Lecture 2 Summary Phys 402 We wrote down the Schrödinger equation in spherical coordinates and proceeded to solve it by separation of variables. We will solve it for a Hydrogen atom in which there is a (conservative) electrostatic (central) force between a proton and an electron, with potential , where is the electronic charge, is the permittivity of free space, and )4/()(02rerVπε−=e0εris the radial coordinate, representing the distance from the proton to the electron (we shall discuss the QM 2-body problem in more detail during a discussion section later in the semester). We look for a solution with constant energy Esuch thath/),,(),,,(iEterutr−=φθφθψ. The resulting time-independent Schrödinger equation in spherical coordinates is given by Eq. [4.14] of Griffiths. Separate variables as),()(),,(φθφθYrRru=to get an equation that has r-dependent terms (only) on one side, and φθ,-dependent terms (only) on the other side (Griffiths, p. 134). Each side of the equation must separately equal a constant “α” (i.e. something independent of φθ,,r), yielding the radial and angular equations, Griffiths [4.16] and [4.17], respectively. Starting with the definition of the angular momentum operator , the angular momentum squared operator in spherical coordinates is: )( ∇−×=×=rhrrrrirprL222222sinsinsinφθθθθθ∂∂−⎟⎠⎞⎜⎝⎛∂∂∂∂−=•=hhrrLLL . By comparison with Eq. [4.17] one sees that it contains the angular momentum squared operator: , which is a nice eigenvalue problem. The eigenvalues of will turn out to be , where is zero or a positive integer, and the eigenfunction is the ‘spherical harmonic’ , where is another positive or negative integer or zero with YYLα22h=2L)1(2+llhl),(φθmYlm m≥l. The radial equation has an infinite number of bound states (E<0) for any given value of l. )()()1(24)(22202222rRErRrmredrrRdm=⎥⎦⎤⎢⎣⎡++−+− llhhπε The solutions are proportional to a finite polynomial called the Laguerre polynomial, which we will find later. The solution to this equation also results in a quantization condition for the energy:20222421⎟⎟⎠⎞⎜⎜⎝⎛−=πεemnEnh, where is the electron (reduced) mass, and is an integer that is bigger than l, i.e. mn 1−≤nl. One also finds a characteristic length for the Hydrogen atom, called the Bohr radius:22004meahπε= , which is about 0.5 Angstroms.The full solution of the time-independent Schrödinger equation for the H-atom is found by multiplying the R(r) solution with the angular solution and properly normalizing the entire wavefunction: ()()[]hlllllll/0121/0330),(22!2!12),,,(0tiEmnnarmnneYnarLenarnnnnatr−+−−−⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+−−⎟⎟⎠⎞⎜⎜⎝⎛=φθφθψ There are three quantum numbers: n(principal), (ang. mom.) and m(magnetic). They have possible values given by: l lllll,1,...0,...1,1,...2,1,0,...4,3,2,1−+−−=−==mnnThe Hydrogen atom wavefunctions are orthonormal, Griffiths [4.90]. We shall examine the angular and radial equations in more detail
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