Blog of Veikko M.O.T. Nyfors, Hybrid Quantum ICT consultant

Quantum Mechanics demystified, a try

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Differential Equations

Oh dear!

Trying to grasp essence of Schrödinger’s wave equation went high over my head. What an earth does it mean to say equation


describes quantum mechanical system’s state and dynamics.

Turned out that is a differential equation. Those define dependencies and correlations of certain function $\Psi$, it’s derivates and other parameters. They are good for modeling behavior and evolvement of various systems over time and parameters.

Differential equations for a system are deduced by using rules, properties and other known attributes of the system, which describe the features, functionality and other aspects of the traits we are interested in the system.
Once you have a good enough equation, you can try to solve it to get the function to predict how the system behaves over time.

Specifically, by solving wave function \(\Psi\) out from Schrödinger’s wave equation, one can predict whereabouts of the quantum mechanical entity.
Quantum mechanical entity itself is very much described by the Hamiltonian operator \(\hat{H}\).

Well, anyway, one more path to catch on the quantum way: how to solve differential equations.
In this article I will present some of the methods for solving differential equations.

Simple differential equations

Simple differential equations are ones, that only contain derivates of the function, not the function itself. Like

\[y'+2x-1 = 0 \implies y'=-2x+1\]

These can be solved by simply integrating both sides of the equation

\[\int y' = \int -2x+1dx \implies y = -x^2 + x +C\]

where C is constant. C can have an arbitrary value, thus there is indefinite amount of solutions.
Often one has an intial condition specified, which limits the number of solutions.
In our sample, intial condition

\[y(-1) = 1\]

will give

\[y = -x^2 + x +3\]

as solution.

Separating differential equations

Linear first-order differential equations