Archive of posts with tag “complex”
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Regularizing the partition function of a hydrogen atom
The partition function of a hydrogen atom diverges (only considering bound states). However, we can regularize it to get finite answers. Different regularizations give the same result. They largely agree with the physical arguments for the case of the hydrogen atom at room or cold temperature, but this should be considered a mere coincidence. The results from regularized partition functions cannot generally be trusted. -
The duality between two plane trajectories related by a conformal map
The conformal map transforms the trajectory with energy in potential into the trajectory with energy in potential . I will prove this beautiful result and show some implications of it. -
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 -fold symmetry is compatible with translational symmetry is the same as whether . -
You can replace with in the Schrödinger equation?
When someone asks you why it is here instead of 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. -
Drawing a heart using Joukowsky transformation
Joukowsky transformation of a circle centered at of radius 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: . It turns out that we can write canonical equations , Poisson brackets
, and canonical transformations
in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.
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