Changing bases (with bra-ket notation)

[iScience]

http://i.imgur.com/ORtBJdT.jpg

i don't understand why the old base is written in terms/as a linear combination of the new bases. wouldn't i want to map my coordinates from old to new not new to old?..

here's what my textbook says about it, can you guys interpret this for me, i still don't get it..

(it's the stuff of the red outline)

http://i.imgur.com/n5HJ7F2.jpg

 

[jcsd]

Let's say you have an arbitrary vector written in terms of the old basis. Is it easier to write it in terms of the new basis knowing what each of old basis vectors are in the new basis, or is it easier to start by knowing what the new basis vectors are in the old basis?

 

[iScience]

i don't follow; how are the two any different?

in both cases it sounds like I'm writing the new basis in terms of the old. can you elaborate on what you mean?

 

[jcsd]

Given a vector:

[tex]|v\rangle = \sum^n_{j=1} x_j|e_j\rangle[/tex]

and that

[tex]|e_j\rangle = \sum^n_{i=1} S_{ij}|f_i\rangle[/tex]

then

[tex]|v\rangle = \sum^n_{j=1}\sum^n_{i=1}x_j S_{ij}|f_i\rangle[/tex]

Or in other words knowing how to express the old basis vectors in terms of the new basis vectors tells you how to express a vector in the new basis given it's expression in the old basis.

 

[CellCoree]

I can understand that this concept may seem confusing at first. Let me try to provide a clearer explanation.

In quantum mechanics, bases are used to represent the state of a system. These bases can be chosen arbitrarily, but some bases may be more convenient to use than others. When we change bases, we are essentially changing the way we represent the state of the system. This can be useful in solving certain problems or making calculations easier.

In the image provided, the old bases are written in terms of the new bases because this allows us to map the coordinates from the old bases to the new bases. Think of it as a transformation, similar to converting from Cartesian coordinates to polar coordinates. We are not mapping from new to old, but rather from old to new.

The red outline in the textbook image is simply showing the relationship between the old and new bases. The new bases are expressed as a linear combination of the old bases, which means that the new bases can be written as a combination of the old bases. This is important because it allows us to map the coordinates from the old bases to the new bases.

I hope this helps to clarify the concept of changing bases in quantum mechanics. It may take some time to fully understand, but with practice and further study, it will become clearer.