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...
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...
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...
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?