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| classes:2009:fall:phys4101.001:lec_notes_1106 [2009/11/06 16:20] – x500_liux0756 | classes:2009:fall:phys4101.001:lec_notes_1106 [2009/11/07 21:36] (current) – yk |
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| ===== Nov 06 (Fri) ===== | ===== Nov 06 (Fri) Legendre polynomials, Radial wave equation ===== |
| ** Responsible party: liux0756, Dagny ** | ** Responsible party: liux0756, Dagny ** |
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| === Part I Legendre polynomial === | ==== Part I: Legendre polynomial ==== |
| * | In the last class the power series method is used: |
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| | <math>U(\xi)=\sum_{n=0}^\infty a_n \xi^2</math> |
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| | Substitute it into differential equation derives the recursive relation: |
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| | <math>a_k=\frac{(\nu+1)(\nu+2)...(\nu+k)(-\nu)(-\nu+1)...(-\nu+k-1)}{(k!)^2} a_0</math> |
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| | <math>a_0</math> is determined by normalization. |
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| | Now consider the convergence requirement. Just as the simple harmonic oscillator, for normalizable solutions the power series must terminate. And if <math>\nu</math> is an integer, <math>a_k</math> will be zero if k is large enough. If <math>\nu</math> is negative, there exists a corresponding positive <math>\nu</math> that leads to the same recursive relation. So <math>\nu</math> can be limited to non-negative numbers: |
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| | <math>\nu=0,1,2,...</math> |
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| | <math>\alpha=\nu(\nu+1)=0,2,6,12,...</math> |
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| | Generally <math>P_\nu (z) </math> is called Legendre function, <math>\nu</math> is any real number. If convergence is important, <math>\nu=l</math> is integer, we deal with the Legendre polynomial <math>P_l (z) </math> |
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| | ==== Part II: Radial part of Schrodinger equation ==== |
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| | The equation is written as: |
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| | <math>\frac{d}{dr} (r^2 \frac{dR}{dr})- \frac{2mr^2}{\hbar^2}[V(r)-E]R=l(l+1)R</math> |
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| | <math>l(l+1)</math> is related with the square of angular momentum: |
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| | <math>L^2=l(l+1)\hbar^2</math> |
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| | Define <math> R(r)=\frac{u(r)}{r}</math>, then the differential equation is transformed from <math>R(r)</math> to <math>u(r)</math>: |
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| | <math> -\frac{\hbar^2}{2m} \frac{d^2u}{dr^2}+[V+\frac{\hbar^2}{2m} \frac{l(l+1)}{r^2}]u=Eu </math> |
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| | This equation is similar to 1D Schrodinger equations discussed in Chapter 2. |
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| | The equation above cannot be solved further before one knows the potential distribution in the system. |
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| | === Example === |
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| | Since we cannot go much further without specifying potential energy, now consider a 3D infinite square well. |
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| | When <math>r>a</math>, the potential goes to infinity. |
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| | When <math>r<a</math>, the potential is 0. |
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| | The radial differential equation is: |
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| | <math>\frac{d^2u}{dr^2}-[\frac{l(l+1)}{r^2}-k^2]u=0</math> |
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| | where <math>k^2=\frac{2mE}{\hbar^2} </math> |
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| | The solution is Bessel function <math>j_l(kr)</math> |
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| | Now take a look at the 0th order solution. |
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| | When <math>l=0</math>, the differential equation is <math>\frac{d^2u}{dr^2}=-k^2u</math> |
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| | The solution is: <math>u(r)=Ae^{ikr}+Be^{-ikr}=A'sinkr+B'coskr</math> |
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| | The boundary condition is <math>u(a)=0</math> |
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| | At r=0, if u is not 0, then <math>R=\frac{u}{r}</math> will go to infinity. So <math>u(0)=0</math> should be satisfied. (However, this is not quite correct because even if the value of wave function is infinite, the wave function may still be able to be normalized: <math>\int_0^a \frac{1}{r^2}dV = \int_0^a \frac{1}{r^2} r^2 sin\theta dr d\theta d\phi=\int_0^a sin\theta dr d\theta d\phi</math> is normalizable - finite) |
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| | So the <math>B'coskr</math> term should be 0. (In fact this term is allowed) |
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