Harmonic Oscillator Energy Levels Using Commutators

Introduction
A mass attached to a spring, when stretched and released, executes a simple harmonic motion. The energy of a harmonic oscillator is given by
\begin{equation}
      H = \frac{p^2}{2m} + \frac{1}{2}m \omega^2 x^2
\end{equation}
where $p$ is the momentum of the particle at position $x$. Classically the energy depends on the amplitude and all positive values are allowed. Quantum mechanically, energy is quantized and all energies are not allowed. The allowed values are the eigenvalues of the Hamiltonian operator.
\begin{equation}
\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m \omega^2 \hat{x}^2 \label{EQ02}
\end{equation}
Here $\hat{x},\hat{p}$ denote the position and momentum operators and $\hat{H}$ has been written down by replacing $x,p$ in \EqRef{EQ01} by the corresponding operators $\hat{x}$ and $\hat{p}$. The energy levels can be derived by solving the Schr\"{o}dinger equation or using commutation relations. The use of algebraic methods, employing the commutation relations only, followed here underlines the power of the canonical quantization rule listed as one of the postulates.

Goals 

1.  The energy levels of harmonic oscillator \eqref{EQ02} are given by
                   $$  E_n = (n+\tfrac{1}{2}) \hbar \omega $$      
2.  If $\ket{n}$ is an eigenvector of $H$ with eigenvalue $E_n$, then  $a, a^\dagger$ are raising lowering ladder operators and
   \begin{eqnarray}
         a \ket{n} &=& \sqrt{n} \ket{n-1}\\
          a^\dagger \ket{n} &=&\sqrt{n+1}\ket{n+1}
   \end{eqnarray}
3.  The minimum energy possible for harmonic oscillator is not zero; it is $\frac{1}{2}\hbar\omega$. This can be understood from uncertainty principle.

Milestones
 

1. For one degree of freedom the canonical quantization rule
   $$  [ \hat{x},\hat{p}] = i \hbar;  $$
2. The operators $a, a^\dagger, N$ are introduced by
   \begin{eqnarray}
       a &=& \frac{1} {\sqrt{2m\omega \hbar}}( p-im\omega x) ,\\
       a^\dagger &=& \frac{1}{\sqrt{2m\omega \hbar}}( p+im\omega x).\\
       N &=& a^\dagger a.
   \end{eqnarray}
3. These operators satisfy commutation relations
       \begin{equation}
             [ a, a^\dagger ] =1,\qquad [N,a]=-a,\qquad
                [N,a^\dagger]=a^\dagger.
   \end{equation}
4. The harmonic oscillator Hamiltonian
   \begin{equation}
   H = \frac{p^2}{2m} + \frac{1}{2}\, m \omega^2 q^2 .
   \end{equation}
   expressed in terms of $a$ and $a^\dagger$ takes the form
       \begin{equation}     H = ( a^\dagger a +\frac{1}{2} ) \hbar \omega .\end{equation}
5. The operator $N=a^\dagger a$ is a positive definite operator and its eigenvalues are positive.
    
6. $a^\dagger$ is a raising operator for $N$. If $\nu$ is an eigenvalue of $N$ with eigenvector $\ket{\psi}$ then, for $r=1,2,3,\cdots$  the vector $(a^\dagger)^r \ket{\psi}$  is an eigen vector of $N$ with eigenvelues $\nu+r$.
7. The operator $a$ is lowering operator. If $\nu$ is an eigenvalue of $N$ with eigenvector $\ket{\psi}$, $a$ lowers the eigenvalue of $N$ by one unit and $r$ successive application of $a$ on $\ket{\psi}$ will produce eigenvector with eigenvalue $\nu-r$.
8. Eigenvalues of $N$ cannot be negative, implying $ \ket{\chi_r} =0 $ for $r> \nu$. Taking largest $r$ such that  $r \le \nu$, gives an eigenvector of $N$ with eigenvalue $0$
9. The eigenvalues of $N$ are all non negative integers and the energy levels are $ E_n= \Big( n+\frac{1}{2}) \hbar \omega$

Tasks for you

• Derive  algebraic relations, \eqref{EQ09}  and \eqref{EQ11}, using the commutation relations of $\hat{x},\hat{p}$.
• Follow all the steps outlined in the milestones from any book of your choice[1,2,3,4], or else see my lecture notes [5].
• Look at the flow chart[6] and verify  that it represents the logic of derivation correctly and that you have understood each process listed in the flowchart.
• Write your comment to give feed back whether you found the flowchart useful or not and whether you will like to see more flowcharts everywhere.

  

General Instructions

• Ask as many questions on the above points and try to find answers from your own sources.
• Keep posting comments at suitable stages. Indicate when you are ready with this unit of learning.
• This will be followed by detailed stock taking of your efforts.\\[1mm]
• Please indicate  corrections/changes/comments  in a the next tab  Thanks for your efforts.

References

1.  E. Merzbacher, ``Quantum Mechanics"
2.  L. I. Schiff, ``Quantum Mechanics"
3.  Mathews and Venkatesan,``A text Book of Quantum Mechanics"
4.  David J. Griffiths, ''Quantum Mechanics"
5.  A. K. Kapoor, "Energy Levels of Harmonic Oscillator Using Commutation Relations"
6. A. K. Kapoor, "Flow Chart for Harmonic Oscillator Energy Levels"
     
   
   

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