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- A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in spherical polar coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)Show that the total energy eigenfunctions ψ210(r, θ, φ) and ψ211(r, θ, φ) are orthogonal. Doyou have to integrate over all three variables to show this?Obtain the value of the Lagrange multiplier for the particle above the bowl given by x^2+y^2=az
- Let f(x)= 4xex - sin(5x). Find the third derivative of this function. Note ex is denoted as e^x below. Select one: (12+4x^3)e^x + 125sin(5x) 12e^x + 125cos(5x) not in the list (12+4x)e^x + 125cos(5x) (8+4x)e^x + 25sin(5x)1. a. For a free particle, write the relations between the wave vector k and itsmomentum vector p and angular frequency ω and its energy E.b. What is the general form in one dimension of the wave function for a freeparticle of mass m and momentum p?c. Can this wave function ever be entirely real? If so, show how this ispossible. If not, explain why not.d. What can you say about the integral of the |Ψ (x; t)|^2 from - ∞ to + ∞ ?e. Is this a possible wave function for a real, physical particle? Explain whyor why not.The Hamiltonian of a spin in a constant magnetic field B aligned with the y axis is given by H = aSy, where a is a constant. a) Use the energies and eigenstates for this case to determine the time evolution psi(t) of the state with initial condition psi(0) = (1/root(2))*matrix(1,1). (Vertical matrix, 2x1!) b) For your solution from part (a), calculate the expectation values <Sx>, <Sy>, <Sz> as a function of time. I have attached the image of the orginial question!
- Consider a particle of spin s = 3/2. (a) Find the matrices representing the operators S^ x , S^ y ,S^ z , ^ Sx 2 and ^ S y 2 within the basis of ^ S 2 and S^ z (b) Find the energy levels of this particle when its Hamiltonian is given by ^H= ϵ 0 h 2 ( Sx 2−S y 2 )− ϵ 0 h ( S^ Z ) where ϵ 0 is a constant having the dimensions of energy. Are these levels degenerate? (c) If the system was initially in an eigenstate Ψ0=( 1 0 0 0) , find the state of the system at timeThe spherical harmonics are the eigenfunctions of ?̂2 and ?̂ ? for the rigid rotor and thehydrogen atom (and other spherically symmetric problems). In this problem, we willexamine the nature of the angular nodes for these systems.Since the spherical harmonics include a factor of eim, which never has magnitude zero, forthis exercise we will construct some linear combinations of the spherical harmonics so weare working with real-valued functions. Two of the real-valued spherical harmonics are:12 (?1−1 + ?11) = 12 √ 32? sin ? cos ? 12? (?32 − ?3−2) = 14 √1052? sin2 ? cos ? sin 2?(a) Determine the angles at which nodal surfaces will occur for each of these functions, anddescribe the nature of the nodal surfaces that they represent. In other words, identifythe locations of nodal planes and other surfaces in the Cartesian axis system.(b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what isthe total number of distinct angular nodal surfaces?Using the condition (3.027) of Lect. 16, prove that the mo- mentum operator p is Hermitian. HINT: Use the periodic boundary conditions for the functions g(r) and s(x).