I have been working through Leonard Susskind's book Qunatum Mechanics: The Theoretical Minimum. I have decided that I like Dirac notation. I have always found matrices easy to understand so I have been translating Dirac notation into my language. I decided to work in the ordinary 3D Cartesian space. This illustrates what is going on although it is not proof that it works in any coordinate system (basis). So here is the x-axis unit vector. I've called it i. Dirac writes it as |i>
But there is also a line vector version. This is called the DUAL. Dirac writes it as
. Dirac called this bracket notation because of the outside brackets, or bra-ket notation. The dual is called the ket. The inner product gives just a number.
I then multiplied them the other way round and found out that I got a matrix. This is called taking the OUTER PRODUCT. In quantum mechanics, a matrix is called an OPERATOR. I did it with all 3 basis vectors for x, y and z coordinates.
When you add the three matrices together you get the identity matrix I. It is like the number 1 for matrices. If you multiply any matrix by I, you get the same matrix back. I did it for the limited case of 3D Cartesian space but here is the general result for any orthogonal basis. This is what I was trying to understand. More on orthogonal later but in 3D space it means the vectors are at 90 degrees to each other.
