- ) 2 ⟩ = lim T → ∞ 1 T ∫ − T 2 T 2 V ( t ) 2 d t . {\displaystyle \left\
langle V(t)^{2}\right\rangle =\lim _{T\to \infty }{\frac {1}{T}}\int _{-{\frac...
- \mathbf {v} } of an
inner product space where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \
langle \cdot ,\cdot \rangle } is the
inner product.
Examples of
inner products...
- ⟨ y , y ⟩ , {\displaystyle \
langle x+y,x+y\rangle =\
langle x,x\rangle +2\operatorname {Re} (\
langle x,y\rangle )+\
langle y,y\rangle ,}
where Re {\displaystyle...
- {\begin{bmatrix}1&\
langle x\rangle &\
langle p\rangle \\\
langle x\rangle &\
langle x\star x\rangle &\
langle x\star p\rangle \\\
langle p\rangle &\
langle p\star x\rangle...
-
denote quantum states. The
notation uses
angle brackets, ⟨ {\displaystyle \
langle } and ⟩ {\displaystyle \rangle } , and a
vertical bar | {\displaystyle |}...
- , ⟨ ψ ∣ ⟩ H . {\displaystyle \left\
langle \,\
langle \psi \mid \,\mid g\right\rangle _{H}=\left\
langle g,\,\
langle \psi \mid \right\rangle _{H}.} With...
- ⟨ x , A ∗ y ⟩ , {\displaystyle \
langle Ax,y\rangle =\
langle x,A^{*}y\rangle ,}
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \
langle \cdot ,\cdot \rangle } is the inner...
- is a
subset of a
group G {\displaystyle G} , then ⟨ S ⟩ {\displaystyle \
langle S\rangle } , the
subgroup generated by S {\displaystyle S} , is the smallest...
- {\begin{aligned}\
langle X\rangle _{\psi }&=\
langle \psi |X|\psi \rangle =\
langle \psi |\mathbb {I} X\mathbb {I} |\psi \rangle \\&=\iint \
langle \psi |x\rangle \
langle...
- )={\frac {\
langle \mathbf {v} ,\mathbf {u} \rangle }{\
langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,}
where ⟨ v , u ⟩ {\displaystyle \
langle \mathbf...
