How to Solve the Schrödinger Equation for Free Particles
How to Shift the Symmetric Square Well Around the Origin
How Particles Pass Through Potential Barriers That Have Less Energy

How to Find the Normalized Wave Function for a Particle in an Infinite Square Well

In quantum physics, if you are given the wave equation for a particle in an infinite square well, you may be asked to normalize the wave function. For example, start with the following wave equation:

image0.png

The wave function is a sine wave, going to zero at x = 0 and x = a. You can see the first two wave functions plotted in the following figure.

Wave functions in a square well.
Wave functions in a square well.

Normalizing the wave function lets you solve for the unknown constant A. In a normalized function, the probability of finding the particle between

image2.png

adds up to 1 when you integrate over the whole square well, x = 0 to x = a:

image3.png

Substituting for

image4.png

gives you the following:

image5.png

Here’s what the integral in this equation equals:

image6.png

So from the previous equation,

image7.png

Solve for A:

image8.png

Therefore, here’s the normalized wave equation with the value of A plugged in:

image9.png

And that’s the normalized wave function for a particle in an infinite square well.

  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus
How to Find the Eigenfunctions of L2 in Spherical Coordinates
How to Find a Wave-Function Equation in an Infinite Square Well
How to Find the Energy Level of a Harmonic Oscillator: An Example
How to Determine Harmonic Oscillator Eigenstates of a System
How Particles Tunnel Through Potential Barriers That Have Greater Energy
Advertisement

Inside Dummies.com