Adjoint operator

Okay kiddo, let’s start with a quick math lesson. In math, we work with things called operators which are basically fancy ways of describing how numbers are transformed. One example of an operator is multiplication – it takes one number and multiplies it by another number to get a new number. Another example is differentiation – it takes a function and gives us another function that tells us how fast the first one is changing.

An adjoint operator is a special kind of operator that works with something called a Hilbert space. Now, a Hilbert space is just a special type of space that contains an infinite number of vectors, and is often used in math to describe things like functions or waves.

When we have an operator in a Hilbert space, the adjoint is like the operator’s “mirror image”. It’s a new operator that does something similar, but with the roles of the input and output switched.

Now, let's take a simple example to understand. Suppose we have a baby piano with only 2 keys. You hit the first key, and a different sound comes out than when you hit the second key. Let’s say we have an operator that takes the sound that comes out when you hit the first key and transforms it into the sound that comes out when you hit the second key.

If we want to find the adjoint of this operator, we would first switch the input and output. So, now our operator takes the sound that comes out when we hit the second key and transforms it into the sound that comes out when we hit the first key. This new operator is the adjoint of the original one.

So, basically an adjoint operator just switches the inputs and outputs of another operator when it’s operating in a special kind of space called a Hilbert space.