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I have been working out of the beginning of Shankar lately, and I wanted to address some confusion I have had with regard to Dirac notation.

I know that physicists tend to love the notation, but to me, so far, it is confusing, inconsistent, and even occasionally contradictory. Here is a short list of some of what bothers me about it. I guess I am hoping that some of you can explain to me why some of these problems are not as bad as they seem, or even why they aren't actually problems.

1. Labels:

Kets are used to denote vectors, and a label is placed on the ket to name it. This is okay for me except when it comes to performing operations on vectors. For instance, let V be a vector space over C, and take |A> and |B> to be vectors in V. Also, let a and b be complex numbers. Then, Shankar uses the notation

a |A> + b |B> = |a A + b B>.

Therefore, a A + b B becomes a label for the new ket. This is okay with me except for cases like this: Let |1>, |2>, ..., |n> be a basis for V. Then, (3+2i) |1> - 4 |2> = |(3+2i) 1 - 4 2>.

I know that one shouldn't get hung up on notation, but that justkillsme.

2. Inner products:

This is basically the same thing as the first, but inner product labeling gets even worse. We denote the inner product of |V> with |W> by <V|W>. This seems to imply that the inner product of |LABEL 1> with |LABEL 2> is <LABEL 1|LABEL 2>. However, Shankar also writes things like

<V| (aW + bZ) which is decent except that it has the same labeling issues.

3. Dual space:

Thanks to the Riez Representation Theorem, every element of the dual space can be written as a bra. Because of this, I think writing things like <V| is very good notation. However, Shankar feels that ANYTHING acting on or acted on by a bra should go after the bra.

For instance, instead of 2 <V|, Shankar writes <V| 2 which is okay I guess if we know ahead of time that 2 is in the field and not the vector space. But what about this: let T be a linear transformation acting on bras. Then shankar writes <V| T instead of T <V| or T(<V|). Then he awful things like this:

<V|aT =<V|Ta when what he means is T(a<V|) = a T<V|

What about <V| T L where T and L are linear operators? Does it mean LT(<V|) or TL(<V|)?

Maybe I'm overreacting to all of this, but this really seems to be a big mess!

Anyway, thanks for reading! I would appreciate any discussion!

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# Bra-ket notation

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