A system is in an eigenstate |m, l) of the angular momentum operators L2 and L2. Calculate the expectation values of (L) and (L2), and (L,) and (L).
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- a) Show explicitly (by calculation) that the <p> = <p>* is fulfilled for the expectation value of themomentum. b) The three expressions xp, px and (xp+px)/2 are equivalent in classical mechanics.Show that for corresponding quantum mechanical operators in the orders shown, that <Q> = <Q>* isfulfilled by one of these operators, but not by the other two.Using the eigenvectors of the quantum harmonic oscillator Hamiltonian, i.e., n), find the matrix element (6|X² P|7).Plot the first three wavefunctions and the first three energies for the particle in a box of length L and infinite potential outside the box. Do these for n = 1, n = 2, and n = 3
- Suppose that the wave function for a system can be written as 4(x) = √3 4 · Φι(x) + V3 2√₂ $2(x) + 2 + √3i 4 $3(x) and that 1(x), 2(x), and 3(x) are orthonormal eigenfunc- tions of the operator Ekinetic with eigenvalues E₁, 2E₁, and 4E₁, respectively. a. Verify that (x) is normalized. b. What are the possible values that you could obtain in measuring the kinetic energy on identically prepared systems? c. What is the probability of measuring each of these eigenvalues? d. What is the average value of Ekinetic that you would obtain from a large number of measurements?The operator în · ở measures spin in the direction of unit vector f = (nx, Ny, N₂) nx = sin cosp ny = sinesino nz = cose in spherical polar coordinates, and ở = (x, y, z) for Pauli spin matrices. (a) Determine the two eigenvalues of û.o.