quantum harmonic oscillator and Lie algebraQuantumHarmonicOscillatorAndLieAlgebra2008-09-28 16:01:042009-05-01 03:47:57Topicquantum harmonic oscillatorquantum anharmonic oscillatorLie algebra in quantum theoryLie algebra of a quantum harmonic oscillator% this is the default PlanetPhysics preamble. as your knowledge
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% define commands here\subsection{Lie Algebra of a Quantum Harmonic Oscillator}
One wishes to solve the time-independent Schr\"odinger equation of motion in order to determine the stationary states of the quantum harmonic oscillator which has a quantum
Hamiltonian of the form:
\begin{equation}
\mathbf{H} = (\frac {1}{2m})\cdot P^2 + \frac{k}{2}\cdot X^2~,
\end{equation}
where X and P are, respectively, the coordinate and conjugate momentum operators. X and P
satisfy the Heisenberg commutation/'uncertainty' relations $[X,P]
= i\hbar I$~, where the identity operator $I$
is employed to simplify notation. A simpler, equivalent form of the above Hamiltonian
is obtained by defining physically dimensionless coordinate and momentum:
\begin{equation}
\mathbf{x} = (\frac{X}{\alpha})~, ~ \mathbf{p}= (\frac{\alpha
P}{\hbar}) ~\text{and}~ \alpha = \sqrt {\frac{\hbar}{mk}}~.
\end{equation}
With these new dimensionless operators, $\mathbf{x}$ and
$\mathbf{p}$, the quantum Hamiltonian takes the form:
\begin{equation}
\mathbf{H}= (\frac{\hbar \omega}{2})\cdot (\mathbf{p}^2 +
\mathbf{x}^2)~,
\end{equation}
which in units of $\hbar \cdot \omega$ is simply:
\begin{equation}
\mathbf{H}' = (\frac {1}{2})\cdot (\mathbf{p}^2 +
\mathbf{x}^2)~.
\end{equation}
The commutator of $\mathbf{x}$ with its conjugate operator
$\mathbf{p}$ is simply $[\mathbf{x}, \mathbf{p}] = i$~.\\
Next one defines the superoperators $S_{Hx} = [H, x] = -i \cdot
p$, and $S_{Hp} = [H, p] = i \cdot \mathbf{x}$ that will lead to
new operators that act as generators of a Lie algebra for this
quantum harmonic oscillator. The eigenvectors $Z$ of these
superoperators are obtained by solving the equation $S_H \cdot Z =
\zeta Z$, where $\zeta$ are the eigenvalues, and $Z$ can be
written as $(c_1 \cdot x + c_2 \cdot p)$~. The solutions are
\begin{equation}
\zeta = \pm 1 ~, \text{and} ~~ c_2 = \mp i \cdot c_1~.
\end{equation}
Therefore, the two eigenvectors of $S_H$ can be written as:
\begin{equation}
a^\dagger = c_1* (x-ip)~, \text{and}~ a = c_1 (x+ip)~,
\end{equation}
respectively for $\zeta = \pm 1$~. For $c_1 =\surd {2}$ one
obtains normalized operators $H, a$ and $a \dagger$ that generate
a $4$--dimensional Lie algebra with commutators:
\begin{equation}
[H,a] = -a~,~[H, a^\dagger ]= a^\dagger~, ~ \text{and}~ [a,
a^\dagger]= I ~.
\end{equation}
The term $\mathbf{a}$ is called the \emph{annihilation} operator
and the term $a\dagger$ is called the \emph{creation} operator.
This Lie algebra is solvable and generates after repeated
application of $a\dagger$ all the eigenvectors of the quantum
harmonic oscillator:
\begin{equation}
\Phi_n = (\frac{(a\dagger)^n}{\surd(n!)})\cdot \Phi_0 ~.
\end{equation}
The corresponding, possible eigenvalues for the energy, derived
then as solutions of the Schr\"odinger equations for the quantum
harmonic oscillator are:
\begin{equation}
E_n = \hbar \cdot \omega (n+ \frac{1}{2}) ~, ~\text{where}~ n =
0,1, \ldots, N~.
\end{equation}
The position and momentum eigenvector coordinates can be then also computed by iteration from %%@
\emph{(finite)} matrix representations of the \emph{(finite)} Lie
algebra, using, for example, a simple computer programme to
calculate linear expressions of the annihilation and creation
operators. For example, one can show analytically that:
\begin{equation}
[a, x^k] = (\frac{k}{\surd 2})\cdot (x_{k-1})~.
\end{equation}
One can also show by introducing a \emph{coordinate}
representation that the eigenvectors of
the harmonic oscillator can be expressed as \emph{Hermite polynomials} in terms of the coordinates. In the coordinate representation the quantum
\emph{Hamiltonian} and \emph{bosonic} operators have,
respectively, the simple expressions:
\begin{equation}
\begin{aligned}
H &= (\frac{1}{2})\cdot[-\frac{d^2}{dx^2}) + (x^2)]~, \\ a &=
(\frac{1}{\surd 2})\cdot (x + \frac{d}{dx})~, \\ a\dagger &=
(\frac{1}{\surd 2})\cdot (x - \frac{d}{dx})~.
\end{aligned}
\end{equation}
The ground state eigenfunction normalized to unity is obtained from solving the simple %%@
first-order differential equation $a\Phi_0 (x) = 0$ and which
leads to the expression:
\begin{equation}
\Phi_0 (x)= (\pi^{-\frac{1}{4}})\cdot \exp(-\frac{x^2}{2})~.
\end{equation}
By repeated application of the creation operator written as
\begin{equation}
a\dagger = (-\frac{1}{\surd 2})\cdot
(\exp(\frac{x^2}{2}))\cdot(\frac{d}{dx^2})\cdot
\exp(-\frac{x^2}{2}) ~,
\end{equation}
one obtains the $n$-th level eigenfunction:
\begin{equation}
\Phi_n(x) = (\frac{1}{(\surd\pi) 2^n n!)})\cdot
(\mathbf{He}_n (x))~,
\end{equation}
where $\mathbf{He}_n(x)$ is \emph{the Hermite polynomial} of order
$n$~. With the special generating function of the Hermite
polynomials
\begin{equation}
F(t,x) = (\pi^{-\frac{1}{4}})\cdot (\exp((-\frac{x^2}{2}) + tx
-(\frac{t^2}{4}))~,
\end{equation}
one obtains explicit analytical relations between the
eigenfunctions of the quantum harmonic
oscillator and the above special generating
function:
\begin{equation}
F(t,x) = \sum_{n=0} (\frac{t^n}{\surd (2^n \cdot n!)})\cdot
\Phi_n(x) ~.
\end{equation}
Such applications of the \PMlinkname{Lie algebra}{LieAlgebra}, and the related algebra of the \emph{bosonic} operators as defined above are quite numerous
in theoretical physics, and especially for various quantum field
carriers in QFT that are all \emph{bosons}. (Please note also
additional examples of special `Lie' superalgebras for
gravitational and other fields in Section 6, such as gravitons and
Goldstone quanta that are all \emph{bosons} of different spin
values and \emph{`Penrose homogeneity'}).\\
In the interesting case of a \emph{two-mode} bosonic quantum
system formed by the tensor (direct) product of \emph{one-mode}
bosonic states: $\mid m,n> := \mid m> \otimes \mid n>$, one can
generate a $3$--dimensional Lie algebra in terms of \emph{Casimir}
operators. \emph{Finite}-- dimensional Lie algebras are far more
tractable, or easier to compute, than those with an infinite basis
set. For example, such a Lie algebra as the $3$--dimensional one
considered above for the two-mode, bosonic states is quite useful
for numerical computations of vibrational (IR, Raman, etc.)
spectra of two--mode, \emph{diatomic} molecules, as well as the
computation of scattering states. Other perturbative calculations
for more complex quantum systems, as well as calculations of exact
solutions by means of Lie algebras have also been developed (see
for example Fernandez and Castro,1996).