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I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state ($\psi(x,0)$) for the free particle wavefunction and interpret it such that I say that the particle is exactly at $x=0$ during time $t=0$? No, because the delta function is not compliant with the Born interpretation of the function $\psi$. Evolving function that is ...

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That is indeed how you would go about it. Note, however, that there is nothing to guarantee that the solution is going to be reasonable, or that the integral even exists. In fact, because the Schrödinger equation is time reversible to a large extent, you are essentially guaranteed to not end up in physical states. One thing to note is that the frequency ...

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Consider evolution of gaussian wave packet. Its wave function in position representation looks like: $$\Psi(\vec r,t)=\left(\frac a{a+i\hbar t/m}\right)^{3/2}\exp\left(-\frac{\vec r\cdot \vec r}{2(a+i\hbar t/m)}\right).\tag1$$ Corresponding relative probability density is $$P(r)=|\Psi|^2=\left(\frac a{\sqrt{a^2+(\hbar ... 0 See, you are required to find volume current density J_\phi. Though its name is volume current density, you know it is the current flowing per unit surface area. Now the subscript \phi in J_\phi denotes it is flowing in the \hat{\phi} direction. Now in the spherical polar co-ordinate the infinitesimal length elements along the direction ... 0 I found a compact version of the course notes, re-written in english by the same professor who gave the original lectures. You can find them at this link. 2 If you note that$$\delta(\cos\theta)=\frac{\delta(\theta-\pi/2)}{\sin\theta} Then you can see that the sine terms actually cancel out.

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EDITED ANSWER: The delta distribution $\delta(x)$ is not unique. It is invariant under transformations of the form $\delta(x) \to f(x)\delta(x)$ where $f(0) = 1$. This is because it is really a distribution and not a function. It is mathematically improper to talk about $\delta(x)$ instead of $\int \delta(x)dx$. Derivations of the term you're interested in ...

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I found following books useful: A Guide to Distribution Theory and Fourier Transforms By Robert S. Strichartz. Not very rigorous and not much content either. But good book to start from. Generalized Functions: Theory and Applications By Ram P. Kanwal. Not very rigorous. This book starts with chapter on Dirac delta function and then slowly builds the ...

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