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I'm not an expert teacher or lecturer of chemistry. I was only a student from SMA NEGERI 15 SURABAYA who had been one of the Bronze Medalist Participants of Olimpiade Sains Nasional X (2011) of Chemistry In Manado, North Sulawesi, 11 - 16 September 2011 and graduated in 2012. Now, I'm studying at Universitas Airlangga in Surabaya, Indonesia. I do love chemistry and I would like to help them who had difficulties in studying chemistry. That's why, please understand me if you found some misconcepts in my entries. Suggestions are always necessary in order to develop this blog. And I'm sorry because my English isn't so well.

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Wednesday, April 11, 2012

Particle in a Ring

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:




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!






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:



Thank you.

Source:
Documents of Pelatnas 2 IChO 2012
Preparatory Problems IChO 2012


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