Quantum Mechanics
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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
\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) , \] \[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) v(t, \mathbf r) = \hbar {\frac {\partial }{\partial t}} u(t, \mathbf r) , \]
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
\[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)\]
Particle in a box
A theoretical one dimensional potential
\[\displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise,}}\end{cases}}\]
is known as an infinite potential well. Its time independent eigenfunctions are
\[\sqrt{\frac{2}{L}}\psi_n(x) = \sin\left(k_n x \right), \qquad n = 1,2,3,...\]
where $k_n = \frac{\pi n}{L}$. With a time dependency similar to Harmonic oscilator
\[\psi_n(t, x) = \mathrm e ^ {-i \omega_n t} \psi_n(x),\]
where $\omega_n$ and $k_n$ are connected through dispersion relation through energy $E_n$
\[{\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}={\frac {\hbar ^{2} k_n^2}{2m}}}.\]
