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:

For instance, the expression

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,

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

Secondly, if you take the product of the bra,

then you get the answer,

Now that you know this you can test to see if

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

to a linear combination of kets, like so,

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,

Then, as you know,

you can finally write this as,

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

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

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