The variance σX^2=⟨(X^−⟨X^⟩)^2⟩ of an operator, X^, is a measure of how large a range its possible values are spread over (the standard deviation is given by σ=sqrt(σ2)). Suppose that |X⟩ is an eigenstate of some operator X^, what is the variance of X^ in this state? You may assume that |X⟩ is normalized (⟨X|X⟩=1).
Note that the expectation value of an operator in the state |ψ⟩ is given by ⟨O^⟩≡⟨ψ|O^|ψ⟩.
To find the variance of an operator X^ in the state |X⟩, we can use the following formula:
Var(X) = ⟨(X - ⟨X⟩)²⟩
Here, ⟨X⟩ represents the expectation value of the operator X^ in the state |X⟩.
In this case, we are given that |X⟩ is an eigenstate of the operator X^. An eigenstate of an operator is a state in which the operator acts as a scalar multiple. Therefore, if |X⟩ is an eigenstate of X^, then X^|X⟩ = λ|X⟩, where λ is the corresponding eigenvalue.
Using the given normalization condition ⟨X|X⟩ = 1, we can compute the expectation value of X^ in |X⟩:
⟨X⟩ = ⟨X|X⟩ = 1
Now we can calculate the variance:
Var(X) = ⟨(X - ⟨X⟩)²⟩
= ⟨(X - 1)²⟩
Since |X⟩ is an eigenstate of X^, we can substitute X^ for X in the above expression:
Var(X) = ⟨(X^ - 1)²⟩
At this point, we have the variance expression in terms of the operator X^. To evaluate it further, we need to know more about the operator X^, such as its form or the basis it is expressed in.