Physlib.QuantumMechanics.DDimensions.Operators.StateObservables.Variance

Given a partial linear map $T$ on a complex inner product space $H$ and a vector $\psi$ in the domain of $T$, the variance is defined as the squared norm of the centered vector: $$\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$$ where $\langle T \rangle_\psi$ is the expectation value of $T$ with respect to $\psi$, defined as the real part of the inner product $\langle \psi, T\psi \rangle$.

For a partial linear map $T$ on a complex inner product space $H$ and a vector $\psi$ in the domain of $T$, the variance $\text{Var}_\psi(T)$ is equal to the squared norm of the centered vector: $$\text{Var}_\psi(T) = \|\text{centered}_\psi(T)\|^2$$ where the centered vector is defined as $\text{centered}_\psi(T) = T\psi - \langle T \rangle_\psi \psi$, and $\langle T \rangle_\psi$ is the expectation value of $T$ with respect to $\psi$.

Let $H$ be a complex inner product space. For any partial linear map $T$ on $H$ and any vector $\psi$ in the domain of $T$, the variance of $T$ with respect to $\psi$ is equal to the squared norm of the difference between $T\psi$ and the expectation value $\langle T \rangle_\psi$ scaled by $\psi$: $$\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$$ where $\langle T \rangle_\psi$ denotes the expectation value of $T$ in the state $\psi$.

Let $H$ be a complex inner product space and $T$ be a symmetric partial linear map on $H$. If $\psi$ is a unit vector (i.e., $\|\psi\| = 1$) in the domain of $T$, then the variance of $T$ in the state $\psi$ satisfies the identity: $$\text{Var}_\psi(T) = \|T\psi\|^2 - \langle T \rangle_\psi^2$$ where $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ is the expectation value of $T$ with respect to $\psi$.

Let $H$ be a complex inner product space, $T$ be a partial linear map on $H$, and $\psi$ be a vector in the domain of $T$. The variance of $T$ with respect to $\psi$, denoted as $\text{Var}_\psi(T)$, is non-negative: $$0 \leq \text{Var}_\psi(T)$$

Let $H$ be a complex inner product space, $T$ be a partial linear map on $H$, and $\psi$ be a vector in the domain of $T$. The variance of $T$ in the state $\psi$, defined as $\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$ (where $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$), is zero if and only if the centered vector $T\psi - \langle T \rangle_\psi \psi$ is zero.

Let $H$ be a complex inner product space and $T$ be a partial linear map on $H$. For any $\psi$ in the domain of $T$, the variance $\text{Var}_\psi(T)$ is equal to $0$ if and only if $T\psi = \langle T \rangle_\psi \psi$, where $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ denotes the expectation value of $T$ in the state $\psi$.

Let $H$ be a complex inner product space and $T$ be a partial linear map on $H$. For any vector $\psi$ in the domain of $T$ with unit norm $\|\psi\| = 1$, the variance of $T$ in the state $\psi$ is zero if and only if $\psi$ is an eigenvector of $T$ with eigenvalue equal to the expectation value $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$.

Given a partial linear map $T$ on a complex inner product space $H$ and a vector $\psi$ in the domain of $T$, the standard deviation is defined as the square root of the variance: $$\sigma_\psi(T) = \sqrt{\text{Var}_\psi(T)}$$ where $\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$ is the variance of $T$ in the state $\psi$.

Let $H$ be a complex inner product space. For any partial linear map $T$ on $H$ and any vector $\psi$ in the domain of $T$, the standard deviation $\sigma_\psi(T)$ is equal to the square root of the variance $\text{Var}_\psi(T)$: $$\sigma_\psi(T) = \sqrt{\text{Var}_\psi(T)}$$

Let $H$ be a complex inner product space, $T$ be a partial linear map on $H$, and $\psi$ be a vector in the domain of $T$. The standard deviation of $T$ in the state $\psi$, denoted as $\sigma_\psi(T)$, is non-negative: $$0 \le \sigma_\psi(T)$$

Let $H$ be a complex inner product space, $T$ be a partial linear map on $H$, and $\psi$ be a vector in the domain of $T$. The square of the standard deviation $\sigma_\psi(T)$ of $T$ in the state $\psi$ is equal to the variance $\text{Var}_\psi(T)$, namely $$\sigma_\psi(T)^2 = \text{Var}_\psi(T)$$

Let $H$ be a complex inner product space. For a partial linear map $T$ on $H$ and a vector $\psi$ in the domain of $T$, the standard deviation $\sigma_\psi(T)$ is zero if and only if the centered vector $T\psi - \langle T \rangle_\psi \psi$ is the zero vector, where $\langle T \rangle_\psi$ is the expectation value of $T$ with respect to $\psi$.

Let $H$ be a complex inner product space and $T$ be a partial linear map on $H$. For any vector $\psi$ in the domain of $T$, the standard deviation $\sigma_\psi(T)$ is $0$ if and only if $T\psi = \langle T \rangle_\psi \psi$, where $\langle T \rangle_\psi = \operatorname{Re} \langle \psi, T\psi \rangle$ is the expectation value of $T$ in the state $\psi$.

Let $H$ be a complex inner product space and $T$ be a partial linear map on $H$. For any unit vector $\psi$ in the domain of $T$ (i.e., $\|\psi\| = 1$), the standard deviation $\sigma_\psi(T)$ is zero if and only if $\psi$ is an eigenvector of $T$ with eigenvalue equal to the expectation value $\langle T \rangle_\psi = \operatorname{Re} \langle \psi, T\psi \rangle$.

Let $T$ be a symmetric partial linear map on a complex Hilbert space $H$. If $\psi$ is a vector in the domain of $T$ such that $T\psi$ is also in the domain of $T$, then the real part of the inner product of $\psi$ and $T(T\psi)$ is equal to the squared norm of $T\psi$, i.e., $$\operatorname{Re} \langle \psi, T(T\psi) \rangle = \|T\psi\|^2.$$

Let $H$ be a complex inner product space and $T: H \to H$ be a partial linear map. For any vector $\psi$ in the domain of $T$ such that $T\psi$ is also in the domain of $T$, the variance of $T$ in the state $\psi$ is given by the formula: $$\text{Var}_\psi(T) = \text{Re} \langle \psi, T(T\psi) \rangle - \langle T \rangle_\psi^2$$ where $\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$ and $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ is the expectation value of $T$ with respect to $\psi$.