13 4 points The mass.frequency and amplitude of one oscillator are each twice that of a...
One bullet has twice the mass of a second bullet. If both are fired so that they have the same speed, which has more kinetic energy? What is the ratio of the kinetic energies of the two bullets? Options: - The one with the greatest mass has more kinetic energy but the reason for these energies cannot be predicted - the one with twice the mass has greater kinetic energy and the reason for these energies is equal 2. -...
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...
A spring is attached to the ceiling and pulled 18 cm down from equilibrium and released. After 4 seconds the amplitude has decreased to 4 cm. The spring oscillates 13 times each second. Assume that the amplitude is decreasing exponentially. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t. Preview Get help: Video License Points possible: 1 This is attempt 1 of 3
A spring is attached to the...
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...
A particle with mass m is in a one dimensional simple harmonic oscillator potential. At timet0 it is described by the superposition state where Vo, 1 and Vz are normalised energy eigenfunctions of the harmonic oscillator potential corresponding to energies Eo, E1 and E2 (a) Show that the wavefunction is normalised (b) If an observation of energy is made, what is the most likely value of energy and with what probability would it be obtained? (c) If the experiment is...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
4 A nonlinear oscillator Consider a perturbed harmonic oscillator. Using x p2 H + ke? 2 + ex4 2m 1. write this Hamiltonian in terms of â and at 2. At what frequency or frequencies could this system absorb radiation if € = 0, i.e. the oscillator is unperturbed 1 3. Qualitatively, what do the states look like for the perturbed Hamiltonian? Write the new states as a sum of the unperturbed states, without worrying too much about the amplitude...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
10.0 points 003 A "seconds" pendulum is one that moves through its equilibrium position once each second. The length of a seconds pendulum is 0.9921 m at Tokyo and 0.9966 m at Cam- bridge, England. What is the Tokyo: Cambridge ratio of the free-fall acceleration at these two locations, respectively? 004 (part 1 of 2) 10.0 points The frequency of oscillation of the wave emit- ted by an FM radio station is 98.7 MHz. Find the period of vibration. Radio...
6. Fluids (16 points). At one point in a vertical pipeline the water s d is 3.00 m/s and the pressure is the line, 12.0 m lower than the first, the diameter of the pipe is at the first (Pwater 1000 kg/m3 the second point. (b, points) Find the pressure at the second point. 5.00 × 105 Pa. At a second point in twice that kg/m3). ( a, 8 points) Find the water's speed at