Construct the explicit form of the lowest three eigenfunctions of the harmonic oscillator.

Expert Solution

genius_generous answered 1 year ago

1) What is ? for a harmonic oscillator (in terms of the period)?

1) What is ? for a harmonic oscillator (in terms of the period)? 2) What is ? for a spring? A) If you increase the amplitude, what happens to the period? B) If you increase the mass, what happens to the period? C) If you increase the spring constant, what happens to the period? 3) What is ? for a simple pendulum? A) If you increase the amplitude, what happens to the period? B) If you increase the mass,...

Consider a three-dimensional isotropic harmonic oscillator for which the Hamiltonian is given by H = p2...

Consider a three-dimensional isotropic harmonic oscillator for which the Hamiltonian is given by H = p2 2m+ 1/2mω2r2. Use the variational method with the trial function u(r) = 1πa2 3/4 exp(−r2/2a2) and obtain E. Minimizing E with respect to a2, show that the upper bound for the ground-state energy reproduces the exact result for the energy given by a =(mω and Ea = 32ω. Substitute the above value of a in the trial function and show that it also reproduces...

consider Three-Dimensional harmonic oscillator with the same frequencies along all three directions. a) determine the wave...

consider Three-Dimensional harmonic oscillator with the same frequencies along all three directions. a) determine the wave function and the energy of the ground state. b) how many quantum numbers are needed to describe the state of oscillation? c) the degeneracy of the first excited state. express the wave function involved in the schrodinger equation as a product given by x, y, z and separate the variables.

Solve schroedinger's equation for a three dimensional harmonic oscillator and obtain its eigen values and eigen...

Solve schroedinger's equation for a three dimensional harmonic oscillator and obtain its eigen values and eigen functions.Are the energy levels degenerate? Explain what is the minimum uncertainty in its location in the lowest state.

Consider a system of three non-interacting particles confined by a one-dimensional harmonic oscillator potential and in...

Consider a system of three non-interacting particles confined by a one-dimensional harmonic oscillator potential and in thermal equilibrium with a total energy of 7/2 ħw. (a) what are the possible occupation numbers for this system if the particles are bosons. (b) what is the most probable energy for a boson picked at random from this system.

Solve the quantum simple harmonic oscillator in three dimensions i.e. find the energy and eigenkets. It's...

Solve the quantum simple harmonic oscillator in three dimensions i.e. find the energy and eigenkets. It's solution will include hermite polynomials.

Method A: Based on the frequency of a simple harmonic oscillator. The angular frequency of a...

Method A: Based on the frequency of a simple harmonic oscillator. The angular frequency of a mass on a spring is given by ω=(k/m)1/2ω=(k/m)1/2 where m is the mass and k is the spring constant. The period of a harmonic oscillator is T=2π/ωT=2π/ω . If you do a little math, you can get a formula for the spring constant in terms of the mass and the period. a) Design a procedure to measure the spring constant based on Method A...

As in the figure below, a simple harmonic oscillator is attached to a rope of linear...

As in the figure below, a simple harmonic oscillator is attached to a rope of linear mass density 5.4 ✕ 10−2 kg/m, creating a standing transverse wave. There is a 3.5-kg block hanging from the other end of the rope over a pulley. The oscillator has an angular frequency of 44.1 rad/s and an amplitude of 255.0 cm. (a) What is the distance between adjacent nodes? m (b) If the angular frequency of the oscillator doubles, what happens to the...

a) a simple harmonic oscillator consists of a glass bead attached to a spring. it is...

a) a simple harmonic oscillator consists of a glass bead attached to a spring. it is immersed in a liquid where it loses 10% of its energy over 10 periods. compute the coefficient of kinetic friction b assuming spring constant k=1 N/m and mass m=10 grams; b) suppose the spring is now disconnected and the bead starts failing in the liquid, accelerating until it reaches the terminal velocity. compute the terminal velocity.

Solve the quantum harmonic oscillator problem by using the matrix method.

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