# Is f an Eigenfunction of Operator A Given the Equality <f|A f>² = <f|A² f>?

• TeCTeP
In summary, the conversation discusses proving that a function f is an eigenfunction of an arbitrary operator A. The equation <f|f> = 1 is rewritten as <f|f|f> = |f>, and it is noted that the Hilbert subspace is assumed to be unidimensional with a basis formed by the vector |f>. The concept of a projector onto the vector |f> is introduced and it is mentioned that it is generally equivalent to the unit operator on this subspace, but in this case, due to orthonormalization, it is equal to the unit operator. It is also noted that the operator is named after the mathematician Charles Hérmite.
TeCTeP
where A is arbitrary operator (not only ermiton), f is function, that < f | f > = 1. How to prove, that f is eigenfunction of operator A?

Try to rewrite the equation.

i've prooved it when operator is ermiton, but i can't proove for arbitrary operator...

What do you know about this baby $|f\rangle\langle f|$...?

Daniel.

dextercioby said:
What do you know about this baby $|f\rangle\langle f|$...?

Daniel.
i've never used it before.

$$\langle f|f\rangle =1 \Rightarrow |f\rangle\langle f|f\rangle =|f\rangle \Rightarrow |f\rangle\langle f|=\hat{1}$$

Does that help...?

Daniel.

dextercioby said:
$$\langle f|f\rangle =1 \Rightarrow |f\rangle\langle f|f\rangle =|f\rangle \Rightarrow |f\rangle\langle f|=\hat{1}$$

Does that help...?

Daniel.
Thank you. I've prooved it with your help. The puzzle is solved

Some details:

1.$|f\rangle\langle f|$ is called the projector onto the vector $|f\rangle$.I assumed the Hilbert subspace is unidimensional and that the basis is formed by this vector $|f\rangle$.Therefore,the projector is generaly ~ to the unit operator on this subspace,but in this case,due to the orthonormalization,it coincides with the unit operator.

2.It's called HERMITEAN (or symmetric) operator,after the name of the 19-th cent.French mathematician Charles Hérmite.

Daniel.

## 1. What does the equation < f | A f >^2 = < f | A^2 f > mean?

This equation is a mathematical representation of the relationship between a function f and an operator A. It states that the squared inner product of f with A f is equal to the inner product of f with the squared operator A^2 f.

## 2. How is this equation used in scientific research?

This equation is commonly used in quantum mechanics to calculate the expectation value of a physical quantity, such as energy or momentum. It helps to determine the probability of a certain outcome for a given system.

## 3. What is the significance of the squared inner product in this equation?

The squared inner product represents the probability amplitude for a specific outcome. This means that the larger the squared inner product, the higher the probability of that outcome occurring.

## 4. Can this equation be applied to any type of function and operator?

Yes, this equation can be applied to any type of function and operator as long as they satisfy the necessary mathematical properties, such as being Hermitian and having a well-defined inner product.

## 5. Are there any practical applications of this equation outside of quantum mechanics?

Yes, this equation has been used in other fields of science, such as signal processing and image recognition, to analyze and process data. It can also be applied in engineering and economics for optimization problems.

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