1.16: El Conmutador
- Page ID
- 84464
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Hay que tener cuidado de observar el orden correcto de los operadores. Por ejemplo,
\[ \hat{x}\hat{k} \neq \hat{k}\hat{x} \nonumber \]
pero
\[ \hat{x}\hat{\omega} = \hat{\omega}\hat{x} \nonumber \]
En mecánica cuántica definimos el conmutador:
\[ [\hat{q},\hat{r}]=\hat{q}\hat{r}-\hat{r}\hat{q} \nonumber \]
Nos encontramos con que los operadores\(\hat{r}\) y\(\hat{\omega}\) conmutan porque\([\hat{x},\hat{\omega}]=0\).
Considerando a los operadores\(\hat{x}\) y\(\hat{k}\):
\[ [\hat{x},\hat{k}]=-ix\frac{d}{dx}+i\frac{d}{dx}x \nonumber \]
Para simplificar esto aún más necesitamos operar en alguna función, f (x):
\[\begin{align*} [\hat{x},\hat{k}]f(x) &=-ix\frac{df}{dx}+i\frac{d}{dx}(xf) \\[4pt] &= -ix\frac{df}{dx}+if\frac{dx}{dx}+ix\frac{df}{dx} \\[4pt] &=if \end{align*} \nonumber \]
Por lo tanto, los operadores\(\hat{x}\) y\(\hat{k}\) no conmutan, i.e.
\[ [\hat{x},\hat{k}] = i \nonumber \]
Aunque se utilizaron las transformadas de Fourier, la Ecuación (1.10.13) también se puede derivar de la relación (1.16.5) para los operadores no conmutantes\(\hat{x}\) y operadores\(\hat{k}\). De ello se deduce que todos los operadores que no conmutan están sujetos a un límite similar en el producto de sus incertidumbres. Veremos en la siguiente sección que este límite se conoce como “el principio de incertidumbre”.
\(^{†}\)Hemos aplicado el teorema de Parseval; ver los Conjuntos de Problemas.