The variance σ2X=⟨(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=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⟩.
Well, well, well, if it isn't my old friend, the variance! So, you want to know about the variance of an operator in an eigenstate? Let me juggle some concepts for you!
If |X⟩ is an eigenstate of some operator X^, then we can write X^|X⟩ = x|X⟩, where x is the corresponding eigenvalue. Now, let's calculate the variance of X^ in this state.
The variance σ^2X is defined as the expectation value of (X^ - ⟨X^⟩)². Here, ⟨X^⟩ represents the average value of X^ in the given state. But hold your clown nose, my friend! In an eigenstate, the expectation value of X^ is simply the eigenvalue x. Fancy, huh?
So, ⟨X^⟩ = x. Now, let's compute the variance:
σ^2X = ⟨(X^ - ⟨X^⟩)²⟩
= ⟨(X^ - x)²⟩
= ⟨X^² - 2xX^ + x²⟩
= ⟨X^²⟩ - 2x⟨X^⟩ + x²
Now, my dear interlocutor, let's put some makeup on this equation. Since |X⟩ is an eigenstate of X^, we know that X^|X⟩ = x|X⟩. Therefore, we can rewrite the equation above as:
σ^2X = ⟨X^²⟩ - 2x⟨X^⟩ + x²
= ⟨X^² - 2xX^ + x²⟩
= ⟨0⟩
= 0
Oh my sides! It turns out that in an eigenstate, the variance of X^ is zero! That means all the possible values of X^ are concentrated perfectly around the eigenvalue x. No spreading, no wide range, just a nice and cozy spot on the clown plane.
So, my friend, in an eigenstate, the variance of X^ is as scarce as a clown without a funny wig. Hope that brings a smile to your mathematical face!
To find the variance of X^ in the eigenstate |X⟩, we need to calculate the expectation value of (X^ - ⟨X^⟩)^2 in the state |X⟩.
Let's start by finding the expectation value of X^ in the state |X⟩. Using the definition provided, the expectation value is given by:
⟨X^⟩ = ⟨X|X^|X⟩
Since |X⟩ is an eigenstate of X^, X^|X⟩ will simply produce the eigenvalue X|x⟩, where x represents the eigenvalue corresponding to the eigenstate |X⟩. Therefore, we have:
⟨X^⟩ = ⟨X|X|x⟩ = x⟨X|X⟩
Since |X⟩ is normalized, ⟨X|X⟩ = 1. Thus, we have:
⟨X^⟩ = x
Now, let's calculate the expectation value of (X^ - ⟨X^⟩)^2 in the state |X⟩:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|(X^ - ⟨X^⟩)^2|X⟩
Expanding the square and applying the definition of expectation value, we get:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|(X^ - ⟨X^⟩)(X^ - ⟨X^⟩)|X⟩
Using the linearity of the inner product, we can write this as:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|X^2 - X^⟨X^⟩ - ⟨X^⟩X^ + ⟨X^⟩^2|X⟩
Since X^ is a Hermitian operator, X^|X⟩ will also produce the eigenvalue times the eigenstate. Therefore, we have:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|X^2|x⟩ - x⟨X|X^|X⟩ - x⟨X|X^|X⟩ + x^2⟨X|X⟩^2
Using the normalization condition, ⟨X|X⟩ = 1, we can simplify this further:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|X^2|x⟩ - 2x^2 + x^2
Finally, the variance is given by:
σ^2X = ⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|X^2|x⟩ - x^2
Therefore, the variance of X^ in the eigenstate |X⟩ is given by ⟨X|X^2|x⟩ - x^2.
To find the variance of X^ in the state |X⟩, we need to calculate the expectation value of the operator (X^ - ⟨X^⟩)^2 in that state.
First, let's find the expectation value ⟨X^⟩ in the state |X⟩:
⟨X^⟩ = ⟨X|X^|X⟩
Since |X⟩ is an eigenstate of X^, we know that X^|X⟩ = x|X⟩, where x is the eigenvalue corresponding to the state |X⟩.
Therefore, ⟨X^⟩ = ⟨X|x|X⟩
Since |X⟩ is normalized, we have ⟨X|X⟩ = 1, so ⟨X^⟩ = ⟨X|x|X⟩ = x⟨X|X⟩ = x.
Now, let's find the expectation value of (X^ - ⟨X^⟩)^2 in the state |X⟩:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X| (X^ - x)^2 |X⟩
Using the identity (A - B)^2 = A^2 - 2AB + B^2, we can expand the expression:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X| X^2 - 2X^x + x^2 |X⟩
Since X^|X⟩ = x|X⟩, we can replace X^ in the above expression with x:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X| x^2 - 2x^2 + x^2 |X⟩
Simplifying further:
⟨(X^ - ⟨X^⟩)^2⟩ = ⟨X|0|X⟩
Since any operator multiplied by the zero operator gives the zero operator, we have ⟨X|0|X⟩ = 0.
Therefore, the variance of X^ in the state |X⟩ is zero (σ^2X = 0), which indicates that all measurements of X^ in this state will yield the same value x.