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| classes:2009:fall:phys4101.001:lec_notes_1019 [2009/10/21 19:54] – x500_spil0049 | classes:2009:fall:phys4101.001:lec_notes_1019 [2009/10/21 21:17] (current) – x500_spil0049 |
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| ∴<math>\sum f_i^*g_i\delta_{ij}= \sum f_i^*g_i</math> whooooo!\\ | ∴<math>\sum f_i^*g_i\delta_{ij}= \sum f_i^*g_i</math> whooooo!\\ |
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| | *This time we claim that <math> <Q> =\int\psi^*\hat{Q}\psi dx</math> which is the same as <math> <\psi|Q|\psi></math> here we recognize that both <math>\psi's</math> are vectors and that Q is like a matrix. |
| | We begin by examining <math>|\psi>=\sum c_n\psi_n = \sum c_n|e_n></math>\\ |
| | We want to check that <math> <Q>>=(c_1^*.....c_n^*)(matrix)\coloum{c_1.......c_n}</math> |
| | So we speculate <math> \hat{Q} =Q_{mn} =<c_m\hat{Q} c_n>=\int\psi_m^*Q\psi_n dx</math>\\ |
| | With this in mind <math><Q_\psi>=\int\psi^*\hat{Q}\psi dx=\int\sum c_i^*\psi_i^*\hat{Q}</math>\\ |
| | So we speculate <math> \hat{Q} =Q_{mn} =<c_m\hat{Q} c_n>=\int\psi_m^*Q\psi_n dx</math>\\ |
| | With this in mind <math><Q_\psi>=\int\psi^*\hat{Q}\psi dx=\int\sum c_i^*\psi_i^*\hat{Q}\sum c_j\psi_jdx</math>\\ |
| | =<math>\sum_{ij} c_{i}^*c_{j}\int\psi_i^*\hat{Q}\psi_jdx</math>\\ |
| | =<math><e_i|\hat{Q}|e_j>=Q_ij</math>\\ |
| | = ∴ <math>\sum_{ij} c_i^*Q_{ij}c_j</math> whooooo\\ |
| | |
| | *There was also a question asked in class. What exactly the question was i dont remember could somebody help?\\ |
| | However the discussion went something like this:\\ |
| | In general <math>\psi(x)=1/2\pi \int\phi(k)e^ikx</math> analogous to <math>\sum c_n\psi_n(x)</math>\\ where <math>|c_n|^2=E</math> and <math> <E>= \sum|c_n|^2 E_n</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 |
| | <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]]**\\ |