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Unformatted text preview: PHY251 Lecture Notes  Set 8 PHY251 Lecture Notes Set 8 file:///C/Documents/phy_courses/PHY251/PHY251_lecture_08.HTML (1 of 6) [4/3/2001 12:30:05 PM] PHY251 Lecture Notes Set 8 file:///C/Documents/phy_courses/PHY251/PHY251_lecture_08.HTML (2 of 6) [4/3/2001 12:30:05 PM] PHY251 Lecture Notes Set 8 file:///C/Documents/phy_courses/PHY251/PHY251_lecture_08.HTML (3 of 6) [4/3/2001 12:30:05 PM] The Schrödinger equation describes the behavior of a particle as function of position and time. The Schrödinger equation with a static potential V = V ( r ) is an example of an eigenvalue problem, with the Operator (called the Hamiltonian or energy operator in this case)  h ² / 2 m ( ∂ / ∂ x ² + ∂ / ∂ x ² + ∂ / ∂ x ² ) + V ( r ) acting on the wave function ψ ( r ), giving the energy E (a constant) times the wavefunction: H op ψ ( r ) = [ h ² / 2 m ( ∂ ² / ∂ x ² + ∂ ² / ∂ x ² + ∂ ² / ∂ x ² ) + V ( r )] ψ ( r ) = E ψ ( r ) (1) For the Hydrogen atom the potential is simply the Coulomb potential, which depends only on distance: V = V ( r ) =  e ²/(4 πε r ) = α hc / r , where α ≡ e ²/(4 πε hc) . Because the potential in equation (1) depends only on the distance r to the atom's nucleus, and not on any direction, we profit from rewriting equation (1) in terms of spherical coordinates r , θ , and φ . That implies that we need to reexpress the derivatives in terms of derivatives with respect to these new variables as well, a tedious process as we have seen before! The ultimate result is given below: H op ψ ( r ) = [ h ² / 2 m { ∂ ² / ∂ r ² + 2 ∂ / r ∂ r + 1 / r ² ( ∂ ² / ∂ θ ² + cot θ ∂ / ∂ θ + ∂ ² / sin² θ ∂ φ ² )} + V ( r )] ψ ( r , θ , φ ) = E ψ ( r , θ , φ ) (2) This secondorder differential equation can be solved using the fact that the solutions factorize, i.e. the wavefunction is a product of three separate functions that depend each on a different variable: ψ ( r , θ , φ ) = R ( r ) T ( θ ) F ( φ ) (3) The equation (2) can be solved; it is technically complicated but not impossible. One finds:The equation (2) can be solved; it is technically complicated but not impossible....
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 Spring '01
 Rijssenbeek
 Physics, Atomic orbital, pn, Spin quantum number, Lecture Notes Set, ms. We

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