HW12 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #12 Due Wednesday, December 3 at 2pm Problem 12.1 : Analytic solution of radial equation for hydrogen. (20 points) Stationary states for the hydrogen atom that are also eigenstates of L 2 and L z were found to take the form ψ n±m ( r, θ, ϕ ) = R ( r ) Y m ± ( θ, ϕ ) u ( r ) r Y m ± ( θ, ϕ ) , (1) where the Y m ± are the spherical harmonics, and u ( r ) solves the radial equation, - ¯ h 2 2 m e d 2 u dr 2 + ± - ke 2 r + ¯ h 2 2 m e ± ( ± + 1) r 2 ² u = Eu , (2) where we have written m e for the mass of the electron to avoid confusion with the azimuthal angular momentum quantum number m , and k = 1 / 4 π² 0 . We will solve this equation using the method of Frobenius, the same method we explored for the analytic solution of the harmonic oscillator. a) We begin by introducing a dimensionless variable to replace r , which we’ll call ρ . Divide the radial equation (2) by E and define a variable ρ r/ ¯ r , where ¯ r is for you to determine, such that the first and last terms take the form d 2 u 2 + . . . = u . (3) What is ¯ r ? Check that it has units of length. What is the sign of E appropriate to bound states, and given this, is ¯ r real and positive? Next define a constant ρ 0 to absorb most of the remaining constants, so that the equation can be written d 2 u 2 = ³ 1 - ρ 0 ρ + ± ( ± + 1) ρ 2 ´ u . (4) What is ρ 0 ? What are its units? b) Next we will study the asymptotics of the solution. Unlike the harmonic oscillator case, where x → ∞ and x → -∞ had the same behavior and could be examined at the same time, here we will separately consider r → ∞ and r 0. Explain why in the r → ∞ limit (which implies ρ → ∞ ), the radial equation reduces to d 2 u 2 u . (5) 1
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Verify that the solution in this limit is u ( ρ ) Ae - ρ + Be + ρ . (6) Explain what constraint must we put on A or B to make sure the wavefunction can be normalized. Now show that in the r 0 limit (implying ρ 0) the radial equation becomes d 2 u 2 ± ( ± + 1) ρ 2 u . (7) This equation has a simple solution: consider u ( ρ ) α for some number α . What two values of α satisfy the equation? Which one must we throw out to prevent the wavefunction from blowing up? c) We will now extract both asymptotic behaviors from u ( ρ ) to define a new function to work with, v ( ρ ), as: u ( ρ ) ρ ± +1 e - ρ v ( ρ ) . (8) Verify that the radial equation becomes, in terms of v ( ρ ), ρ d 2 v 2 + 2( ± + 1 - ρ ) dv + [ ρ 0 - 2( ± + 1)] v = 0 . (9) d) To solve this diFerential equation, we will postulate a series solution for v : v ( ρ ) = ± j =0 c j ρ j , (10) where the c j are constants. Show that the result of part c) implies the recursion relation for the constants: c j +1 = ² 2( j + ± + 1) - ρ 0 ( j + 1)( j + 2 ± + 2) ³ c j . (11) e) Let us explore what happens if the series goes on forever. Write down the large-
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HW12 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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