About solving differential equations

[ice109]

i'm in my second semester of qm and we've now solved the quantum harmonic oscillator, the spherical well, and the hydrogen atom and all these problems are solved by the frobenius method ( series ). so I'm left wondering if most of the differential equations physicists solve using this method, first separation of variables on the pde and then the frobenius method, power series ansatz, on the resulting odes? if not can someone post some other ones?

note i know that the harmonic oscillator can be solved using operator splitting so if anyone knows of some other problems which are solved like that please post those as well.

 

[CompuChip]

Often physicists make use of series, but that depends on the problem. When possible, they will always start by a separation of variables, because it makes problems much easier. However, if the resulting equation is something like $u''(t) + u(t) = c$ then there is no need to use a power series to solve it (e.g. it is immediately clear that plugging in a solution like $u(t) = e^{\lambda t}$ will do). The Schroedinger equation for a free particle, finite square well, and harmonic oscillator can be solved perfectly well without using a series Ansatz. On the other hand, using a series solution can be easier when you pre-suppose certain symmetries. For example, if you have a potential which is rotationally invariant around the z-axis or even spherically symmetric, you expect the same symmetry for the solution. While it is perfectly possible to solve for the wave function in Cartesian coordinates, it is much more convenient to express the wave-function as a series of spherical harmonics. Such a series is completely determined by its coefficients, and finding them should be easy precisely because of the presence of a symmetry (for example, we can immediately see that those with $\ell \neq 0$ should not contribute or that a certain coefficient will be zero because the integral which we need to calculate to determine it is anti-symmetric).