We get the time-independent Schrödinger equation by substituting this into Equation "time-independent Schrödinger equation for a one dimensional particle":
Now, there are two ways to go about this.
The first is the stupid "here's a guess" + "hey this family of solutions forms a complete basis"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.
The second is the much celebrated ladder operator method.
TODO are there experiments, or just theoretical?
Show up in the solution of the quantum harmonic oscillator after separation of variables leading into the time-independent Schrödinger equation, much like solving partial differential equations with the Fourier series.
I.e.: they are both:
- solutions to the time-independent Schrödinger equation for the quantum harmonic oscillator
- a complete basis of that space
Not the same as Hermite polynomials.
www.physics.udel.edu/~jim/PHYS424_17F/Class%20Notes/Class_5.pdf by James MacDonald shows it well.
And then we can prove the ladder properties easily.
The commutator appear in the middle of this analysis.
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The quantum harmonic oscillator is a fundamental concept in quantum mechanics that describes the behavior of a particle subject to a restoring force that is proportional to its displacement from an equilibrium position. This model is essential for understanding various physical systems, such as vibrations in molecules, phonons in solid-state physics, and quantum field theory. Here's a detailed overview of the quantum harmonic oscillator: ### 1.