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[QUE/QM-02001]

Node id: 1973page

       Let \(q,p\) be the coordinate and momentum operators and define
\(q,q^\dagger\) by $$ a = {1\over \sqrt{2m\omega \hbar}}( p-im\omega q) $$ $$
a^\dagger = {1\over \sqrt{2m\omega \hbar}}( p+im\omega q) $$ and $ N= a^\dagger
a $
                

  1.  Compute the commutator $ [ a, a^\dagger ]$ Use results in part (1)} and find the commutators $$[N,a]   \mbox{ and } [ N,a^\dagger]  $$
  2.  Express the harmonic oscillator Hamiltonian \begin{equation*} H = \frac{p^2}{2m} + \frac{1}{2}\, m \omega^2 q^2 \end{equation*} in terms of $a$ and $a^\dagger$.
  3. Show that \(H = \hbar\omega\Big(a^\dagger a + \frac{1}{2}\Big)\)

 

 kapoor's picture 22-04-14 08:04:05 n

[QUE/QM-02002]

Node id: 1972page

\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)

  1.       Use position representation        \[ \hat{q} \to q ; \qquad p\to -i\hbar \pp{q} \]      for the operators  \(\hat{p},\hat{q}\)  and prove the following relations    \begin{equation}\label{qm-que-02001;1}      [ \hat{q},\hat{p}^N] = i N\hbar \hat{p}^{N-1}; \qquad\qquad       [\hat{p},\hat{q}^N]  =-i N\hbar \hat{q}^{N-1}.      \end{equation}       Note you could have equally well used the momentum representation  \[ \hat{q} \to  i\hbar \pp{p}\qquad \hat{p} \to p \]   
  2.       Prove the above commutators  \eqref{qm-que-02001;1} using canonical commutation relations,  \([q,p]=i\hbar\), only. Do  not use any  representation  for position or momentum operator.
 kapoor's picture 22-04-14 08:04:33 n

[QUE/QM-02003]

Node id: 1974page

  Let
      $$L_x =yp_z-zp_y,\ \  L_y=zp_x-xp_z, \ \ \mbox{ and } L_z=xp_y-yp_x $$  be the angular momentum operators. Prove any one of the following.
          $$ [ L_x, L_y] = i \hbar L_z $$
          $$ [ L_y, L_z] = i \hbar L_x $$
          $$ [ L_z, L_x] = i \hbar L_y $$
      Use the fundamental commutators,      \[[x,p_x]=i\hbar,\quad [y,p_y]=i\hbar,\quad [z,p_z]=i\hbar,\]        and the identities involving commutators.

 kapoor's picture 22-04-14 07:04:50 n
 
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