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Additional resources for An extensible model of the electron
No. All three components of the angular momentum operators can be specified precisely at the same time if the expectation values of all the commutators of the angular momentum operators are precisely zero in a particular state. This is the case when the hydrogen atom is in the ground state, or the s-level ( l = 0). 6-4. Show that the n=2, l = 1 , and ml = 1 wave function indeed satisfies the time- independent Schroedinger’s equation given in the text for the hydrogen atom: Ψ211(r,θ,φ) = R21 (r)Y11 (θ,φ) = [(2a0 )−3 / 2 6-2 1 r −r / 2a 0 3 iφ e ] [− e sinθ] , 8π 3 a0 r −r / 2a 0 r ∂ 2 ∂ r r2 −3 / 2 1 −3 / 2 1 (r )[(2a0 ) e ] = (2a0 ) [2 − 2 + ] e−r / 2a 0 , 2 ∂r ∂r a0 (2a0 ) 3 a0 3 a0 ⎛ 1 ∂ ∂ 1 ∂2 ⎞ iφ e sinθ = −2e iφ sinθ .
C) Since the complex dielectric constant based on the oscillator strength f using either the quantum mechanical model or the classical harmonic oscillator model gives the same result for ε” , it is obvious that the same should be true for ε’. The classical harmonic oscillator model, therefore, can be used to characterize the dispersion and absorption characteristics of linear optical media with only three phenomenological parameters: the oscillator strength f, that characterizes the strength of the charge e2 , the resonance frequency ω21 , and the damping constant Γ associated with the bound particle in the harmonic oscillator model.
For l = 0 , it is: h2 1 ∂ 2 ∂ Hˆ = − [ (r ) ] + V (r ) 2m r 2 ∂ r ∂r . The corresponding Schroedinger’s equation is ⎫ ⎧ h2 1 ∂ 2 ∂ [ 2 (r ) ] + V (r )⎬ Rno (r ) = E n Rno (r ) , ⎨− ∂r ⎭ ⎩ 2m r ∂ r for r≤a , ⎧ h2 1 ∂ 2 ∂ ⎫ [ 2 (r ) ]⎬ Rno (r ) = E n Rno (r ) ⎨− ∂r ⎭ ⎩ 2m r ∂ r for r≥a . and , The equation for r ≤ a can be converted to: 2mE d2 U (r ) = − 2 U (r ) 2 h dr , where U(r) = r R(r) . The general solution of this equation is: U (r ) = A cos kr + B sin kr where k = 2mE . To satisfy the boundary condition that Rn0(r) must be finite at r =0, h A must be equal to 0, or 6-7 U (r ) = B sin kr Similarly, , for for r ≥ a , U (r ) = C e α r + D e − α r where α = r ≤ a.