Unitary Operators: Proving <Af,Ag>=<f,g>

In summary: Thanks again.In summary, the person is trying to show that a unitary operator obeys <Af,Ag>=<f,g>, where A is a unitary operator. However, they are not given the fact that the adjoint of A is equal to its inverse, which is the problem. They have no clue how to prove the given task without using the mentioned statement.
  • #1
Physicsdudee
14
2
Homework Statement
Prove properties of unitary operator
Relevant Equations
Unitary maps are bijective and have the property of <Af,Ag>=<f,g>
Hello folks,

I need to show that a unitary operator obeys <Af,Ag>=<f,g>, where A is a unitary operator. However, I am technically not yet given the fact, that the adjoint of A is equal to its inverse, and that is the problem. I have no clue how to prove the given task without using the mentioned statement.
The space the scalaroduct is defined in is the $L^2$ space.
 
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  • #2
What is your definition of a unitary operator?
 
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  • #3
PeroK said:
What is your definition of a unitary operator?
Well, my definiton of a unitary operator is the following: Let A be a unitary operator, then A is bijective and <Af,Ag>=<f,g> where f,g is an element of L2.
 
  • #4
Physicsdudee said:
Well, my definiton of a unitary operator is the following: Let A be a unitary operator, then A is bijective and <Af,Ag>=<f,g> where f,g is an element of L2.
And what are you trying to prove?
 
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  • #5
PeroK said:
And what are you trying to prove?
I need to prove one aspect of the given definition, namely the aspect that <Af,Ag>=<f,g>
 
  • #6
Physicsdudee said:
I need to prove one aspect of the given definition, namely the aspect that <Af,Ag>=<f,g>
That's the definition you gave. You can't prove a definition!
 
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  • #7
PeroK said:
That's the definition you gave. You can't prove a definition!
Yeah, well I was thinking the same thing, that is why I was confused about it, the question itself seemed a bit unclear.
 
  • #8
Physicsdudee said:
Yeah, well I was thinking the same thing, that is why I was confused about it, the question itself seemed a bit unclear.
There are generally two possible (and, of course, equivalent) definitions of a unitary operator.

1) It preserves the inner product.

2) Its adjoint is its inverse.

Whatever one you choose, you have to prove that the other is equivalent.

In fact, Wikipedia gives three equivalent definitions, but the thrid is more technical.

https://en.wikipedia.org/wiki/Unitary_operator
 
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  • #9
PeroK said:
There are generally two possible (and, of course, equivalent) definitions of a unitary operator.

1) It preserves the inner product.

2) Its adjoint is its inverse.

Whatever one you choose, you have to prove that the other is equivalent.

In fact, Wikipedia gives three equivalent definitions, but the thrid is more technical.

https://en.wikipedia.org/wiki/Unitary_operator
Thanks for the help. I have read into the article already some time ago. I am noticing that I have misunderstood the assignment given, I think, at least the way I understand the assignment now makes more sense. I will give it a go.
 
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1. What is the definition of a unitary operator?

A unitary operator is a linear transformation on a complex vector space that preserves the inner product between two vectors. In other words, the inner product of any two vectors remains the same after the transformation.

2. How do you prove that = for a unitary operator A?

To prove that = for a unitary operator A, we use the definition of a unitary operator and the properties of inner products. We first show that A preserves the inner product by showing that =. Then, we use the fact that A is unitary to show that =. Combining these two results, we can conclude that =.

3. What is the significance of proving = for a unitary operator A?

Proving = for a unitary operator A shows that A preserves the inner product, which means it preserves the geometric properties of the vectors. This is important in many applications, such as quantum mechanics, where unitary operators represent physical transformations.

4. Can you provide an example of a unitary operator and how = can be proven for it?

One example of a unitary operator is the identity operator, which leaves vectors unchanged. To prove = for the identity operator, we simply substitute A=I (identity operator) in the definition and we get =, which is true since the inner product of any vector with itself is equal to its norm squared.

5. Is it possible for a non-unitary operator to satisfy =?

No, it is not possible for a non-unitary operator to satisfy =. The definition of a unitary operator requires that it preserves the inner product, which means that = must hold for all vectors f and g. If a non-unitary operator satisfies this equation for some vectors, it will not hold for all vectors and therefore cannot be a unitary operator.

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