Particle in a 1-Dimensional box: Difference between revisions

From Physics Book
Jump to navigation Jump to search
No edit summary
No edit summary
Line 24: Line 24:
We will begin to solve the Schrodinger equation with the boundary conditions as shown above.  
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>
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>
Isolating the second partial derivative term gives us the following:
<math>-\frac{\partial^2\psi(x)}{\partial x^2}= -\frac{2mE}{\hbar^2}\psi(x)</math>


The general solution to this partial differential equation is very well known and is given here: <math> \psi(x) = Asin(kx) + Bcos(kx) </math>
The general solution to this partial differential equation is very well known and is given here: <math> \psi(x) = Asin(kx) + Bcos(kx) </math>
Notice how when the potential is 0, the particle behaves as classical physics would predict.
===A Computational Model===
===A Computational Model===



Revision as of 23:35, 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]

Isolating the second partial derivative term gives us the following: [math]\displaystyle{ -\frac{\partial^2\psi(x)}{\partial x^2}= -\frac{2mE}{\hbar^2}\psi(x) }[/math]

The general solution to this partial differential equation is very well known and is given here: [math]\displaystyle{ \psi(x) = Asin(kx) + Bcos(kx) }[/math]

Notice how when the potential is 0, the particle behaves as classical physics would predict.

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Simple

Middling

Difficult

Connectedness

  1. How is this topic connected to something that you are interested in?
  2. How is it connected to your major?
  3. Is there an interesting industrial application?

History

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

Internet resources on this topic

References

This section contains the the references you used while writing this page