Recent content by Robin04

I am writing my application to the PSI program of Perimeter Institute. As I am not a native English speaker, I am not sure whether my statement of purpose is strong enough. In the application form, there is a list of topics that I can comment on in my letter, and I tried to write something about...

So let the wavefunction be $$\psi(x)=A(\tanh(x)-ik)e^{ikx}+B(\tanh(x)+ik)e^{-ikx}$$ The limit in ##x\rightarrow -\infty## is $$\psi(x\rightarrow -\infty)=-A(1+ik)e^{ikx}+B(ik-1)e^{-ikx}$$ and in ##x\rightarrow \infty## $$\psi(x\rightarrow \infty)=A(1-ik)e^{ikx}+B(ik+1)e^{-ikx}$$ What should...

We have the potential $$V(x)=-\frac{1}{\cosh^2 (x)}$$ Show that the Schrödinger equation has the solution $$\psi(x)=(\tanh(x)-ik)e^{ikx}$$ and calculate the transmission and reflection coefficients for the scattering process. It is easy to show that the given wavefunction indeed solves the...

Seeking for a general formula might sound too optimistic to me. I only need to generate lists for small values of S, let's say up to 10-20. If no explicit formula could be given, I would already be satisfied with a more efficient algorithm to generate the lists. My current code, for S = 5, has...

Yes, ##K## is a constant and ##S## runs from 1 to K.

Let us consider two sequences: $$n_k \in \Omega,\,k=1,2,...K,$$ $$m_k \in \Omega,\,k=1,2,...K,$$ where $$\Omega:=\{n\in\mathbb{N}|\,n\leq K\}.$$ Let us define $$\sigma_n:=\sum_{k=1}^K k\, n_k,\,\sigma_m:=\sum_{k=1}^K k\,m_k$$ The task is to find all possible ##(n_k,\,m_k)## pairs such that...

Thank you!

Is is true that the dihedral group ##D_n## does not have an irreducible representation with a dimension higher than two?

Another idea: classically, the bound states in a square-well potential are independent of the depth of the well, while in quantum mechanics for a given level the wave functions take different values at the classical turning points for different depths. In zeroth-order, no quantum correction...

Right wave function for harmonic oscillator? Those should be Hermite-polynomials, not simple exponentials, no? By correct result I mean eigenvalues. That is true, but the ##\pm## in the argument means that it is a linear combination of a ##+## and a ##-## term. My writing was a bit...

That is true, but I assume there's a (hidden) step in this calculation that assumes that the wavefunction at the turning points is zero.

The Wilson-Sommerfeld quantization rule claims (##\hbar=1##) $$\frac{1}{2\pi} \oint p(x)\,dx=n,\,n=1, 2, ...$$ where integration is done in the classically allowed region. Applying this to a square-well potential with a depth of ##V_0## and width ##a##, we get $$E=\frac{\pi^2 n^2}{2a^2}$$ This...

If I understood well, cosmology makes a difference between matter moving in spacetime and the expansion of spacetime itself. Are these concepts experimentally distinguishable, or this distinction is only in our theories?

How do you obtain a recursion for an integral? :O

Mathematica gives it with elliptic functions. I wonder how it found the solution to the original form without them... Always a mystery :/