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Download Some Additional Math and Physics Rules - Lecture Notes | PHYS 211 and more Study notes Physics in PDF only on Docsity!
Some additional math and physics rules Series expansions. An exponential function can be written as a series as follows: : x 3 x! x x? x! OSL ete te pe ee tet 2) 3 44 St 6t 7 Similarly, the sine and cosine functions can be written as follows: 3 5 7 sing 9- S42 Z 3 oF 7 2 4 6 cos @ =1-— ge 2 4 él The complex exponential. In general, e” =cosP+ising e' =cos@—isind Cosines and sines can be written in terms of the complex cxponcntial as io ie e te cos 9 = ——_ 2 ie i? . e@e —€é sind = - 2 Time-dependent slates. Assume that a system is initially in the state |) = a,|¢,)+a,|¢,)+a,|¢,)+--- where the states \¢) , \¢.) . lds), ... are states with definite cnergies Ey, Ez, E3,, .... Then state of the system at any arbitrary time ¢ is given by |yee)) = ye |S + age) + age | de This is important because systems change in time. A system might be in a particular state at some time, but that state will typically (but not always) change. For instance, if you locate an electron and find it to be at a certain location, then immediately after that measurement, you know the electron is there, But it won't typically stay al that location, nor will its location even remain determined. The rule stated here (although not derived) shows how to determine how a system will change in time after a measurement. All the other math and physics rules still hold!!! You will use the same rules as before. to determine probabilities and expectation values. 
