2019 May 21 | Quantum Science Philippines

## Archive for May 21st, 2019

### 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,       […]

### 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|Â = μ⟨φ|     (3) Multiplying […]

### 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 […]

### 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 […]

### Derive Bhor’s quantization condition from De Broglie’s relation.

Tuesday, May 21st, 2019

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 […]

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