Question

In: Physics

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 the exact ground-state wave function.

Solutions

Expert Solution

The trial wave function:

The expectation of energy:

For each of the components:

The total expectation energy from all components:

Therefore,

This is exactly the ground state energy of a 3D Harmonic oscilator:

And the wave function:

This is also exactly the ground state wave function of the harmonic oscillator.


Related Solutions

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.
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.
Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P...
Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P 2 + V (X) where X and P are the position and linear momentum operators, and they satisfy the commutation relation: [X, P] = i¯h The eigenvectors of H are denoted by |φn >; where n is a discrete index H|φn >= En|φn > (a) Show that < φn|P|φm >= α < φn|X|φm > and find α. Hint: Consider the commutator [X, H] (b)...
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 particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)=...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)= 1/2 mω^2 (x^2+y^2 ) a. Use separation of variables in Cartesian coordinates to solve the Schroedinger equation for this particle. b. Write down the normalized wavefunction and energy for the ground state of this particle. c. What is the energy and degeneracy of each of the lowest 5 energy levels of this particle? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose a particle of mass m and charge q is in a one-dimensional harmonic oscillator potential...
Suppose a particle of mass m and charge q is in a one-dimensional harmonic oscillator potential with natural frequency ω0. For times t > 0 a time-dependent potential of the form V1(x) = εxcos(ωt) is turned on. Assume the system starts in an initial state|n>. 1. Find the transition probability from initial state |n> to a state |n'> with n' ≠ n. 2. Find the transition rate (probability per unit time) for the transition |n>→|n'>.
(4) Consider a system described by the Hamiltonian H, H = 0 a a 0 !,...
(4) Consider a system described by the Hamiltonian H, H = 0 a a 0 !, where a is a constant. (a) At t = 0, we measure the energy of the system, what possible values will we obtain? (b) At later time t, we measure the energy again, how is it related to its value we obtain at t = 0 ? (c) If at t = 0, the system is equally likely to be in its two possible...
Construct the explicit form of the lowest three eigenfunctions of the harmonic oscillator.
Construct the explicit form of the lowest three eigenfunctions of the harmonic oscillator.
2. For a system of non-interacting, one-dimensional, distinguishable classical particles in a harmonic oscillator potential, V...
2. For a system of non-interacting, one-dimensional, distinguishable classical particles in a harmonic oscillator potential, V = kx2 in contact with a particle reservoir with chemical potential µ and a thermal reservoir at temperature T. (a) Calculate the grand partition function Z for the system. Note that there is no fixed "volume" for this system. (b) Obtain N (number of particles) and U as functions of µ and T and show that U satisfies the equipartition theorem
Demonstrate that the ground-state wave function for the one-dimensional harmonic oscillator satisfies the appropriate Schrodinger's equation
Demonstrate that the ground-state wave function for the one-dimensional harmonic oscillator satisfies the appropriate Schrodinger's equation
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT