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catinabox
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I am trying to work out the wavefunctions for a particle in a box between -a/2 and a/2.I have already gone through the solution for a box between 0 and a and got the solution [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a} )[/tex]So I can see that for -a/2 to a/2 I have [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi(x+\frac{a}{2})}{a})[/tex]Which by some trig leads to [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})cos(\frac{n\pi}{2})+\sqrt{\frac{2}{a}}cos(\frac{n\pi x}{a})sin(\frac{n\pi}{2})[/tex]Now i can see it differs for even and odd n as for even n [tex]sin(\frac{n\pi}{2})=0[/tex] for odd n [tex]cos(\frac{n\pi}{2})=0[/tex].
(NOT SURE WHATS HAPPENED WITH LATEX HERE :()Therefore even n leads to [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})cos(\frac{n\pi}{2})[/tex] odd n leads to [tex]\sqrt{\frac{2}{a}}cos(\frac{n\pi x}{a})sin(\frac{n\pi}{2})[/tex]From research I have found that the wavefunction for n even is in fact just [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})[/tex] and odd n just [tex]\sqrt{\frac{2}{a}}cos(\frac{n\pi x}{a})[/tex]This is were I am confused because the [tex]cos(\frac{n\pi}{2})[/tex] for even n is positive or negative 1 and [tex]sin(\frac{n\pi}{2})[/tex] for odd n is positive or negative 1.Why is only the positive chosen, is this to do with normalistion?Any help is much appreciated.Thank you.
(NOT SURE WHATS HAPPENED WITH LATEX HERE :()Therefore even n leads to [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})cos(\frac{n\pi}{2})[/tex] odd n leads to [tex]\sqrt{\frac{2}{a}}cos(\frac{n\pi x}{a})sin(\frac{n\pi}{2})[/tex]From research I have found that the wavefunction for n even is in fact just [tex]\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})[/tex] and odd n just [tex]\sqrt{\frac{2}{a}}cos(\frac{n\pi x}{a})[/tex]This is were I am confused because the [tex]cos(\frac{n\pi}{2})[/tex] for even n is positive or negative 1 and [tex]sin(\frac{n\pi}{2})[/tex] for odd n is positive or negative 1.Why is only the positive chosen, is this to do with normalistion?Any help is much appreciated.Thank you.
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