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How to Use Linear Operators in Quantum Physics

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:

image0.png

For instance, the expression

image1.png

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,

image2.png

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

image3.png

Secondly, if you take the product of the bra,

image4.png

then you get the answer,

image5.png

Now that you know this you can test to see if

image6.png

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

image7.png

to a linear combination of kets, like so,

image8.png

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,

image9.png

Then, as you know,

image10.png

you can finally write this as,

image11.png

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

image12.png

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

image13.png

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

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