I was just talking with Matt several times in the last few weeks about the Schrödinger equation - I had never calculated solutions for it before, and he didn't know what it meant that the set of possible wave functions was an infinite-dimensional vector space.
Anyway, I thought it was interesting that even for the toy models (the completely free particle, the particle in a box, the particle in a finite potential well), calculating the solutions requires some awful hand-waving. At least, in the discussions in wikipedia, they claimed that there are sinusoidal solutions to the equation for the completely free particle, even though the sinusoids are just as non-normalized as the exponentials. And for the other two, we have to assume that the space of possible wave-functions includes ones that have discontinuities in certain derivatives at certain points, in a way that intuitively makes sense, but I saw no mathematical justification for.
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timeasmymeasure
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Date: 2010-09-15 05:32 pm (UTC)Anyway, I thought it was interesting that even for the toy models (the completely free particle, the particle in a box, the particle in a finite potential well), calculating the solutions requires some awful hand-waving. At least, in the discussions in wikipedia, they claimed that there are sinusoidal solutions to the equation for the completely free particle, even though the sinusoids are just as non-normalized as the exponentials. And for the other two, we have to assume that the space of possible wave-functions includes ones that have discontinuities in certain derivatives at certain points, in a way that intuitively makes sense, but I saw no mathematical justification for.