classes:2009:fall:phys4101.001:lec_notes_1019
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| classes:2009:fall:phys4101.001:lec_notes_1019 [2009/10/21 21:09] – x500_spil0049 | classes:2009:fall:phys4101.001:lec_notes_1019 [2009/10/21 21:17] (current) – x500_spil0049 |
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| Discribe a plane wave function. If <math> f(x)=e^ikx</math> which can be thought of as a stationary state of momentum. Which implies that\\ <math> <f|f>=\int_{-\infty}^{\infty}|f(x)|^2dx = \infty</math>\\ | Discribe a plane wave function. If <math> f(x)=e^ikx</math> which can be thought of as a stationary state of momentum. Which implies that\\ <math> <f|f>=\int_{-\infty}^{\infty}|f(x)|^2dx = \infty</math>\\ |
| Consider then,<math> f_k(x)=e^ikx then \int f_k(x)^*f_k(x) dk</math> which is proportional to | Consider then,<math> f_k(x)=e^ikx then \int f_k(x)^*f_k(x) dk</math> which is proportional to |
| <math>\delta(k-k')~\delta_{ij}</math> and normalizing in this fashion.\\Next, stationary states are Hamiltonian eigenstates and <math>e^ikx</math>⇔ momentum eigenstate. This means that if <math>\hat{p}e^ikx=khe^ikx</math> | <math>\delta(k-k')~\delta_{ij}</math> and normalizing in this fashion.\\Next, stationary states are Hamiltonian eigenstates and <math>e^ikx</math>⇔ momentum eigenstate. This means that if <math>\hat{p}e^ikx=khe^ikx</math>\\ |
| | So <math>|\phi(k)|^2</math> is the probability density i.e. the likely hood of finding a momentum value.\\ |
| | therefor if you let <math> f_k(x)=1/2\pi e^ikx</math> then <math> f_k(x)=e^ikx</math> then <math>\int f_k(x)^*f_k(x) dk</math> is now equal to |
| | <math>\delta(k-k')=\delta_{ij}</math> |
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| **To go back to the lecture note list, click [[lec_notes]]**\\ | **To go back to the lecture note list, click [[lec_notes]]**\\ |
classes/2009/fall/phys4101.001/lec_notes_1019.1256177383.txt.gz · Last modified: 2009/10/21 21:09 by x500_spil0049