q-oscillator from the q-Hermite Polynomial

In this Letter, the explicit forms of the generators of a q-oscillator algebra are derived from the quantum mechanical Hamiltonian [1,2] of the q-Hermite polynomial [3], the q-analogue of the Hermite polynomial constituting the eigenfunctions of the harmonic oscillator. This is in sharp contrast to the common approach to q-oscillators [4], which assumes certain forms of the algebras without any dynamical/analytical contents behind them. On the other hand, the ordinary harmonic oscillator algebra generated by the annihilation/creation operators has rich analytical structure of differential operators related with the classical analysis of the Hermite polynomial together with the coherent and squeezed states, etc.

Since the annihilation/creation operators of the harmonic oscillator and their algebra are the cornerstone of modern quantum physics, their good deformation is bound to play an important role, as evidenced by the representation theory of the quantum groups in terms of the q-oscillators. Thus our new results are expected to enrich the subject by stimulating the interplay between (quantum) algebra and analysis through new coherent/squeezed states etc, which would find applications in quantum optics and quantum information theory. Here we discuss only Rogers’ q-Hermite polynomial [3], or the so-called continuous q-Hermite polynomial [5,6] for the parameter range 0 < q < 1. Like the Hermite polynomial, the q-Hermite polynomial has no parameter other than q.

This Letter is organized as follows. The factorized Hamiltonian for the q-Hermite polynomial is presented and the q-oscillator commutation relation is shown to be a simple consequence of their structure. After brief exploration of the eigenfunctions, the exact Heisenberg operator solution [7] is presented. A second set of q-oscillator algebra is derived from the explicit forms of the annihilation/creation operators which are the positive/negative energy parts of the exact Heisenberg operator solution. These q-oscillators reduce to the ordinary harmonic oscillator in the q → 1 limit. Relationship to various forms of q-oscillator algebras is explained. The Letter concludes with some historical comments and a summary.

The Hamiltonian of the ‘discrete’ quantum mechanics for one degree of freedom has the general structure [2,1]

in which x ∈ R is the coordinate and p = -i∂ x is the conjugate momentum. The constant γ in the present case is γ def = log q, 0 < q < 1 and the potential function for the dynamics of the q-Hermite polynomial is given by

with

It is a special case of the Askey-Wilson polynomial [1,5]. The Hamiltonian is factorized as

With the explicit form of the potential function V , (3), it is straightforward to derive the q-oscillator commutation relation

Sometimes it is written as [A,

The q-oscillator commutation relation ( 7) is also a consequence of the shape invariance without shifting parameter [8] among the general Askey-Wilson potentials [1,5]. One could also say that the commutation relation of the harmonic oscillator aa †a † a = 1 is a manifestation of the shape-invariance.

The groundstate wavefunction φ 0 is annihilated by the operator A:

in which the standard notation of q-Pochhammer symbol (a ; q) n is used:

including the limiting case n → ∞. With this choice of the groundstate wavefunction, we can show that the Hamiltonian (1) is hermitian with respect to the inner product (f, g)

. By using the factorization (4) and the qoscillator relation (7), it is straightforward to demonstrate that (A † ) n φ 0 is an eigenstate of the Hamiltonian with the geometric sequence spectrum:

3 The q-Hermite polynomial

The analytical approach to the Schrödinger equation

which is a difference equation instead of a second order differential equation, goes as follows.

By similarity transformation in terms of the groundstate wavefunction φ 0 , one introduces

which acts on the polynomial part of the eigenfunction P n (η(x)):

It is elementary to show

since the residues at z = ±1, z = ±q ±1/2 , and z = ±q ∓1/2 all vanish. Thus one can find the eigenpolynomial in η(x) = cos x = (z + 1/z)/2, which is called the continuous q-Hermite polynomial introduced by Rogers [3, 5]

It has a definite parity. Reflecting the orthogonality of the eigenfunctions of the Hamiltonian H, (φ n , φ m ) ∝ δ nm , it is orthogonal with respect to the weight function φ 0 (x) 2 :

satisfying the three term recurrence relation

The action of the creation A † and annihilation A operators on the polynomial H n (cos x|q)

is

The similarity transformed A (22) is proportional to the divided difference operator.

The harmonic oscillator is a typical example for which the Heisenberg operator solution is known and the annihilation/creation operators can also be extracted as the positive/negative frequency parts of the Heisenberg operator solution. The situation is parallel but slightly different for the q-oscillator. The exact Heisenberg operator solution is derived and its pos

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