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What is the reason Schroedinger equation is quoted in terms of potential energy instead of force?

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Schrödinger's Wave Equation is an application of Hamiltonian Mechanics. Unlike Newtonian Mechanics, Hamiltonian Mechanics relies on knowing about the things that contribute to the energy of the system. If you know the things which contribute to the energy of a system, then you can determine things like forces, accelerations, and positions. (All through the miracle that is calculus!) These forces and accelerations can also change over time, and we can better understand how they evolve over time using Lagrangian and Hamiltonian Mechanics. Even better, with Lagrangian and Hamilitonian Mechanics, we don't have to know how these forces change over time, because we can figure it out with math!

This approach makes life easier, because we often want to know what the energy of a particle is. This is especially true of small particles because a lot of their behaviors depend on the particle's energy. The ability to simply set a term to 0 allows us to solve for simple situations, (such as a "particle in a box" and "free particle" situations, where the potential energy is very often 0) which then let us solve more complicated situations later using various techniques, like Perturbation Theory.

This approach is super awesome because humans are get pretty good at identifying the various things that contribute energy to a system. Forces in a system, however, can be tricky, and it can be hard to guess which forces happen when.

To concisely answer your question, the Schrödinger Wave Equation has values for energies because of Hamiliton (and Legrange), and it's super useful.

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A less theoretical reason to consider is that the schrodinger equation usually describes charged particles. It's in general easier to create and measure potential energies (e.g voltages) than forces. Additionally, solving the schrodinger equation gives you energy information, which is observable as an emission spectrum.

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