Complex Inner Products: Skew Symmetry & Linearity

In summary, there are two definitions of the complex inner product - one that follows skew symmetry and linearity in the first component, and one that follows linearity in the ket vector. This is a matter of convention between mathematicians and physicists, but it does not change the results as long as there is consistency. However, the alternate definition does result in the complex conjugate appearing in different places. This is seen in the definition of "bar" and is used in Griffiths's "Introduction to Quantum Mechanics". However, this definition is specific to a function space in the position representation and may not apply to other spaces. This causes confusion when trying to make general statements using specific examples.
  • #1
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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?
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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》.
 
  • #5
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.
 
  • #6
quanjia said:
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.
 

1. What are complex inner products?

Complex inner products are mathematical operations that take two complex numbers and produce a single complex number as a result. They are similar to regular inner products, which are used to measure the angle between two vectors, but they are defined on complex vector spaces instead of real vector spaces.

2. How do complex inner products relate to skew symmetry?

Complex inner products are said to be skew symmetric if they satisfy the property of skew symmetry, which means that the inner product of two vectors is equal to the complex conjugate of the inner product of the second vector with the first vector. This property is important in defining the orthogonality of complex vectors.

3. What is the significance of linearity in complex inner products?

Linearity is an important property of complex inner products, as it allows for the use of complex inner products in linear algebra and other mathematical applications. Linearity means that the inner product is distributive and follows the rules of addition and scalar multiplication, making it a useful tool for solving complex equations and systems of equations.

4. How are complex inner products calculated?

Complex inner products can be calculated using the formula:

〈u,v〉 = u*v = u1v1 + u2v2 + ... + unvn


where u and v are complex vectors and u* is the complex conjugate of u.

5. What are some real-world applications of complex inner products?

Complex inner products have many applications in various fields, including signal processing, quantum mechanics, and digital communications. They are also used in image processing and data compression algorithms, as well as in the analysis of network traffic and data streams. In addition, complex inner products are used in machine learning and artificial intelligence for tasks such as pattern recognition and data classification.

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