quantum harmonic oscillator and Lie algebra
One wishes to solve the time-independent Schrödinger equation of motion in order to determine the stationary states of the quantum harmonic oscillator which has a quantum Hamiltonian
of the form:
 |
(1) |
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:
and |
(2) |
With these new dimensionless operators,
and
, the quantum Hamiltonian takes the form:
 |
(3) |
which in units of
is simply:
 |
(4) |
The commutator
of
with its conjugate operator
is simply
.
Next one defines the superoperators
, and
that will lead to
new operators that act as generators
of a Lie algebra
for this
quantum harmonic oscillator. The eigenvectors
of these
superoperators are obtained by solving the equation
, where
are the eigenvalues, and
can be
written as
. The solutions are
and |
(5) |
Therefore, the two eigenvectors of
can be written as:
and |
(6) |
respectively for
. For
one
obtains normalized operators
and
that generate
a
-dimensional Lie algebra with commutators:
and![$\displaystyle ~ [a, a^\dagger]= I ~.$](img36.png) |
(7) |
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:
 |
(8) |
The corresponding, possible eigenvalues for the energy, derived
then as solutions of the Schrödinger equations for the quantum
harmonic oscillator are:
where |
(9) |
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:
![$\displaystyle [a, x^k] = (\frac{k}{\surd 2})\cdot (x_{k-1})~.$](img43.png) |
(10) |
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:
The ground state eigenfunction normalized to unity is obtained from solving the simple
first-order differential equation
and which
leads to the expression:
 |
(12) |
By repeated application of the creation operator written as
 |
(13) |
one obtains the
-th level eigenfunction:
 |
(14) |
where
is the Hermite polynomial of order
. With the special generating function
of the Hermite
polynomials
 |
(15) |
one obtains explicit analytical relations between the
eigenfunctions of the quantum harmonic
oscillator and the above special generating
function:
 |
(16) |
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).
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.