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).

 

[sir-pinski]

The Frobenius method is a powerful tool for solving differential equations that arise in quantum mechanics. It is commonly used for solving problems involving the harmonic oscillator, spherical well, and hydrogen atom, as you have mentioned. However, it is not the only method used by physicists to solve differential equations.

In fact, there are many other methods that can be used depending on the specific problem at hand. Some examples include separation of variables, Laplace transforms, Green's functions, and numerical methods such as finite difference or finite element methods.

In addition, as you have mentioned, operator splitting is another approach that can be used for solving the harmonic oscillator. This method involves breaking down the differential equation into smaller, more manageable equations that can then be solved separately.

Ultimately, the choice of method depends on the specific problem and the preferences of the physicist. It is always important to have a variety of tools in your mathematical toolbox to approach different types of problems.

Some other problems that can be solved using the Frobenius method include the anharmonic oscillator, the Coulomb potential, and the Schrödinger equation for a particle in a one-dimensional potential well. I encourage you to explore these examples and other methods for solving differential equations in your studies.