# Proof: Operators with same expectation value

A} = \hat{B} is a strong condition, and it's very unlikely to happen unless the operators are very simple.In summary, proving that \hat{A} is equal to \hat{B} depends on the state of \psi and the operators being compared. This is a strong condition and is unlikely to happen unless the operators are simple.
Given some state $\left|\psi\right\rangle$, and two operators $\hat{A}$ and $\hat{B}$, how do you prove that if $\langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle$ then $\hat{A} = \hat{B}$ ?

Is it true for every state?

Given some state $\left|\psi\right\rangle$, and two operators $\hat{A}$ and $\hat{B}$, how do you prove that if $\langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle$ then $\hat{A} = \hat{B}$ ?

Proving $$\hat{A}$$ is the same as $$\hat{B}$$ depends on the state of $$\psi$$. You've made some notation $$<\psi|\psi>$$ is a probability amplitude. Your operators are fine, but if they are equal, it depends on the state of $$|\psi>$$. Note though $$\mathcal{O} \cdot x \ne x \cdot \mathcal{O}$$.

Given some state $\left|\psi\right\rangle$, and two operators $\hat{A}$ and $\hat{B}$, how do you prove that if $\langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle$ then $\hat{A} = \hat{B}$ ?

Well, you can't; you have the equality of the images for only one vector in the common domain. You must have much more than that, of course.

## 1. What is the concept of "Proof: Operators with same expectation value"?

The concept of "Proof: Operators with same expectation value" refers to the mathematical proof that two operators in quantum mechanics can have the same expectation value without being identical, as long as they have the same eigenvalues and corresponding eigenvectors.

## 2. What is an operator in quantum mechanics?

An operator in quantum mechanics is a mathematical representation of a physical observable, such as position, momentum, or energy. Operators act on quantum states to produce a measurable value, known as an eigenvalue.

## 3. How is expectation value calculated for operators?

The expectation value for an operator is calculated by taking the inner product of the operator with the quantum state and then normalizing by the inner product of the state with itself. This can be written as 〈ψ|A|ψ〉, where ψ represents the quantum state and A represents the operator.

## 4. Can two operators have the same expectation value for all quantum states?

Yes, two operators can have the same expectation value for all quantum states if they have the same eigenvalues and corresponding eigenvectors. This can be proven mathematically using the concept of completeness, which states that any quantum state can be expressed as a linear combination of eigenvectors of a given operator.

## 5. What are some real-world applications of the concept of "Proof: Operators with same expectation value"?

The concept of "Proof: Operators with same expectation value" is fundamental in understanding and predicting the behavior of quantum systems, such as atoms, molecules, and subatomic particles. It is also used in the development of quantum algorithms for quantum computers and in various fields of quantum technology, such as quantum cryptography and quantum sensing.

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