Particle in a 1-Dimensional box: Difference between revisions
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===A Mathematical Model=== | ===A Mathematical Model=== | ||
We will begin to solve the Schrodinger equation with the boundary conditions as shown above. | |||
From the points <math> x = 0 </math> to <math> x = L </math>, there is 0 potential energy. Using the time-independent Schrodinger Equation with <math> V(x) = 0 </math>, we arrive at the following Schrodinger Equation: <math>-\frac{\hbar^2}{2m}\frac{\partial^2\psi(x)}{\partial x^2}= E \psi(x)</math> | |||
===A Computational Model=== | ===A Computational Model=== | ||
Revision as of 23:24, 22 April 2022
claimed by Eathan 4/14/2022 The time-independent Schrodinger Equation is a partial differential equation whose solutions describe the wave function of a quantum system. It is given in the following form. [math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{\partial^2\psi(x)}{\partial x^2} + V(x)\psi(x)= E \psi(x) }[/math]
where
[math]\displaystyle{ \hbar }[/math] is the reduced Planck constant
[math]\displaystyle{ m }[/math] is the mass of the particle
[math]\displaystyle{ \psi(x) }[/math] is the wave function
[math]\displaystyle{ V(x) }[/math] is the potential of the system
[math]\displaystyle{ E }[/math] is the energy of the system
The particle in a 1-Dimensional box is a quantum system in which a particle is bounded in a well with infinite energy at the barrier. Classically, the particle in this system cannot escape. However, in this quantum system, we can demonstrate the particle's quantum tunneling by solving the time-independent Schrodinger Equation.
The Main Idea
Imagine the quantum system shown above, where the particle is bounded by infinite potential energy at [math]\displaystyle{ x = 0 }[/math] and [math]\displaystyle{ x = L }[/math]. The system also has 0 potential energy within the boundaries. We will use these as our boundary conditions to solve for the wave function [math]\displaystyle{ \psi(x) }[/math] below.
A Mathematical Model
We will begin to solve the Schrodinger equation with the boundary conditions as shown above. From the points [math]\displaystyle{ x = 0 }[/math] to [math]\displaystyle{ x = L }[/math], there is 0 potential energy. Using the time-independent Schrodinger Equation with [math]\displaystyle{ V(x) = 0 }[/math], we arrive at the following Schrodinger Equation: [math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{\partial^2\psi(x)}{\partial x^2}= E \psi(x) }[/math]
A Computational Model
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