Archive of posts in 2023

Feb 2, 2023

Even solutions to bound states in an odd number of $\delta$ potential wells

Article cover image with no practical usage We try solving the even function solutions to the time-independent Schrödinger equation for the potential $V=-\alpha\sum_{j=-n}^n\delta(x-ja)$ such that $E<0$ (bound states).

/physics #from zhihu #quantum mechanics

Feb 1, 2023

Defend our earth against aliens’ bullets!

Article cover image with no practical usage The aliens intiated their attack to the earth! They shoot bullets with mass $m$ and speed $v$ from a far-awar planet. To defend, humans built a field $U=\alpha/r$ that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.

/physics #calculus #classical mechanics #kepler problem #from zhihu

Jan 18, 2023

This is what will happen after you get a haircut

Article cover image with no practical usage Denote the length distribution of one’s hair to be $f(l,t)$ , where $l$ is hair length, and $t$ is time. Considering that each hair may be lost naturally from time to time (there is a probability of $\lambda\,\mathrm dt$ for each hair to be lost within time range from $t$ to $t+\mathrm dt$ ) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.