If you remember my previous post about how to prove the formula of Particle in a Box, we will see the primary difference between the concept of Particle in a Box and Particle in a Ring. Particle in a Box is a calculation in order to find the energy of a particle in a non-cyclic conjugated polyene compound. Otherwise, Particle in a Ring is a calculation in order to find the energy of a particle in a cyclic conjugated polyene, like Benzene.

To prove the formula, we have to back to our Hamiltonian Operator first:

To prove the formula, we have to back to our Hamiltonian Operator first:

There's a little change about what are the variables that will be substituted to Del. This is our challenge: What are the variables?

First, we have to know that in this calculation, we will assume that particle in a ring is in two dimensional room and we will use the cartecian coordinate so that:

To determine the x and y, we need to make another assumption: Function of f is operated with d/dx and d/dy so that:

Okay, let's rock!

First, we have to know that in this calculation, we will assume that particle in a ring is in two dimensional room and we will use the cartecian coordinate so that:

To determine the x and y, we need to make another assumption: Function of f is operated with d/dx and d/dy so that:

Okay, let's rock!

Okay, that's the formulas of the first coordinate (The X coordinate) now, we will find the second coordinate:

Okay, It's almost done!

Combining the two coordinates with Del:

Okay, we now get our variables, now let's back to the Hamiltonian Operator:

Now, you can continue this formula by your self with the same procedure like we did in the previous post (Particle in a Box) and the last, we will get the equation of Particle in a Ring as follows:

Okay, It's almost done!

Combining the two coordinates with Del:

Okay, we now get our variables, now let's back to the Hamiltonian Operator:

Now, you can continue this formula by your self with the same procedure like we did in the previous post (Particle in a Box) and the last, we will get the equation of Particle in a Ring as follows:

Thank you.

Source:

Documents of Pelatnas 2 IChO 2012

Preparatory Problems IChO 2012

Source:

Documents of Pelatnas 2 IChO 2012

Preparatory Problems IChO 2012

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