Revision as of 14:38, 11 January 2018

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

\[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)\]