Question

In: Physics

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 x) x > L

Which of the following statements is not correct for this solution?

C cos(-k2 L) + D sin(k2 L) = F exp(-k3 L)

The wave function is continuous at x=0 and x=L    

k_1=sqrt(2m( U0-E ))/hbar

k1=k3

B=0

The first derivative of the wave function is continuous at x=0 and x=L

Solutions

Expert Solution


Related Solutions

Calculate all of the energy levels for an electron in the finite potential well of width...
Calculate all of the energy levels for an electron in the finite potential well of width a) L = 10 Å, b) L = 50 Å, c) L = 100 Å and L = 1000 Å using the actual mass of an electron for the conduction band of the AlGaAs/GaAs/AlGaAs quantum well. Repeat problem using a) the effective mass of an electron in GaAs (electron effective mass meff = 0.067*mass of an electron)
An electron is bound to a finite potential well. (a) If the width of the well...
An electron is bound to a finite potential well. (a) If the width of the well is 4 a.u., determine numerically the minimum depth (in a.u.) such that there are four even states. Give the energies of all states including odd ones to at least 3 digits. (b) Repeat the calculation, but now keep the depth of the well at 1 a.u., determine the minimum width (in a.u.)
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
Estimate the ground state for an electron confined to a potential well of width 0.200 nm...
Estimate the ground state for an electron confined to a potential well of width 0.200 nm and height 100 eV. What is the effective well width of the (infinite) well? (Hint: Consider an iterative approach to approximate the penetration depth ? by initially assuming E?V).
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...
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
Consider a particle of mass ? in an infinite square well of width ?. Its wave...
Consider a particle of mass ? in an infinite square well of width ?. Its wave function at time t = 0 is a superposition of the third and fourth energy eigenstates as follows: ? (?, 0) = ? 3i?­3(?)+ ?­4(?) (Find A by normalizing ?(?, 0).) (Find ?(?, ?).) Find energy expectation value, <E> at time ? = 0. You should not need to evaluate any integrals. Is <E> time dependent? Use qualitative reasoning to justify. If you measure...
Consider an electron confined in a one-dimensional infinite potential well having a width of 0.4 nm....
Consider an electron confined in a one-dimensional infinite potential well having a width of 0.4 nm. (a) Calculate the values of three longest wavelength photons emitted by the electron as it transitions between the energy levels inside the well [3 pts.]. (b) When the electron undergoes a transition from the n = 2 to the n = 1 level, what will be its emitted energy and wavelength [2 pts.]. To which region of the electromagnetic spectrum does this wavelength belong?...
Consider an electron confined to an infinite well of width 2ao (Bohr radius). Compare its energy...
Consider an electron confined to an infinite well of width 2ao (Bohr radius). Compare its energy in its three lowest energy states to the three lowest energy states of the Bohr model of the hydrogen atom. Repeat for a proton confined to a well of width 2x10-15 m (a nucleus).
An electron is confined by some potential energy well centered about the origin, and is represented...
An electron is confined by some potential energy well centered about the origin, and is represented by the wave function ψ(x) = Axe−x2/L2, where L = 4.48 nm. The electron's total energy is zero. (a) What is the potential energy (in eV) of the electron at x = 0? eV (b) What is the smallest value of x (in nm) for which the potential energy is zero? nm
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT