3-21-11_Notes - Quantum Mechanics in 3D: The Hydrogen Atom...

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Unformatted text preview: Quantum Mechanics in 3D: The Hydrogen Atom and Angular Momentum QM I Substitute Lecture Notes Chris Mueller Dept. of Physics, University of Florida Last Updated: March 23, 2011 Contents 1 The Schroedinger Equation in 3D 1 1.1 The Angular and Radial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 The Hydrogen Atom 3 2.1 Step 1. Simplify Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Step 2: Strip Off Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Step 3: Power Series Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Properties of the Radial Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Some Comments on the Hydrogenic Energy Levels . . . . . . . . . . . . . . . . . . 6 3 Angular Momentum 6 3.1 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Eigenvalues of L 2 and L z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Action of the Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.5 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 The Schroedinger Equation in 3D Before moving on the the hydrogen atom, lets briefly recap how we broke down the 3D Schroedinger equation. The Schroedinger equation in its coordinate independent form is i ~ t =- ~ 2 2 m 2 + V . (1) This form is coordinate independent because we have yet to choose a specific expression for the Laplacian operator 2 . As usual, we will look for stationary state solutions of the form n ( ~ r,t ) = n ( ~ r ) e- i Ent ~ which brings us to the time independent Schroedinger equation,- ~ 2 2 m 2 + V = E. (2) This is still a coordinate independent expression. To actually solve this equation we will need to choose a specific coordinate system. Since most potentials are spherically symmetric, a nat- ural choice of coordinate system is spherical coordinates. The Laplacian operator in spherical 1 coordinates is 2 = 1 r 2 r r 2 r + 1 r 2 sin sin + 1 r 2 sin 2 2 2 . Plugging this into equation (2) above gives us a multi-variable differential equation which we will now attempt to solve. 1.1 The Angular and Radial Equations The physicists first method of attack for a multi-variable differential equation is most often the method of separation of variables. In this case we will look for solutions of the form ( r,, ) = R ( r ) Y ( , ) . (3) Plugging this into (2) and dividing the entire equation through by ( r,, ) gives us two indepen- dent equations, both of which must equal a constant (see Griffiths for a more detailed analysis), 1 sin sin 2 Y ( , ) + 1 sin 2 Y ( , ) 2 =- ( + 1) Y ( ,...
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This note was uploaded on 10/26/2011 for the course PHY 2049 taught by Professor Any during the Fall '08 term at University of Florida.

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3-21-11_Notes - Quantum Mechanics in 3D: The Hydrogen Atom...

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