i guess I mean that, while one is accustomed to saying "there's two sign choices in taking a square root", that doesn't necessarily mean they're both valid.
one observation: it makes a lot more sense to me for the wavefunction to have $\frac{1}{\sqrt{k(x)}}$ as prefactor than $\sqrt{k(x)}$
simply because $k(x)\to 0$ at a turning point.
so the former implies that this solution must break down in the vicinity of a turning point, whereas the latter wouldn't
and that's indeed the case.
but, to the matter at hand...hmm
one can write equation 2.5.44 as $$W_1'(x)^2 = W_0'(x)^2+i \hbar W_0''(x)=W_0'(x)^2\left(1+i\hbar \frac{W_0''(x)}{W_0'(x)^2}\right)\approx W_0'(x)^2\left(1+\frac{1}{2}i\hbar \frac{W_0''(x)}{W_0'(x)^2}\right)^2$$
yeah. my point is more that sakurai's presentation is really quite similar to shankar's, but shankar's approach avoids the sign issue entirely whereas sakurai creates one and doesn't explain it.
so if you trust Shankar, then you don't have to worry about it in the first place.
I don't find that terribly satisfying, but i'm also a bit tired of thinking about it :/
i'll confess, though, that I find WKB derivation to be tiresome in general. i really like the results they let you obtain---tunneling, quantization, etc.
what do you think the best way to convince someone the earth i is a sphere? assuming they have knowledge of math but no physics other than what they can directly observe
ground up cant trust newton, cant trust anything type style-
lets say they believe the earth is flat and is the stereo-graphic projection of the sphere such that the arctic surrounds the outside
just vector bundles for now :) I attended a seminar where a grad student gave an exposition on the Bott periodicity from VBKT. I didn't find it exciting.
it seemed like a technical theorem. I think it admits a homotopy-theoretic translation which does sound interesting however
but Hatcher's exposition doesn't go through that line
what properties of a sphere are different from a plane in which the top and bottom "borders" are connected and the right and left "borders" are connected
Need some help in understand the proof here: http://dbfin.com/topology/munkres/chapter-1/section-3-relations/problem-13-solution/ > Besides, for any other lower bound x′ of S , x′∈L , therefore, x′≤x .
@danu But aren't we already showed that $x\leq x'$ since x is the lowest upper bound of $L$ (which is also a lower bound of $S$), thus how can $x' \leq x$ ?
assuming that's what you mean, the sphere and the torus have a whole world of differences. for a start - the former is simply connected, the latter isn't.
the original question was how to prove with math and no physics that the earth is not flat, the easiest of course is to measure a bunch of shadows at the same time... but any finite number seems to be also represenntable by a displaced sun on a flat earth.
ok http://dbfin.com/topology/munkres/chapter-1/section-3-relations/problem-13-solution/
> Besides, for any other lower bound x′ of S , x′∈L , therefore, x′≤x .
@shaihorowitz But aren't we already showed that $x\leq x'$ since x is the lowest upper bound of $L$ (which is also a lower bound of $S$), thus how can $x' \leq x$ ?
Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways:
$$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$
$$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$
$$u_1=v_1=0$$
$$u_2=v_2=1$$
$$\lim_{n \to \infty} \frac{n}{u_n}=e$$
$$\lim_{n \to \infty} \frac{2n}{v_n^2}=\pi$$
...
I do need to beef up my knowledge on sequences later, so that I will be less confused on the symmetries and other things in the partial sums that are responsible for the outcome of the sequences
While I like the poetic nature of Grothendieck's description of his views on mathematics a lot, I get the impression that he had an overly grand view about mathematics. I can't say I share that feeling (I should mention I haven't read much of Grothendieck, his mathematical or mathematico-literary works alike).
I need a function that has the same values as
$f(x)=\sin\left(\frac{x}{2}\right)\cdot\sin\left(x^2\right)$
between $x=0$ and the following root of $f(x)$ at $x > 0$. At any other point, the function may differ from f (the reason I need this function is that I need to get the area under $f$ betw...
Is it true that among all rectangles inscribed in a right angle triangle,square has the largest area?Does it always have to be a square?Can't it be any other rectangle?
For any nontrivial algebraic structures with additive identity 0 and multiplicative identity 1 (and binary operation defined by "juxaposition of its arguments"), and at least one sided distributive law holds, one can easily prove that 0 is an absorber, i.e. $0x=0$ for $x=\{0,1\}$ as follows:
$$0...
Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways:
$$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$
$$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$
$$u_1=v_1=0$$
$$u_2=v_2=1$$
$$\lim_{n \to \infty} \frac{n}{u_n}=e$$
$$\lim_{n \to \infty} \frac{2n}{v_n^2}=\pi$$
...
I need a function that has the same values as
$f(x)=\sin\left(\frac{x}{2}\right)\cdot\sin\left(x^2\right)$
between $x=0$ and the following root of $f(x)$ at $x > 0$. At any other point, the function may differ from f (the reason I need this function is that I need to get the area under $f$ betw...
