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- The wavefunction of is Ψ(x) = Axe−αx2/2 for with energy E = 3αℏ2/2m. Find the bounding potential V(x). Looking at the potential’s form, can you write down the two energy levels that are immediately above ??The wavefunction of is Ψ(x) = Axe(−ax^2)/2 for with energy E = 3aℏ2/2m. Find the bounding potential V(x). Looking at the potential’s form, can you write down the two energy levels that are immediately above E?Find the wave function and energy for the infinite-walled well problemCould you explain it to me step by step and in detail?Thanks a lot
- Draw an energy level diagram for a nonrelativistic particle confined inside a three-dimensional cube-shaped box, showing all states with energies below 15· (h2/8mL2). Be sure to show each linearly independent state separately, to indicate the degeneracy of each energy level. Does the average number of states per unit energy increase or decrease as E increases?If you double the width of a one-dimensional infinite potential well, (a) is the energy of the ground state of the trapped electron multiplied by 4, 2, , , or some other number? (b) Are the energies of the higher energy states multiplied by this factor or by some other factor, depending on their quantum number?Calculate the expectation value of x2 in the state described by ψ = e -bx, where b is a ħ constant. In this system x ranges from 0 to ∞.
- For a particle in a box of length L sketch the wavefunction corresponding to the state with the lowest energy and on the same graph sketch the corresponding probability density. Without evaluating any integrals, explain why the expectation value of x is equal to L/2.A free electron has a kinetic energy 13.3eV and is incident on a potential energy barrier of U =32.1eV and width w =0.091nm. What is the probability for the electron to penetrate this barrier (in %)? Check the correct answer and show all work1. a. For a free particle, write the relations between the wave vector k and itsmomentum vector p and angular frequency ω and its energy E.b. What is the general form in one dimension of the wave function for a freeparticle of mass m and momentum p?c. Can this wave function ever be entirely real? If so, show how this ispossible. If not, explain why not.d. What can you say about the integral of the |Ψ (x; t)|^2 from - ∞ to + ∞ ?e. Is this a possible wave function for a real, physical particle? Explain whyor why not.
- Consider the wavefunction for a particle in a one-dimensional box when the level is n = 6. Calculate the total probability of finding the particle between x = 0 and x = L/12? Provide your answer to three significant figures.Please don't provide handwritten solution ..... Determine the normalization constant for the wavefunction for a 3-dimensional box (3 separate infinite 1-dimensional wells) of lengths a (x direction), b (y direction), and c (z direction).The spherical harmonics are the eigenfunctions of ?̂2 and ?̂ ? for the rigid rotor and thehydrogen atom (and other spherically symmetric problems). In this problem, we willexamine the nature of the angular nodes for these systems.Since the spherical harmonics include a factor of eim, which never has magnitude zero, forthis exercise we will construct some linear combinations of the spherical harmonics so weare working with real-valued functions. Two of the real-valued spherical harmonics are:12 (?1−1 + ?11) = 12 √ 32? sin ? cos ? 12? (?32 − ?3−2) = 14 √1052? sin2 ? cos ? sin 2?(a) Determine the angles at which nodal surfaces will occur for each of these functions, anddescribe the nature of the nodal surfaces that they represent. In other words, identifythe locations of nodal planes and other surfaces in the Cartesian axis system.(b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what isthe total number of distinct angular nodal surfaces?