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Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
Electrical Perturbation (bonus problem, 50 pts) An electron moving in a one-dimensional harmonic potential of frequency ω is experiencing a weak electric field E in x-direction, resulting in the Hamiltonian 5. 2mdx2 2 Calculate the energy to the first non-zero correction using perturbation theory and compare with the exact result of e282 2mu2 Hint: Use ladder operators in H and H
Quantum Mechanics. Consider a one-dimensional harmonic oscillator of frequency found in the ground state. At a perturbation is activated. Obtain an expression for the expected value of as a function of time using time-dependent perturbation theory. A step by step process is deeply appreciated. The best handwriting possible, please. Thank you very much. We were unable to transcribe this imageWe were unable to transcribe this imageV (t) = Fox cos (at) We were unable to transcribe this image V (t)...
4. (20 points). 5-function perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at x--a and x-a. We introduce the following δ-function perturbation at V'(x) 00(z). a. Compute the first-order corrections to the energies of the particle induced by the perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
QUANTUM MECHANICS Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be q, mass be m, and the frequency of the oscillator be u. (a) 79 rat t =-oo, the oscillator is in the ground state 10). A uniform electric field E along x axis is applied betweentoo andtoo with the time dependence of E being given by E(t) ー(t/ア Neglect the induced magnetic field. Find the probability that the oscillator goes to the nth excited...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Answer all of the questions: Reference: (Problem 3, don't need to answer this) (25 pts) Consider a Harmonic Oscillator for a particle of mass m in a potential V = kx?, and with an added perturbation H' = Ex4. a. Find the first order correction to the energy of the nth state. b. Are there non-zero 2nd order corrections to the energy? Explain. c. Are there values for n where the first-order approximation breaks down? Explain physically why this makes...