In this video, we take a look at the energy states of a single-particle system in a harmonic potential. The time-independent Schrödinger equation is solved to obtain a result that is characteristic of quantum mechanics: we find that the energy of the particle must be quantized!
00:00 : Intro
00:51: Finding the potential for a harmonic oscillator
01:43: Finding the Hamiltonian
03:00: Simplifying the Schrödinger equation
12:46: Exploring the properties of the simplified Schrödinger equation
14:20: Finding a solution in the limit approaching infinity
16:40: Finding a general solution for all values
18:00: Solving the differential equation for the general solution
27:30: Ensuring normalizability of the wavefunction
33:00: Obtaining the possible energy values
34:08: Reflections on quantization