Complex Inner Products: Skew Symmetry & Linearity

[SeReNiTy]

Previously i learned from maths that the complex inner product satisfied skew symmetry and linearity in the first component, ie - <aA+bB,C> = a<A,C> +b<B,C>

But after studying Shankar in quantum mechanics, he claims the linearity is in the ket vector, ie - <A,bB+cC> = b<A,B> +c<A,C> which would mean that the complex conjugate of the constants b,c appear in the wrong spots?

 

[StatMechGuy]

This is a matter of convention. Mathematicians do it one way, but physicists do it another. It doesn't actually change anything, as long as you remain consistent.

 

[SeReNiTy]

But with this alternate complex inner product you actually get the complex conjugate, so both definitions are not equivalent as with one you get the conjugate of the other.

 

[quanjia]

The definition of "bar" is
integrating{(complex conjugate of[f(x)]) *g(x)} over x,
when <f(x)| hit |g(x)>.

So <aA| equal the complex conjugate of a multiplying <A|.

The definition is used in J.Griffiths's 《Introduction to Quantum Mechanics》.

 

[George Jones]

Yes, it's just convention, and amounts to a change in order of the vectors, i.e., <A,B> is for a physicist what <B,A> is for a mathematician.

 

[StatMechGuy]

The definition of "bar" is
integrating{(complex conjugate of[f(x)]) *g(x)} over x,
when <f(x)| hit |g(x)>.

So <aA| equal the complex conjugate of a multiplying <A|.

The definition is used in J.Griffiths's ?Introduction to Quantum Mechanics?.

Griffiths's book presumes that we're working in a function space in the position representation. I could be working in a discrete space working in the coherent state representation. This is one of the many problems I have with that book: it tries to make general statements using very specific examples, and that leads to confusion.