Uncertainty and eigenstates  Summary and Q&A
TL;DR
Quantum mechanics defines uncertainty using the expectation value of an operator, which can be computed using various formulas.
Questions & Answers
Q: What is the definition of uncertainty in quantum mechanics?
Uncertainty is defined as the expectation value of the square of an operator minus the square of its expectation value. This quantity is always positive.
Q: How is claim 1 proven?
Claim 1 is proven through direct computation. By expanding the expression inside the expectation value, it can be rewritten as the expectation value of the square of the difference between the operator and its expectation value. This result confirms that the uncertainty is positive.
Q: What is claim 2 and how is it proven?
Claim 2 provides an alternative expression for uncertainty. It states that the expectation value of an operator minus its expectation value squared is equal to the integral of the squared norm of the difference between the operator and its expectation value acting on the state. This claim is proven by recognizing that the operator and the state complex conjugate in this integral.
Q: What is the relationship between eigenstates of an operator and uncertainty?
If a state is an eigenstate of an operator, there is no uncertainty. This is because the operator, when acting on the eigenstate, yields the eigenvalue of the operator, which is also equal to the expectation value of the operator on that state.
Summary & Key Takeaways

The analysis begins with the definition of uncertainty in quantum mechanics, which involves the expectation value of the square of an operator minus the square of its expectation value.

Claim 1 is proven through direct computation, showing that the definition of uncertainty is always a positive quantity.

Claim 2 is also proven, providing another expression for uncertainty and showing that if a state is an eigenstate of an operator, there is no uncertainty.