%
% $BNL;RJ*M}3X(B : $B1i=,LdBj(B 6-1 $B2rEz(B
% $B%U%"%$%kL>(B : /home/edu/a-94e/y9410205/tex/quantum_physics/Q6-1.tex
% $B:n(B  $B@.(B  $B<T(B : y9410205 Yuzri Mohd Yusoff
% $B:n@.G/7nF|(B : /96/1/15
% $BEE;R%a!<%k(B : y9410205@edu.uec.ac.jp
%
\mondai
{y9410205 Yuzri Mohd Yusoff}
{/96/1/15}
{y9410205@edu.uec.ac.jp}
{6-1}
{
%**********     $BLdBjJ,(B  ***************%
$B<!$N8r494X78$r5a$a$h!#(B
\begin{eqnarray}
(a)\hspace{2mm} [x,p^3], \hspace{6mm}& (b)\hspace{2mm}
[x^2,p^3], \hspace{6mm} & (c)\hspace{2mm}, [e^{-x},p] \nonumber
\end{eqnarray}
}
{
%**********  $B2sEz(B ****************%

(a)\hspace{2mm} $[x,p^n] = i\hbar np^{n-1} \hspace{6mm}
\Rightarrow \hspace{6mm} [x,p^3] = 3i\hbar p^2 $

(b)
\begin{eqnarray}
[x^2,p^3] & = & -[p^3,x^2] \nonumber \\
          & = & -[p^3,x]x - x[p^3,x]  \nonumber \\
          & = & x[x,p^3] + [x,p^3]x   \nonumber \\
          & = & x(3i\hbar p^2) + (3i\hbar p^2)x  \nonumber \\
          & = & 3i\hbar(xp^2 + p^2x)  \nonumber 
\end{eqnarray}

(c)
\[
[g(x),p] = i\hbar\frac{d}{dx}g(x) \hspace{6mm} \Rightarrow \hspace{6mm}[e^{-x},p] = i\hbar \frac{d}{dx}e^{-x} = -i\hbar e^{-x} 
\]
%********* $B2rEz=*(B *************%
}
