Consider if [Lx, A] = 0 and [Ly, A] = 0 where A is an operator and Lx and Ly are components of angular momentum operator. Obtain the commutator relation of [L₂, A].
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![Consider if [Lx, A] = 0 and [Ly, A] = 0 where A is an operator and L¸ and Ly are
components of angular momentum operator. Obtain the commutator relation of [L₂, A].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F251ca70b-a9fc-49ea-9e37-40284bdaa92b%2F68e192db-6e66-481c-a90e-edf8cba037cc%2Fz7jo9yn_processed.png&w=3840&q=75)

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- The three matrix operators for spin one satisfy sz Sy – Sy 8z = isz and cyclic permutations. Show that s = sz, (sz tisy )³ = 0. For the sane in, m, can have the values im, m – 1, .., -m, while A12 has eigenvalue m(m + 1). Thus M? = m(m + 1) ±2 5 x 1 = 5 times each once 6 15 4 x 8 = 32 times +3/2 ±1/2 3/2 each 8 times 4 ±1 3 x 27 = 81 times 1 each 27 times 3 2 x 18 = 96 times 1/2 ±1/2 each 48 times 1 × 12 = 42 times 0, each 42 times Total 256 eigenvalues A certain state | 4) is an eigenstate of L? and L,: L'|v) = 1(l+ 1) h² |»), mh|v) . For this state calculate (La) and (L²).Using the eigenvectors of the quantum harmonic oscillator Hamiltonian, i.e., n), find the matrix element (6|X² P|7).A diatomic molecule can be modeled as a rigid rotor with moment of inertia I and an electric dipole moment d along the axis of the rotor. The rotor is constrained to rotate in a plane, and a weak uniform electric field & lies in the plane. Write the classical Hamiltonian for the rotor, and find the unperturbed energy levels by quantizing the angular-momentum operator. Then treat the electric field as a perturbation, and find the first nonvanishing corrections to the energy levels.
- Please help meCalculate the 2nd order energy shift to the ground state energy of the one-dimensional harmonic oscillator, when a perturbation of the form H₁ = Є · (²) is added to the original Hamiltonian Ho = p²/2m+ ½ mw²x². Take a ⇒ (ħ/mw) ¹/2, the characteristic length scale of the oscillator. The second order correction to level n is given by E(2) = Σ m#n ||| H₁|v0| |2 m E(0) - EO)Spin/Field Hamiltonian Consider a spin-1/2 particle with a magnetic moment µ = -e/m$ placed in a uniform magnetic field aligned along the z axis. (a) Write the Hamiltonian for this system in matrix form. (b) Verify by explicit matrix calculation that the Hamiltonian does not commute with the spin operators in the r and y directions. Comment on how this affects the expectation values of these operators.
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- 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.Consider the spherical harmonic having l = 2, me = -2. (a) Show that it is an eigenfunction of the operator , for the projection of the angular momentum onto the axis of quantization (the z-axis). What is the eigenvalue? (b) What is the magnitude of the angular momentum for this state in units of h?