# PlanetPhysics/Quantum Harmonic Oscillator and Lie Algebra

### Lie Algebra of a Quantum Harmonic Oscillator

[edit | edit source]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:

where and are, respectively, the coordinate and conjugate momentum operators. and satisfy the Heisenberg commutation/'uncertainty' relations where the identity operator is employed to simplify notation. A simpler, equivalent form of the above Hamiltonian is obtained by defining physically dimensionless coordinate and momentum:

With these new dimensionless operators, and , the quantum Hamiltonian takes the form:

which in units of is simply:

The commutator of with its conjugate operator is simply ~.\\

Next one defines the superoperators </math>S_{Hx} = [H, x] = -i \cdot pS_{Hp} = [H, p] = i \cdot \mathbf{x} of these superoperators are obtained by solving the equation </math>S_H \cdot Z = \zeta Z\zetaZ~. The solutions are

Therefore, the two eigenvectors of can be written as:

respectively for ~. For one obtains normalized operators and that generate a --dimensional Lie algebra with commutators:

The term is called the *annihilation* operator
and the term is called the *creation* operator.
This Lie algebra is solvable and generates after repeated
application of all the eigenvectors of the quantum
harmonic oscillator:

The corresponding, possible eigenvalues for the energy, derived then as solutions of the Schr\"odinger equations for the quantum harmonic oscillator are:

The position and momentum eigenvector coordinates can be then also computed by iteration from
*(finite)* matrix representations of the *(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:

One can also show by introducing a *coordinate*
representation that the eigenvectors of
the harmonic oscillator can be expressed as *Hermite polynomials* in terms of the coordinates. In the coordinate representation the quantum
*Hamiltonian* and *bosonic* operators have,
respectively, the simple expressions:

Failed to parse (syntax error): {\displaystyle 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})~. }

The ground state eigenfunction normalized to unity is obtained from solving the simple first-order differential equation and which leads to the expression:

By repeated application of the creation operator written as

one obtains the -th level eigenfunction:

where is *the Hermite polynomial* of order
~. With the special generating function of the Hermite
polynomials

one obtains explicit analytical relations between the eigenfunctions of the quantum harmonic oscillator and the above special generating function:

Such applications of the Lie algebra, and the related algebra of the *bosonic* operators as defined above are quite numerous
in theoretical physics, and especially for various quantum field carriers in QFT that are all *bosons*. (Please note also
the additional examples of special `Lie' superalgebras for
gravitational and other fields, related to hypothetical particles such as gravitons and Goldstone quanta that are all *bosons* of different spin values and *`Penrose homogeneity'* ).\\

In the interesting case of a *two-mode* bosonic quantum
system formed by the tensor (direct) product of *one-mode*
bosonic states: , one can
generate a --dimensional Lie algebra in terms of *Casimir*
operators. *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 --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, *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).