Show that the Legendre polynomials are orthogonal solutions of the angular part of the Schrodinger equation.
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- L What are the differences and similarities between the quantization of angular momentum in the Bohr model and the Schrödinger theory? Explain in your own words and use examples from both to support your explanation. 111 Responses Reply Showing All Responses ordered by Newest ResponsesReview schrodinger equations not dependent on 3D time in ball coordinates -V² + V(f) ) µ(F) = E Þ(*) 2m F = (r,0,9) and E is the energy system. Assume the potential is only radial function V (f) = V(r). With %3D 'To solve the Schrodinger equation above apply the method ψη, θ φ) -R(r)P (θ) Q (φ) variable separation problem : By defining the separation constant in the angular function 0 as (1 + 1) Also show that P(0) = P"(cos 0) angular function solution P(0) can be written as Polynomial Associated Legendre The combined angular functions of the sphere are known as the "harmonic function of the sphere" (spherical harmonics) Y(0, p) x Pi" (cos 0)e±impE8A.12 At what radius does the probability density of an electron in the H atom fall to 50 per cent of its maximum value? E8A.13 At what radius in the H atom does the radial distribution function
- 7. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4, (hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x = L/4, ifthe electron is in the first excited state (n = 2) for each experiment and you repeat the experiment N, =100,000 times?Consider a particle in a 2-D box having Lx = 10 nm and Ly = 10 nm. a) Make a surface plot of all the wave functions for the first and second energy levels. b) What is the degeneracy of the second energy level? Compare and contrast the wave functions of the second energy level. c) How does the number of nodes in the x-coordinate change as n increases? How does the number of nodes in the y-coordinate change as n, increases? d) Explain whether or not those same states would be degenerate if Lx = 10 nm and Ly = 15 nm.1. A particle is confined to the x-axis between x = 0 and x = L. The wave function 3π of the particle is = A sin (²x) + A sin (37 x) with A E R. 4 2L a. b. C. Determine A. Determine the probability that the particle is in the interval [0,1]. J Determine (x).
- Review schrodinger equations not dependent on 3D time in ball coordinates -v² + V(f) ) µ(F) = E Þ(*) 2m i = (r, 0, 4) V (f) = V(r). and E is the energy system. Assume the potential is only radial function To solve the Schrodinger equation above apply the method With y(r, 0, q) = R(r)P(0)Q(9) variable separation problem : Specify a common solution (r, 0,0) for l = 0 and V(r) = 0B7(x, t) = Ae-iwt e-(mw/ħ).x² which is a solution to Schrödinger's equation. Determine the potential V(x) that is consistent with this wave function, Note: You do not have to normalize V since Schrödinger's equation is linear.