Solve the following questions relating to the Harmonic Oscillator. d.e.Confirm that the state (x) obeys the uncertainty relation forpotential and kinetic energy given by σTσv ≥ |(4₁(x) |⁄2 [Î‚Ñ]| 4(x))|.2iThe Virial Theorem applied to the Harmonic Oscillator demands thatthe average kinetic energy plus the average potential energy is equal to the totalenergy. Confirm that the Virial Theorm holds for the ground state of theHarmonic Oscillator if you equate the average potential energy to (V) and theaverage kinetic energy to (T). 1.The following questions will relate to the HarmonicOscillator with the following Hamiltonian in a system of natural units where k = 1, µ =1,ħ = 1:Ĥ :=1 d² 12 d x² + z ²With ground-state wavefunction given by:Yo (x)- G) ₁=x²2

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Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.
Calculate the energy, corrected to first order, of a harmonic oscillator with potential: