Difference between revisions of "Schrödinger equation"
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| − | A theoretical one dimensional potential | + | 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. A theoretical one dimensional potential |
\displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise,}}\end{cases}} | \displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise,}}\end{cases}} | ||
Revision as of 11:17, 21 May 2019
Solution procedure is still compiling ... so please wait for results :)
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.
Particle in a box
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. 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}}}.
