Operators

Tuesday, May 21st, 2019

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 […]

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Properties of Operator

Tuesday, May 21st, 2019

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 […]

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