Show this integral defines a scalar product.

In summary, the conversation discusses a problem from quantum homework where the goal is to show that an integral involving two single variable complex polynomials is a scalar product. The three criteria for a scalar product are mentioned, and the person seeking help has figured out linearity and positive definiteness but is stuck on showing conjugate symmetry. They have asked for help and provided relevant equations and their solution attempt so far. Another user has pointed out the need to do more in the positive definiteness part and has requested for the first user to contribute on a different thread.
  • #1
PhysStudent12
Hi,

I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2)
is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on showing that they have conjugate symmetry. Anyone know how to show this integral has conjugate symmetry? Thanks!
 
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  • #2
I have moved this thread to the homework forum. @PhysStudent12 please provide the information requested in the homework template--basically, what equations are applicable, and what attempts at solution you have made.
 
  • #3
PeterDonis said:
I have moved this thread to the homework forum. @PhysStudent12 please provide the information requested in the homework template--basically, what equations are applicable, and what attempts at solution you have made.
Sure, sorry about that, I'm new here.

The problem:

I have to show <p1|p2> = ∫-11 dx (p1*)(p2)
is a scalar product (p1 and p2 are single variable complex polynomials).

Relevant Equations:

A scalar product has three criteria:
Linearity: a<x|y> = <ax|y> and <x+y|z> = <x|z> + <y|z>
Positive Definiteness: <x|x> ≥ 0 and <x|x> = 0 iff x = 0
Conjugate symmetry: <x|y> = the conjugate of <y|x>

Solution Attempt:

Linearity is pretty easy, <ap1|p2> = ∫-11 dx a(p1*)(p2) = a∫-11dx (p1*)(p2) = a<p1|p2> and so forth. For positive definiteness, <0|0> = 0 trivially, and <p1|p1> should be positive since the negative parts should multiply by each other in the integral (correct me if I'm wrong here).

The part I need help on is showing conjugate symmetry. I'm not sure how to approach that part.

Thanks!
 
  • #4
PhysStudent12 said:
For positive definiteness, <0|0> = 0 trivially, and <p1|p1> should be positive since the negative parts should multiply by each other in the integral (correct me if I'm wrong here).
You need to do more here. The non-negativity actually follows from the non-negativity of the modulus of a complex number. Perhaps more importantly, remember the "iff" in
PhysStudent12 said:
Positive Definiteness: <x|x> ≥ 0 and <x|x> = 0 iff x = 0
So, indeed ##\langle 0 | 0 \rangle = 0##. But can you also deduce that ##\langle p | p \rangle = 0## implies ##p = 0## identically?
 

1. What is a scalar product?

A scalar product is a mathematical operation that takes two vectors and produces a single scalar quantity. It is also known as a dot product or inner product.

2. How is a scalar product defined?

A scalar product is defined as the product of the magnitudes of two vectors and the cosine of the angle between them.

3. What does it mean for an integral to define a scalar product?

In the context of linear algebra, an integral defines a scalar product if it satisfies the properties of symmetry, linearity, and positive-definiteness, which are necessary for a scalar product to exist.

4. How is an integral used to define a scalar product?

An integral is used to define a scalar product by integrating a function that represents the product of two vectors over a given domain. This results in a single scalar value that represents the magnitude of the two vectors and the angle between them.

5. What is the significance of a scalar product in physics and engineering?

A scalar product has many applications in physics and engineering, such as determining work done by a force, calculating moment of a force, and determining the angle between two vectors. It is also used in vector calculus, which is essential in solving many real-world problems in these fields.

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