%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Expert Solution

please thumb up

genius_generous answered 1 year ago

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'>.

The Simple harmonic oscillator: A particle of mass m constrained to move in the x-direction only...

The Simple harmonic oscillator: A particle of mass m constrained to move in the x-direction only is subject to a force F(x) = −kx, where k is a constant. Show that the equation of motion can be written in the form d^2x/dt2 + ω^2ox = 0, where ω^2o = k/m . (a) Show by direct substitution that the expression x = A cos ω0t + B sin ω0t where A and B are constants, is a solution and explain the...

Consider a Simple Harmonic Oscillator of mass m moving on africtionless horizontal surface about mean...

Consider a Simple Harmonic Oscillator of mass m moving on a frictionless horizontal surface about mean position 'O'. When the oscillator is displaced towards right by a distance 'x' , an elastic restoring force F = -kx is produced which keeps it oscillating. Keeping in view equations of Simple Harmonic Motion show that the displacement x of the oscillator at time t is given by; x =ASin (ωt + θ). (1) where A is the amplitude of oscillation and (ωt...

Consider a particle that is conﬁned by a one dimensional quadratic (harmonic) potential of the form...

Consider a particle that is conﬁned by a one dimensional quadratic (harmonic) potential of the form U(x) = Ax2 (where A is a positive real number). a) What is the Hamiltonian of the particle (expressed as a function of velocity v and x)? b) What is the average kinetic energy of the particle (expressed as a function of T)? c) Use the Virial Theorem (Eq. 1.46) to obtain the average potential energy of the particle. d) What would the average...

Consider a particle moving in two spatial dimensions, subject to the following potential: V (x, y)...

Consider a particle moving in two spatial dimensions, subject to the following potential: V (x, y) = ( 0, 0 ≤ x ≤ L & 0 ≤ y ≤ H ∞, otherwise. (a) Write down the time-independent Schr¨odinger equation for this case, and motivate its form. (2) (b) Let k 2 = 2mE/~ 2 and rewrite this equation in a simpler form. (2) (c) Use the method of separation of variables and assume that ψ(x, y) = X(x)Y (y). Rewrite...

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.

7. Consider two noninteracting particles in a 1D simple harmonic oscillator (SHO) potential, which has 1-particle...

7. Consider two noninteracting particles in a 1D simple harmonic oscillator (SHO) potential, which has 1-particle spatial wavefunctions ψ n ( x), where n = 0, 1, 2, … (we ignore spin by assuming both particles have the same spin quantum number or are spin 0). These wavefunctions are normalized to 1 and satisfy ψ n * ( x)ψ m ( x)dx −∞ ∞ ∫ = 0 when n ≠ m , i.e., they are orthogonal. The energies are !ω0...

Consider a classical harmonic oscillator of mass m and spring constant k . What is the...

Consider a classical harmonic oscillator of mass m and spring constant k . What is the probability density for finding the particle at position x ? How does this compare to the probability density for the ground state of a quantum mechanical harmonic oscillator

1)Consider a particle that is in the second excited state of the Harmonic oscillator. (Note: for...

1)Consider a particle that is in the second excited state of the Harmonic oscillator. (Note: for this question and the following, you should rely heavily on the raising and lowering operators. Do not do integrals.) (a) What is the expectation value of position for this particle? (b) What is the expectation value of momentum for this particle? (c) What is ∆x for this particle? 2) Consider a harmonic oscillator potential. (a) If the particle is in the state |ψ1> =...

Consider a particle of mass m confined to a one-dimensional box of length L and in...

Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction. For a partide in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why < px2>= n2h2/4L2

Question