## Archive of posts with tag “complex”

### The duality between two plane trajectories related by a conformal map

The conformal map $\fc wz$ transforms the trajectory with energy $-B$ in potential $\fc Uz\ceq A\v{\d w/\d z}^2$ into the trajectory with energy $-A$ in potential $\fc Vw\ceq B\v{\d z/\d w}^2$. I will prove this beautiful result and show some implications of it.

- Categories: physics
- Tags: complex, classical mechanics, canonical transformation, kepler problem, mathematical physics, vector analysis, long paper

### Rotational symmetry of plane lattices as a simple example of algebraic number theory

For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an $m$-fold symmetry is compatible with translational symmetry is the same as whether $\varphi(m)\le2$.

- Categories: math
- Tags: complex, condensed matter physics, algebraic number theory, lattice, mathematical physics

### 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

### Drawing a heart using Joukowsky transformation

Joukowsky transformation of a circle centered at $\left(1,1\right)$ of radius $1$ is a curve resembling a heart.

### Use complex numbers as canonical variables

In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: $\mathbf c\coloneqq\alpha\mathbf q+\mathrm i\beta\mathbf p$. It turns out that we can write canonical equations $\frac{\mathrm d\mathbf c}{\mathrm dt}=-2\mathrm i\alpha\beta\frac{\partial\mathcal H}{\partial\mathbf c^*}$, Poisson brackets $\left\{f,g\right\}=-2\mathrm i\alpha\beta \left(\frac{\partial f}{\partial\mathbf c}\cdot \frac{\partial g}{\partial\mathbf c^*}- \frac{\partial f}{\partial\mathbf c^*}\cdot \frac{\partial g}{\partial\mathbf c}\right)$, and canonical transformations $\frac{\partial\mathbf c^*}{\partial\mathbf c'^*}= \frac{\partial\mathbf c'}{\partial\mathbf c}, \frac{\partial\mathbf c}{\partial\mathbf c'^*}= -\frac{\partial\mathbf c'}{\partial\mathbf c^*}$ in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.

- Categories: physics
- Tags: classical mechanics, canonical transformation, hamiltonian, complex, long paper