Quantum Physics For Dummies, Revised Edition
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The Hamiltonian represents the total energy of all the particles in a multi-particle system. You can describe that system in quantum physics terms. The following figure shows a multi-particle system where a number of particles are identified by their position (ignoring spin).

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To find the total energy for this system, begin by working with the wave function. The state of a system with many particles, as shown in the figure, is given by

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And here’s the probability that particle 1 is in d3r1, particle 2 is in d3r2, particle 3 is in d3r3, and so on:

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The normalization of

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demands that

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Okay, so what about the Hamiltonian, which gives you the energy states? That is, what is H, where

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When you’re dealing with a single particle, you can write this as

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But in a many-particle system the Hamiltonian must represent the total energy of all particles, not just one.

The total energy of the system is the sum of the energy of all the particles, so here’s how you can generalize the Hamiltonian for multi-particle systems without spin:

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This, in turn, equals the following:

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Here, mi is the mass of the ith particle and V is the multi-particle potential.

About This Article

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About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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