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• ### 3. A particle of mass m in a one-dimensional box has the following wave function in...

3. A particle of mass m in a one-dimensional box has the following wave function in the region x-0 tox-L: ? (x.r)=?,(x)e-iEy /A +?,(X)--iE//h Here Y,(x) and Y,(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are the energies of these levels. The wave function is zero for x< 0 and forx> L. (a) Find the value of the probability distribution function atx- L/2 as a function of...

• ### Particle in a box. (a) Let H=L?([0,L]) (square integrable wave functions on the interval 0 <...

Particle in a box. (a) Let H=L?([0,L]) (square integrable wave functions on the interval 0 < x <L). Show that the wave functions Yn(x) = eilanx/L, n=0,1,-1,2, -2,... (6) form an orthonormal system in H. Is this system a basis? (b) Show that the wave functions Yn are eigenstates of the momentum operator p on H= L?([0,L]). Hence, show that the variance Ap in the state Yin vanishes. What is the variance Ať in the state Yn? Why is the...

• ### 3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave...

3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave functions Ψ(z, t) which are periodic functions of z: Ψ(z,t) = Ψ(z + L, t). a) Solve the Time-Independent Schroedinger equation. For each allowed energy, En, you will find two solutions, (s). Why does this not contradict the theorem that we proved in class about the non-degeneracy of the solutions to the TISE in one dimension? b) Start with the initial condition Ψ(z,0) sin2(nz/L)....

• ### 8. (a) Consider the following contour plots of the wave functions of a 2-D particle (mass-m)...

8. (a) Consider the following contour plots of the wave functions of a 2-D particle (mass-m) in a box area L. What are the energies of these two states? 60 40 100 (b) Is the transition between these two state allowed or forbidden?

• ### 1. Suppose we didn't actually know the wave functions for a particle in a box. Reasonable...

1. Suppose we didn't actually know the wave functions for a particle in a box. Reasonable guesses for the ground- and first-excited-state wave functions might be functions of the form 1 = a y (1 - y) 02 = by (y-c)(y-1), where y = (x/L), L is the length of the box, and a, b, and care constants. (a) These functions have quite a number of features that make them sensible guesses. Sketch both functions and list these special features....

• ### Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as...

Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...

• ### A particle in a box of length L. has energy E1- 2.0 eV. The same particle...

A particle in a box of length L. has energy E1- 2.0 eV. The same particle in a box of length Lb has energy E,-50.0 eV. a. What is the ratio La/ Lb? Show your working. The figure below shows a standing de Broglie wave b. Does this standing wave represent a particle that travels back and forth between the boundaries with a constant speed or a changing speed? Support your answer with an appropriate explanation. c. If the speed...

• ### 8. Consider one electron in a 1D box of side L. Its wavefunction is given by...

8. Consider one electron in a 1D box of side L. Its wavefunction is given by из where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, 2m dx2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is Ψ(x) an eigenfunction of A? If it is an eigenfunction, what is the 9. A linear polyene contains 8 -electrons, and absorbs light with412 nm. b)...

• ### Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave fu...

Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave function is given by o)-A. Find the normalization constant A function in this basis. Solve for the coeffici You may find useful the integrals in the front of the (6 pt) d) Now consider the finite potential spherical well with V(r)- ing only the radial part...

• ### 1. A particle in an infinite square well has an initial wave function Alsin sin 4 0 < x < L other...

help on all a), b), and c) please!! 1. A particle in an infinite square well has an initial wave function Alsin sin 4 0 < x < L otherwise s(x, t = 0) 0 (a) Find A so that the wavefunction is normalized. (b) Find '(z,t). (c) Find the expectation value(E) of the energy of ψ(x,t = 0). You may use the result mx n 2 0 1. A particle in an infinite square well has an initial wave...