Show that R(r) is a solution of the following differential equation for l = 1, R(r) = (r/ao) * e-r/2ao.
What is the eigenvalue? Using this result, calculate the value of the principal quantum number (n) for this function.
Show that R(r) is a solution of the following differential equation for l = 1, R(r)...
This question is in physical chemistry. 1.) 2.) 3.) Show that the function (r/ao)e-o is a solution of the differential equation for R(r) .dlr2 dR(r) [h21(1 + 1) 4-IR(r)-ER(r) for 1-1 e2 ATEor Match the items in the left column to the appropriate blanks in the equations on the right. Make certain each equation is complete before submitting your answer
1. Show that the wave function V = Ce-r/ao where ao = hc/(mca) is a solution to the time independent Schrodinger equation for the Hydrogen atom. Determine the angular momentum quantum number l and the energy eigenvalue. Show that the normal- ization constant C = (Ta) -3.
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
3. Consider the following differential equation 0o and a series solution to the differential equation of the form a" n-0 (a) Find the recurrence relations for the coefficients of the power series. 3 marks] (b) Determine the radius of convergence of the power series. l mar (c) Write the first eight terms of the series solution with the coefficients written in terms of ao and ai 2 marks] 3. Consider the following differential equation 0o and a series solution to...
#2 ONLY PLEASE 1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
The hydrogenic radial function R(r) are relatively simple for the case l = n-1 (the maximum allowed value for l for given n): R(r) = Arn-1 er/ab (l = n -1) (a) Write down the radial schrodinger equation for this case. (b) Verify that the proposed solution does indeed satisfy this equation if and only if En = -Er/n2 (c) Plot the radial function R(r) for n = 0,1,2, (assume ab=1)
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
Find the stable equilibrium solution of the following differential equation: + y - 1 = e2( y − 1). The stable equilibrium solution is y = Check Find the general solution to the differential equation: x + y - x115 = 0. Answer: y(x) = Check Solve the initial-value problem: dy = e ** - y, yO= dx Answer: y(x) = Check