Quantum Physics For Dummies, Revised Edition
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In quantum physics, you need to know how to use linear operators. An operator A is said to be linear if it meets the following condition:

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For instance, the expression

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is actually a linear operator. In order to understand this, you need to know just a little more about what happens when you take the products of bras and kets. Firstly, if you take the product of the bra,

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where c is a complex number, then you get the answer,

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Secondly, if you take the product of the bra,

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then you get the answer,

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Now that you know this you can test to see if

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is actually a linear operator. Okay then, you can now apply

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to a linear combination of kets, like so,

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where c1 and c2 are complex numbers. Now that you know how the product of a bra with a sum of two kets goes, you can say,

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Then, as you know,

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you can finally write this as,

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This is exactly what a linear operator should do — if you replace A in the above equation defining a linear operator, with

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then the result is the same as the one you just found. So

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is indeed a linear operator — although a pretty funny looking one!

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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