Quantum Physics For Dummies, Revised Edition
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In quantum physics, you can use the Schrödinger equation to see how the wave function for a particle in an infinite square well evolves with time. The Schrödinger equation looks like this:

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You can also write the Schrödinger equation this way, where H is the Hermitian Hamiltonian operator:

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That’s actually the time-independent Schrödinger equation. The time-dependent Schrödinger equation looks like this:

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Combining the preceding three equations gives you the following, which is another form of the time-dependent Schrödinger equation:

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And because you’re dealing with only one dimension, x, this equation becomes

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This is simpler than it looks, however, because the potential doesn’t change with time. In fact, because E is constant, you can rewrite the equation as

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That equation makes life a lot simpler — it’s easy to solve the time-dependent Schrödinger equation if you’re dealing with a constant potential. In this case, the solution is

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Neat. When the potential doesn’t vary with time, the solution to the time-dependent Schrödinger equation simply becomes

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the spatial part, multiplied by

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the time-dependent part.

So when you add in the time-dependent part to the time-independent wave function, you get the time-dependent wave function, which looks like this:

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The energy of the nth quantum state is

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Therefore, the result is

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where exp (x) = ex.

About This Article

This article is from the book:

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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