## Archive of posts with tag “letter”

### You can replace $\mathrm i$ with $-\mathrm i$ in the Schrödinger equation?

When someone asks you why it is $-\mathrm i$ here instead of $\mathrm i$ or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.

- Categories: physics
- Tags: quantum mechanics, letter, complex

### Mapping from Kepler problem to free particle on 3-sphere

There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for $E<0$ is $\mathbf u\coloneqq\frac{p^2-p_0^2}{p^2+p_0^2}\hat{\mathbf n}+\frac{2p_0}{p^2+p_0^2}\mathbf p,$ where $\mathbf u$ is the position of the particle on 3-sphere (a 4-dimensional vector), $\mathbf p$ is the momentum of the original particle in Kepler problem, $\hat{\mathbf n}$ is a vector perpendicular to the 3-dimensional hyperplane where $\mathbf p$ lies, and $p_0\coloneqq\sqrt{-2mE}$.

- Categories: physics
- Tags: classical mechanics, canonical transformation, letter, kepler problem