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henpen
 one year ago
Best ResponseYou've already chosen the best response.0dw:1360436634776:dw\[φ''=\frac{2mE}{ℏ^2} φ\] \[ \varphi= A \sin(\frac{\sqrt{2mE}}{ℏ}x)\] \[\varphi(a)=\varphi(a)=0\] \[ \pm\frac{\sqrt{2mE}}{ℏ}a=n \pi\] \[ \frac{\sqrt{2mE}}{ℏ}=\pm\frac{n \pi}{a}\] \[\lambda= \frac{2 \pi}{\pm\frac{n \pi}{a}}=\pm\frac{2a}{n}\] Which, for n=1, is shorter than the real groundstate lavelength (4a). What have I done incorrectly?

JamesJ
 one year ago
Best ResponseYou've already chosen the best response.1You haven't written down the correct general solution. You can also have terms \[ ... + B \cos(\omega x) \] with \( \omega = \sqrt{2mE}/\hbar \).

JamesJ
 one year ago
Best ResponseYou've already chosen the best response.1This will give you the 4a, as cos(omega.x) = 0 => omega.x = pi/2 + n.pi

henpen
 one year ago
Best ResponseYou've already chosen the best response.0Aren't you assuming A=0 for that solution? \[Asin(\pi/2 \pm n\pi) \ne 0\]

JamesJ
 one year ago
Best ResponseYou've already chosen the best response.1The general solution of y '' + w^2.y = 0 is y(x) = A.sin(wx) + B.cos(wx) or y(x) = A.cos(wx + phi0) Either way, there are two parameters. You are applying one of the boundary conditions. You actually are going to apply another boundary condition to y'(x) in order to completely nail down the solution.

JamesJ
 one year ago
Best ResponseYou've already chosen the best response.1Alternatively, the two conditions you are using phi(a) = 0, phi(a) = 0. From this, you get two families of solutions. One with sin, the other with cos. This equivalent to using the second general solution with the phi0 alternating between 0 and pi/2

henpen
 one year ago
Best ResponseYou've already chosen the best response.0Ah, that's it! Thanks. What are the boundary conditions? That \[\varphi'(a)=\varphi(a)=0\] Because \[\varphi(x)=0\] for \[x>a, x<a\]?

henpen
 one year ago
Best ResponseYou've already chosen the best response.0*\[\varphi'(a)=\varphi'(a)=0\]

henpen
 one year ago
Best ResponseYou've already chosen the best response.0That guess came from avoiding making the first derivative jump about at + a

JamesJ
 one year ago
Best ResponseYou've already chosen the best response.1You have two conditions, which solving for the two parameters in the general solution. I recommend you start with A.cos(wx + phi0) and you'll get what you want.

henpen
 one year ago
Best ResponseYou've already chosen the best response.0\[A \sin( \omega a)+B \cos( \omega a)\] It is harder to make this expression = 0 in this form than yours, as both summands cannot be made = 0 or the negative of the other by eyeballing it, so yours is simpler. Thanks, I think I've got it.

henpen
 one year ago
Best ResponseYou've already chosen the best response.0@JamesJ , sorry to tag you, but I have a (very) brief question: the general solution is \[ \varphi= A e^{\pm i \omega x}\] What is done with \[\Im(\varphi)\]? Is only the real part considered, as is often done in classical mechanics? Thanks again
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