Quantum Mechanics
Introduction
The quantum world is governed by the Schrödinger equation
{\displaystyle {\hat {H}}|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle }
where \hat H is the Hamiltonian, |\psi (t)\rangle is the quantum state function and \hbar is the reduced Planck constant.
The Hamiltonian consists of kinetic energy \hat T and potential energy \hat V. As in classical mechanics, potential energy is a function of time and space, whereas the kinetic energy differs from the classical world and is calculated as
\hat T = - \frac{\hbar^2}{2m} \nabla^2 .
The final version of the single particle Schrödinger equation can be written as
\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) \psi(t, \mathbf r) = i\hbar {\frac {\partial }{\partial t}}\psi(t, \mathbf r)
Quantum state function is a complex function, so it is usually split into the real part and imaginary part
u, v \in C(\mathbb R)\colon \psi = u + i v ,
which for a real V yields a system of two real equations
\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) ,
which may be easier to handle.
Harmonic oscilator
By selecting the potential V(t, \mathbf r) and the initial state \psi(0, \mathbf r) we get a unique solution for time propagation of the quantum state function. Probably the most used and well known example is the quantum harmonic oscilator, where we select a quadratic potential
V(t, \mathbf r) = V(\mathbf r) = \frac{1}{2} m \omega^2 r^2 ,
where m is the mass of the particle and \omega is the angular frequency of the oscilator.
The 1D harmonic oscilator has known eigenstate solutions
\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .
where the functions H_n are the physicists' Hermite polynomials. Time propagation of eigenstates is described with
\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)
