Quantum Chemistry/Example 2

Write a question and its solution that demonstrates the correspondence principle for a particle in a 1D box

QuestionEdit

Starting with the wave equation of a 1D box:

 

Show that the average probability of the particle in a 1D box follows the correspondence principle given that the average probability according to classical mechanics is:

 

Where   is the length of the 1D box,   is the principle quantum number, and   is the position of the particle in a 1D box.

SolutionEdit

The probability distribution of a particle in a 1D box is represented by the wavefunction multiplied by it's complex conjugate   over the full length of the box. The probability of finding a particle in a specific range is determined by integrating the wavefunction multiplied by it's complex conjugate over a distance between two given values:

 

 

 

The correspondence principle states that as quantum numbers become large, quantum mechanics reproduces expected results from classical mechanics. Therefore, the average probability of finding a particle in a 1D box for quantum mechanics should match classical mechanics as the quantum number reaches infinity according to the correspondence principle.

The average value of an integrand is given by the formula:

 

In this example, the function to be integrated is a   function, which is a function with continuously repeating cycles from   to  . Therefore, determining the average over one cycle determines the average over an infinite amount of cycles, going to infinity represents the principle quantum   going to very large numbers. So, the average of   as   goes to infinity is determined between one cycle from   to  .

 , as  

Using the trigonometric relationships below, the solution to the original integral becomes trivial.

 

 

 

The new value of   is inserted into the integral and solved for.

  , as  

 , as  

 , as  

 , as  

Thus, the average value of   as the principle quantum number goes to infinity is equal to  . By plugging that value into the probability distribution formula for a particle in a 1D box, the average probability becomes  .

 , as  

 

 

This matches the average probability of a particle in a 1D box for classical mechanics as given in the question and demonstrates the correspondence principle using the 1D box model.