Understanding Why < x | f > = f ( x )

  • Thread starter maria clara
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In summary, the conversation discusses the concept of the bra-ket notation and why the eigenvalue is written in the bra instead of the eigenfunction. It is explained that the bra-ket notation is an inner product of functions and the product < x' | f > should be written as < delta (x-x') | f >. The reason for writing the eigenvalue in the bra is simply a notational abbreviation.
  • #1
maria clara
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0
Why is it true that
< x | f > = f ( x )
?
shouldn't it be the result of the integral from minus infinity to infinty on xf(x)?
but then it can't even be a function of x...:confused:
 
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  • #2
maria clara said:
Why is it true that
< x | f > = f ( x )
?
shouldn't it be the result of the integral from minus infinity to infinty on xf(x)?
but then it can't even be a function of x...:confused:

Here, x is an "eigenvalue" of the position operator. What you need to use in the integral is not the eigenvaule, but the state that is the "eigenfunction" that corresponds to this eigenvalue.

What states are eigenstates of the position operator?
 
  • #3
detla functions... and it certainly works that way, thanks:smile:

but I'm still confused: the definition of the bra-ket is simply the inner product of the functions you put there:
< f | g > = integral from minus infinity to infinity on f*g (f*=the complex conjugate of f).
So the product < x' | f > should be written as follows:
< delta (x-x') | f >...
why do they write the eigenvalue in the bra, and not the function itself?
 
  • #4
maria clara said:
detla functions... and it certainly works that way, thanks:smile:

but I'm still confused: the definition of the bra-ket is simply the inner product of the functions you put there:
< f | g > = integral from minus infinity to infinity on f*g (f*=the complex conjugate of f).
So the product < x' | f > should be written as follows:
< delta (x-x') | f >...
why do they write the eigenvalue in the bra, and not the function itself?

They write the eigenvalue in the bra because they are too lazy to write the eigenfunction. Really, it's just a notational abbreviation.
 
  • #5
everything makes much more sense now, thanks a lot!:smile:
 

1. What does "Understanding Why < x | f > = f ( x )" mean?

The notation < x | f > represents the inner product between the vector x and the function f. It is a mathematical concept used in functional analysis to measure the similarity or "closeness" between a vector and a function.

2. Why is it important to understand this equation?

This equation is important because it allows us to describe the relationship between vectors and functions in a mathematical framework. It helps us to analyze and understand the properties and behavior of functions in a more rigorous way.

3. How is this equation used in science?

This equation is used in various fields of science, such as physics, engineering, and data analysis. It is used to model physical phenomena, make predictions, and analyze data by representing functions as vectors and using the inner product to measure their similarity.

4. Can you provide an example of how this equation is used?

One example of how this equation is used is in quantum mechanics, where the wave function of a particle is represented as a vector and the inner product is used to calculate the probability of finding the particle in a particular state.

5. What are some common misconceptions about this equation?

One common misconception is that the notation < x | f > represents a multiplication between the vector x and the function f. In reality, it is a more abstract concept that measures the similarity between the two. Another misconception is that this equation only applies to vectors and functions in a finite-dimensional space, when in fact it can be extended to infinite-dimensional spaces as well.

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