"Physics Math" and inner product

topsquark

Forum Staff
Jan 2006
11,568
3,453
Wellsville, NY
I just read the intro chapter of my Quantum Mechanics text and I need someone to double check what I think is an example of "Physics Math"... that is Math done not quite correctly in order to produce a result that works okay.

QM is pretty much all Linear Algebra. Let's assume we have a complex vector space, V. (The vector space of kets in this example. Kets are noted in the form |v> .) There is a vector space dual to V, let's call it V*. (This is the bra vector space, noted as <f|. )

Now, the text introduces an "inner product" such that we have a "bracket" <f|v> which is just a real number.

But this isn't really an inner product? I would expect an inner product to be a mapping: \(\displaystyle \odot ~ : V \times V \to \mathbb{R}\), not \(\displaystyle \odot ~ : V^* \times V \to \mathbb{R}\). The latter is more reminiscent of function notation f(v). (Or more precisely <f| ( |v>).) Now, the property that the text is after is that the bracket represents the following sum: \(\displaystyle \sum_i a^i b^i\) which we can define on V* x V so it gives us correct results. But strictly speaking this isn't an inner product, right?

Thoughts?

-Dan
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Strictly speaking it isn't an inner product. But there is an isomorphism between the vector space (the "kets") and its dual (the "bras") so the rule is "to take the inner product of to vectors, u and v, take the dual to u, u*, and calculate u*(v)".
 
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