4.8. The energy eigenfunctions V1, V2, V3, and 4corresponding to the four lowest energy states for aparticle confined in the finite potential well= {0V(x) =-Vo|x| a/2are sketched in Fig. 4.4. For which of these energyeigenfunctions would the probability of finding theparticle outside the well, that is, in the region [x] > a/2,be greatest? Explain. Justify your reasoning using thesolution to the Schrödinger equation in the regionx > a/2.

Answered: Image /qna-images/answer/48fba535-1108-42eb-b52a-d77ecaa7e795.jpg

Answered: 4.8. The energy eigenfunctions V1, V2,…

Similar questions

There is an electron, in a 1-d, infinitely deep square potential well with a width of d. If it is in ground state, 1. Draw the electron's wavefunction. Show the position of the walls of the potential well. 2. Explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.
By 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 n
2. A simple harmonic oscillator is in the state 4 = N(Yo + λ 4₁) where λ is a real parameter, and to and ₁ are the first two orthonormal stationary states. (a) Determine the normalization constant N in terms of λ. (b) Using raising and lowering operators (see Griffiths 2.69), calculate the uncertainty Ax in terms of .
4.3 A particle with mass m and energy E is moving in one dimension from right to left. It is incident on the step potential V(x) = 0 for x 0, as shown on the diagram. The energy of the particle is E > Vo. = V(x) V = Vo V=0 x = 0 (a) Solve the Schrödinger equation to derive 4(x) for x 0. Express the solution in terms of a single unknown constant. (b) Calculate the value of the reflection coefficient R for the parti- cle.
2.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)
3.7 Suppose that a wave function (r,t) is normalized. Show that the wave function e¹0 W(r,t), where 0 is an arbitrary real number, is also normalized.
(a) Write down the wave functions for the three regions of the potential energy barrier (Figure 5.25) for E < U₁. You will need six coefficients in all. Use complex exponential notation. (b) Use the boundary conditions at x = 0 and at x = L to find four relationships among the six coeffi- cients. (Do not try to solve these relationships.) (c) Sup- pose particles are incident on the barrier from the left. Which coefficient should be set to zero? Why?
Consider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure 6.10 (attached). The particle is in thermal equilibrium with a reservoir at temperature so the probabilities of its various states are determined by Boltzmann statistics.
4.5 Consider reflection from a step potential of height Vo with E > Vo, but now with an infinitely high wall added at a distance a from the step (see diagram): V(x) E V = Vo V = 0 x = 0 x = a x (a) Solve the Schrödinger equation to find v/(x) for x < 0 and 0 ≤ xa. Your solution should contain only one unknown constant. (b) Show that the reflection coefficient at x = 0 is R = 1. This is different from the value of R previously derived without the infinite wall. What is the physical reason that R = 1 in this case? (c) Which part of the wave function represents a leftward-moving particle at x < 0? Show that this part of the wave function is an Solutions of the one-dimensional time-independent Schrödinger equation 103 eigenfunction of the momentum operator, and calculate the eigen- value. Is the total wave function for x ≤ 0 an eigenfunction of the momentum operator?
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.