Quantum Mechanics Question

Consider the result of two consecutive rotation of a vector, first at the angle of 30 degrees then at the angle of 60 degrees. Do it first using multiplication of the respective matrices, and then by using a matrix representing a single rotation by angle of 90 degrees.

Quantum Mechanics Question

Consider linear transformation acting on a 3-d displacement vector, which reflects a vector with respect to its origin. Find a matrix T, which describes this transformation.

Show that eigenvectors of a unitary transformation belonging to distinct eigenvalues are orthogonal.

a) Show that eigenvalues of a unitary transformation have modulus 1. b) Show that eigenvectors of a unitary transformation belonging to distinct eigenvalues are orthogonal.

Consider lowering and rising operators that we encountered in the harmonic oscillator problem.

Consider lowering and rising operators that we encountered in the harmonic oscillator problem. 1. Show that a+ = 2. Find eigenvalues and eigenfunctions of the lowering operator 3. Does the rising operator have normalizable eigenfunctions?

Find the momentum-space wave function

Find the momentum-space wave function See attached file for full problem description.