Solve the Schrodinger equation for a particle incident from the left on a potential step { (z < 0) Vo, (z > 0) 0, V = in the case where E < VO. Show that the reflection coefficient is 1, and explain this with reference to the behaviour of the wave function in the two regions.
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- Show 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 PROCEDUREBy employing the prescribed definitions of the raising and lowering operators pertaining to the one-dimensional harmonic oscillator: x = ħ 2mω -(â+ + â_) hmw ê = i Compute the expectation values of the following quantities for the nth stationary staten. Keep in mind that the stationary states form an orthogonal set. 2 · (â+ − â_) [ pm 4ndx YmVndx = 8mn a. The position of particle (x) b. The momentum of the particle (p). c. (x²) d. (p²) e. Confirm that the uncertainty principle is satisfied for all values of nIn this question we will consider a finite potential well in which V = −V0 in the interval −L/2 ≤ x ≤ L/2, and V = 0 everywhere else (where V0 is a positive real number). For a particle with in the range −V0 < E < 0, write and solve the time-independent Schrodinger equation in the classically allowed and classically forbidden regions. Remember to keep the wavenumbers and exponential factors in your solutions real!
- 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).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.compute d and e please!
- A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: (x,0) = A [v₁ (x) + ₂(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy, if you exploit the orthonormality of 1 and 2. Recall that, having normalized at t = 0, you can rest assured that it stays normalized—if you doubt this, check it explicitly after doing part (b).) (c) (b) Find (x, t) and (x, t)|². Express the latter as a sinusoidal function of time, as in Example 2.1. To simplify the result, let w = ²ħ/2ma². Compute (x). Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a /2, go directly to jail.) (d) Compute (p). (As Peter Lorre would say, "Do it ze kveek vay, Johnny!”) (e) If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them? Find the expectation value of H. How does…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.what are the possible results that may be obtained upon measuring the property lz on a particle in a particular state, if its wavefunction is known to be Ψ, which is an eigenfunction of l2 such that l2Ψ=12ℏΨ? SHOW FULL AND COMPLETE PROCEDURE IN A CLEAR AND ORDERED WAY