A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic osci...
please solve with explanations 3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillator potential ()1ma'. A time dependent uniform electric field E, ()E, os eris 2 applied in the x direction. The particle is in the harmonic oscillator ground state at time a) What is the time dependent perturbation Hamiltonian H'(t) - the potential enegy of the charge in this electric field? b) Find the amplitude ci(t) of finding the particle...
Suppose a particle is in a one-dimensional harmonic oscillator potential. Suppose that a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the particle is in the ground state. Use first order perturbation theory to find the probability that at some time t1 > 0 the particle is in the first excited state of the harmonic oscillator. H' = ext.
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
9.5 A particle of mass m is in the ground state in the harmonic oscillator potential A small perturbation Bx6 is added to this potential (a) What are the units of ?? (b) How small must B be in order for perturbation theory to be valid? (c) Calculate the first-order change in the energy of the particle.
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b) Calculate the energy shift in the ground state and in the degenerate 1t excited state of a 2-dimensional harmonic oscillator H(P2P,2/2m m(x +y)due to the perturbation V 2Axy. (20 pts) (a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b)...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to a 2 weak perturbation: H-ix. You are asked to solve the ground state of the new Hamiltonian - À + in two ways. (a) Solve by using the time-independent perturbation theory. Find the lowest non- vanishing order correction to the energy of the ground state. And find the lowest non vanishing order correction to the wavefunction of the ground state. (b) Find the wavefunction...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator 2 Zm 2 The normalized eigenfunctions for Hare φη (x) with energies E,,-(n + 2) ha. At time t-0 the wavefunction of the particle is given by у(x,0)- (V3іфі (x) + ф3(x)). Now let H' be an operator given by where k is a positive constant. 1) Show that H' is Hermitian. 2) Express H' by the step-operators a+ and a 3) Calculate the...
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state and first excited state respectively. (a) Write the normalization condition for lv(o) and determine the mean value (H) of the energy in terms of co and ci. (b) With the additional requirement (H)-ho. calculate eoand o,p. [4] Consider a harmonic oscillator...