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| classes:2009:fall:phys4101.001:lec_notes_1014 [2009/10/15 00:44] – czhang | classes:2009:fall:phys4101.001:lec_notes_1014 [2009/10/15 23:37] (current) – fixed k and kappa per Yuichi's request prestegard |
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| Since <math>k^2=\frac{-2mE}{h^2}</math> and <math>\kappa^2=\frac{-2m(E+V_0)}{h^2}</math>, | Since <math>\kappa^2=\frac{-2mE}{\hbar^2}</math> and <math>k^2=\frac{2m(E+V_0)}{\hbar^2}</math>, |
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| we define<math>Z_0^2=\frac{-2ma^2V_0}{h^2}</math> and <math>Z^2=k^2a^2=\frac{-2ma^2(E+V_0)}{h^2}</math> | we define<math>Z_0^2=\frac{-2ma^2V_0}{\hbar^2}</math> and <math>Z^2=k^2a^2=\frac{-2ma^2(E+V_0)}{\hbar^2}</math> |
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| Since it is really hard to solve it analytically, we can solve it graphically. | Since it is really hard to solve it analytically, we can solve it graphically. |
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| {{:classes:2009:fall:phys4101.001:qm_lec_pic.jpg|}} | {{:classes:2009:fall:phys4101.001:qm_lec_pic.jpg|}}\\ **the cotangent part of the graph does not look quite right! //Yuichi//** |
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| as we can see from the graph, the periodic functions are CotZ, and the other one is <math>\sqrt{(\frac{Z_0}{Z})^2-1}</math> | as we can see from the graph, the periodic functions are CotZ, and the other one is <math>\sqrt{(\frac{Z_0}{Z})^2-1}</math> |
| The other line is when <math>Z_0=10</math>, which has more intersections denoting more bound states. | The other line is when <math>Z_0=10</math>, which has more intersections denoting more bound states. |
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| Remember that <math>Z_0^2=\frac{2ma^2V_0}{h^2}</math>, the magnitude of Z0 is determined by the production of a^2 and V0, a is the width of potential well, and V0 is the depth of the potential well. If we keep a constant , raise the potential to infinity, Z0 goes to infinity, then we have infinite square well, which corresponds with infinite intersection on graph, and we would have infinite bound states. | Remember that <math>Z_0^2=\frac{2ma^2V_0}{\hbar^2}</math>, the magnitude of Z0 is determined by the production of a^2 and V0, a is the width of potential well, and V0 is the depth of the potential well. If we keep a constant , raise the potential to infinity, Z0 goes to infinity, then we have infinite square well, which corresponds with infinite intersection on graph, and we would have infinite bound states. |
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| If we keep potential constant, and increase the width of the well, it would also increase the number of bound states. | If we keep potential constant, and increase the width of the well, it would also increase the number of bound states. |
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