For the potential well shown below, make a qualitative sketch of the two energy eigenstate wave functions whose energies are indicated (Note that E1,3 represent the ground state and the 2nd excited state respectively.) Your sketch must show E3 Vo E₁ 0 a b X
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- a particle is confined to move on a circle's circumference (particle on a ring) such that its position can be described by the angle ϕ in the range of 0 to 2π. This system has wavefunctions in the form Ψm(ϕ)= eimlϕ where ml is an integer. Show that the wavefunctions Ψm(ϕ) with ml= +1 and +2 are ORTHOGONAL Show full and complete procedure. Do not skip any stepFind the gradient field F = Vo for the potential function o below. P(x,y.z) = In (3x +y° +z²) Vøxy.z) = (O Vo(x.y.z z)3DShow that the following wave function is normalized. Remember to square it first. Limits of integration go from -infinity to infinity. DO NOT SKIP ANY STEPS IN THE PROCEDURE
- Exercise 6.4 Consider an anisotropic three-dimensional harmonic oscillator potential acy = { m (w² x ² + w} y² + w? 2²). V (x, y, z) = = m(o² x² + @z. (a) Evaluate the energy levels in terms of wx, @y, and (b) Calculate [Ĥ, Î₂]. Do you expect the wave functions to be eigenfunctions of 1²? (c) Find the three lowest levels for the case @x = @y= = 2002/3, and determine the degener- of each level.Consider a finite potential step with V = V0 in the region x < 0, and V = 0 in the region x > 0 (image). For particles with energy E > V0, and coming into the system from the left, what would be the wavefunction used to describe the “transmitted” particles and the wavefunction used to describe the “reflected” particles?Solve the problem for a quantum mechanical particle trapped in a one dimensional box of length L. This means determining the complete, normalized wave functions and the possible energies. Please use the back of this sheet if you need more room.
- The essence of the statement of the uniqueness theorem is that if we know the conditions the limit that needs to be met by the potential of the system, then we find the solution of the system , then that solution is the only solution that exists and is not other solutions may be found. If we know potential solutions of a system, can we determine the type of system that generate this potential? If so, prove the statement! If no, give an example of a case that breaks the statement!A particle of mass m is confined within a finite square well of depth V0 and width L.Sketch this potential, together with the form of the wavefunction and probability density for a particle in the lowest energy state. Briefly outline the procedure you would follow to determine the total number of energy eigenstates that can exist within a given finite square well.An electron is confined to a 1-dimensional infinite potential well of dimensions 1.55 nm. Find the energy of the ground state of the electron. Give your answer in units of electron volts (eV). Round your answer to 2 decimal places. Add your answer