Physlib.QuantumMechanics.DDimensions.Operators.Uncertainty

Let $H$ be a complex inner product space. For any vectors $u, v \in H$ and any real number $c$, if the difference between the inner products satisfies the identity $\langle u, v \rangle - \langle v, u \rangle = i c$, then the imaginary part of the inner product is given by $\text{Im}(\langle u, v \rangle) = c / 2$.

Let $H$ be a complex inner product space. For any vectors $u, v \in H$ and any real number $c$, if the condition $\langle u, v \rangle - \langle v, u \rangle = ic$ holds (where $i$ is the imaginary unit), then the squared modulus of the inner product satisfies $$|\langle u, v \rangle|^2 = (\text{Re} \langle u, v \rangle)^2 + \left( \frac{c}{2} \right)^2$$ where $\text{Re} \langle u, v \rangle$ denotes the real part of the inner product.

Let $H$ be a complex inner product space. For any unit vector $\psi \in H$ (with $\|\psi\| = 1$) and any vectors $a, b \in H$, let $\mu_a, \mu_b \in \mathbb{R}$ be real numbers representing the expectation values such that $\langle \psi, a \rangle = \mu_a$, $\langle a, \psi \rangle = \mu_a$, $\langle \psi, b \rangle = \mu_b$, and $\langle b, \psi \rangle = \mu_b$. Then the following identity holds: $$\langle a - \mu_a \psi, b - \mu_b \psi \rangle - \langle b - \mu_b \psi, a - \mu_a \psi \rangle = \langle a, b \rangle - \langle b, a \rangle.$$ Here, $\langle \cdot, \cdot \rangle$ denotes the complex inner product.

Let $H$ be a complex inner product space. Suppose $A$ and $B$ are symmetric partial linear maps on $H$. Let $\psi \in H$ be a vector in the domains of both $A$ and $B$ (denoted $\text{dom}(A)$ and $\text{dom}(B)$) such that $A\psi \in \text{dom}(B)$ and $B\psi \in \text{dom}(A)$. For any real number $c$, if the expectation value of the commutator of $A$ and $B$ on the state $\psi$ satisfies $$\langle \psi, (AB - BA)\psi \rangle = ic$$ where $i$ is the imaginary unit, then the following identity holds: $$\langle A\psi, B\psi \rangle - \langle B\psi, A\psi \rangle = ic.$$ Here $\langle \cdot, \cdot \rangle$ denotes the complex inner product on $H$.

Let $H$ be a complex inner product space. For two partial linear maps $A$ and $B$ on $H$ and a vector $\psi$ in the intersection of their domains, the centered commutator expectation is the complex number defined by: $$\langle u, v \rangle - \langle v, u \rangle$$ where $u = A\psi - \langle A \rangle_\psi \psi$ and $v = B\psi - \langle B \rangle_\psi \psi$ are the centered vectors of $A$ and $B$ at $\psi$, and $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ denotes the expectation value of an operator $T$. Here, $\langle \cdot, \cdot \rangle$ denotes the complex inner product on $H$.

Let $H$ be a complex inner product space. For two partial linear maps $A, B$ on $H$, and a vector $\psi \in H$, suppose $\psi$ is in the domains of both $A$ and $B$ (denoted $\text{dom}(A)$ and $\text{dom}(B)$). If the second-order domain conditions are satisfied—specifically, that $A\psi \in \text{dom}(B)$ and $B\psi \in \text{dom}(A)$—then the raw commutator expectation is the complex number given by the inner product $\langle \psi, (AB - BA)\psi \rangle_{\mathbb{C}}$. Here, $\langle \cdot, \cdot \rangle_{\mathbb{C}}$ denotes the complex inner product on $H$.

Let $H$ be a complex inner product space. For any vectors $u, v \in H$ and any real number $c$, if the identity $\langle u, v \rangle - \langle v, u \rangle = i c$ holds (where $i$ is the imaginary unit), then the following inequality is satisfied: $$\left( \frac{|c|}{2} \right)^2 \le (\|u\| \cdot \|v\|)^2$$ where $\langle \cdot, \cdot \rangle$ denotes the complex inner product and $\|\cdot\|$ denotes the associated norm.

Let $H$ be a complex inner product space. Let $A$ and $B$ be partial linear maps on $H$, and let $\psi$ be a vector in the intersection of their domains. Suppose the centered commutator expectation of $A$ and $B$ with respect to $\psi$ satisfies the identity $$\langle u, v \rangle - \langle v, u \rangle = i c$$ for some $c \in \mathbb{R}$, where $u = A\psi - \langle A \rangle_\psi \psi$ and $v = B\psi - \langle B \rangle_\psi \psi$ are the centered vectors, and $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ is the expectation value. Then the squared Robertson uncertainty bound holds: $$\left( \frac{|c|}{2} \right)^2 \le \text{Var}_\psi(A) \cdot \text{Var}_\psi(B)$$ where $\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$ denotes the variance of a map $T$ in the state $\psi$.

Let $H$ be a complex inner product space, and let $A$ and $B$ be partial linear maps on $H$. For a vector $\psi$ in the intersection of the domains of $A$ and $B$, suppose the centered commutator expectation satisfies the identity $\langle u, v \rangle - \langle v, u \rangle = i c$ for some $c \in \mathbb{R}$, where $u = A\psi - \langle A \rangle_\psi \psi$ and $v = B\psi - \langle B \rangle_\psi \psi$ are the centered vectors and $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ is the expectation value. Then the Robertson–Schrödinger uncertainty bound holds: $$\text{Cov}_\psi(A, B)^2 + \left( \frac{c}{2} \right)^2 \leq \text{Var}_\psi(A) \cdot \text{Var}_\psi(B)$$ where $\text{Var}_\psi(T) = \|T\psi - \langle T \rangle_\psi \psi\|^2$ is the variance of an operator $T$ and $\text{Cov}_\psi(A, B) = \text{Re} \langle u, v \rangle$ is the covariance of $A$ and $B$ in the state $\psi$.

Let $H$ be a complex inner product space. Let $A$ and $B$ be partial linear maps on $H$, and let $\psi$ be a vector in the intersection of the domains of $A$ and $B$. Suppose the centered commutator expectation satisfies the identity $$\langle u, v \rangle - \langle v, u \rangle = i c$$ for some $c \in \mathbb{R}$, where $u = A\psi - \langle A \rangle_\psi \psi$ and $v = B\psi - \langle B \rangle_\psi \psi$ are the centered vectors, and $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ is the expectation value. Then the Robertson uncertainty bound in terms of standard deviations holds: $$\frac{|c|}{2} \le \sigma_\psi(A) \cdot \sigma_\psi(B)$$ where $\sigma_\psi(T) = \sqrt{\|T\psi - \langle T \rangle_\psi \psi\|^2}$ denotes the standard deviation of a map $T$ in the state $\psi$.

Let $H$ be a complex inner product space. Let $A$ and $B$ be symmetric partial linear maps on $H$. For a vector $\psi$ in the intersection of the domains of $A, B, AB,$ and $BA$, let $c$ be a real number such that the raw commutator expectation satisfies $\langle \psi, [A, B] \psi \rangle = i c$. Then the centered commutator expectation is also equal to $i c$, i.e., $$\langle u, v \rangle - \langle v, u \rangle = i c$$ where $u = (A - \langle A \rangle_\psi) \psi$ and $v = (B - \langle B \rangle_\psi) \psi$ are the centered vectors of $A$ and $B$ at $\psi$, and $\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle$ denotes the expectation value of an operator $T$.

Let $H$ be a complex inner product space. Let $A$ and $B$ be symmetric partial linear maps on $H$, and let $\psi \in H$ be a vector in the domains of $A$ and $B$ satisfying the necessary second-order domain conditions. If the expectation of the raw commutator $\langle \psi, [A, B] \psi \rangle$ is given by $ic$ for some $c \in \mathbb{R}$, then the squared Robertson uncertainty bound holds: $$\left( \frac{|c|}{2} \right)^2 \le \text{Var}_\psi(A) \cdot \text{Var}_\psi(B)$$ where $\text{Var}_\psi(T)$ denotes the variance of a partial linear map $T$ in the state $\psi$.

Let $H$ be a complex inner product space and let $A$ and $B$ be symmetric partial linear maps on $H$. Let $\psi \in H$ be a state vector such that $\psi$ is in the domains of both $A$ and $B$, and the second-order domain conditions $A\psi \in \text{dom}(B)$ and $B\psi \in \text{dom}(A)$ are satisfied. If the expectation value of the raw commutator $[A, B] = AB - BA$ in the state $\psi$ is given by $\langle \psi, [A, B] \psi \rangle = ic$ for some $c \in \mathbb{R}$, then the following inequality holds: $$\text{Var}_\psi(A) \cdot \text{Var}_\psi(B) \ge \text{Cov}_\psi(A, B)^2 + \left( \frac{c}{2} \right)^2$$ where $\text{Var}_\psi(T)$ denotes the variance of a map $T$ in state $\psi$, and $\text{Cov}_\psi(A, B)$ denotes the covariance of $A$ and $B$ in state $\psi$.

Let $H$ be a complex inner product space and let $A$ and $B$ be symmetric partial linear maps on $H$. Let $\psi \in H$ be a vector such that $\psi$ is in the domains of both $A$ and $B$, and the second-order domain conditions $A\psi \in \text{dom}(B)$ and $B\psi \in \text{dom}(A)$ are satisfied. If the expectation value of the raw commutator in the state $\psi$ is given by $\langle \psi, [A, B] \psi \rangle = ic$ for some $c \in \mathbb{R}$, then the following uncertainty bound holds: $$\frac{|c|}{2} \le \sigma_\psi(A) \cdot \sigma_\psi(B)$$ where $\sigma_\psi(T)$ denotes the standard deviation of the map $T$ in the state $\psi$.