Question

In: Physics

Consider the particle-in-a-box problem in 1D. A particle with mass m is confined to move freely...

Consider the particle-in-a-box problem in 1D. A particle with mass m is confined to move freely between two hard walls situated at x = 0 and x = L. The potential energy function is given as

(a) Describe the boundary conditions that must be satisfied by the wavefunctions ψ(x) (such as energy eigenfunctions).

(b) Solve the Schr¨odinger’s equation and by using the boundary conditions of part (a) find all energy eigenfunctions, ψn(x), and the corresponding energies, En.

(c) What are the allowed values of the quantum number n above? How did you decide on that?

(d) What is the de Broglie wavelength for the ground state?

(e) Sketch a plot of the lowest 3 levels’ wavefunctions (ψn(x) vs x). Don’t forget to mark the positions of the walls on the graphs.

(f) In a transition between the energy levels above, which transition produces the longest wavelength λ for the emitted photon? What is the corresponding wavelength?

Expert Solution

genius_generous answered 2 years ago

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

Consider a particle of mass m that can move in a one-dimensional box of size L...

Consider a particle of mass m that can move in a one-dimensional box of size L with the edges of the box at x=0 and x = L. The potential is zero inside the box and infinite outside. You may need the following integrals: ∫ 0 1 d y sin ⁡ ( n π y ) 2 = 1 / 2 , for all integer n ∫ 0 1 d y sin ⁡ ( n π y ) 2 y = 1...

The particle has a mass of 0.6 kg and is confined to move along the smooth...

The particle has a mass of 0.6 kg and is confined to move along the smooth horizontal slot due to the rotation of the arm OA. Assume the particle contacts only one side of the slot at any instant. The arm has an angular acceleration of θ¨ = 3 rad/s2 when θ˙ = 2 rad/s at θ = 30∘. (Figure 1) Part A Determine the magnitude of the force of the rod on the particle when θ = 30∘. Express...

1. Consider a particle of mass m in a box of length L with boundaries at x...

1. Consider a particle of mass m in a box of length L with boundaries at x = 0 and x = L. At t = 0 the wavefunction is where A is the normalization constant. (a) Determine the basis eigen states for a particle in the box. (b) Determine the normalization constant A. (c) Determine the probability of finding the particle in the ground state at t ≠ 0. (d) Show that the sum of probabilities of finding the...

3. Assume that a particle is confined to a box of length L, and that the...

3. Assume that a particle is confined to a box of length L, and that the system wave function is ψ(x)=sqrt(2/L)*sin[(π*x)/(L)] (1) Is this state an eigenfunction of the momentum operator? Show your work. (2) Calculate the average value of the momentum <p> that would be obtained for a large number of measurements. Explain your result. (3) Calculate the probability that the particle is found between 0.31 L and 0.35 L.

A particle of mass m is confined to a finite potential energy well of width L....

A particle of mass m is confined to a finite potential energy well of width L. The equations describing the potential are U=U0 x<0 U=0 0 < x < L U=U0 x > L Take a solution to the time-independent Schrodinger equation of energy E (E < U0) to have the form A exp(-k1 x) + B exp(k1 x) x < 0 C cos(-k2 x) + D sin(k2 x) 0 < x < L F exp(-k3 x) + G exp(k3...

particle of mass m, which moves freely inside an infinite potential well of length a, is...

particle of mass m, which moves freely inside an infinite potential well of length a, is initially in the state Ψ(x, 0) = r 3 5a sin(3πx/a) + 1 √ 5a sin(5πx/a). (a) Normalize Ψ(x, 0). (b) Find Ψ(x, t). (c) By using the result in (b) calculate < p2 >. (d) Calculate the average energy

a- Consider an electron that moves freely in metal and a particle moves freely in vacuum....

a- Consider an electron that moves freely in metal and a particle moves freely in vacuum. Briefly explain differences between the energy of these particles. b- Briefly explain the temperature dependent conductivity of metal, semiconductor and insulator. (Please write your answer by asking your self “How and Why change” questions )

Consider a very small particle of mass, m, dropped in a fluid. The particle experiences a...

Consider a very small particle of mass, m, dropped in a fluid. The particle experiences a drag force, FD. The constant that relates the drag force to the velocity V is K. Determine the distance covered as the partcile accelerate from rest to 50 percent of its terminal velocity, Vt, in terms of K, m, and acceleration due to gravity, g.

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...