Let us consider the infinite-dimensional space of ket vectors |n〉 (n = 0, 1, 2,…,∞), which are eigenvectors of the Hamiltonian H,

We will focus on the re-scaled dimensionless Hamiltonian spectrum e_n with the properties: e₀ = 0 and e_n < e_{n + 1}. We will use the following notation:

Now, we define two arbitrary partner hermitic conjugate densely defined and dimensionless lowering A₋ and raising A₊ operators satisfying the following relations:

We have to mention, from the beginning, that the operators A₋ and A₊ are simple lowering and raising operators, and they should not be confused with the group generators of quantum groups related to the concrete studied system.

After the repeatable application of the raising operator A₊ on the vacuum state |0〉, we obtain their counterpart

In a previous paper,^[1] we have introduced a new ordering technique for quantum operators which we have called the diagonal ordering operation technique (DOOT), denoted by the symbols # #. Generally, this technique differs from and may yield different results to those of the normal ordering case in the ordinary sense^[2] or the IWOP technique.^[3]

We introduce and adopt the following rules for the DOOT calculus.^[1]

Our aim is to apply the DOOT to the Gazeau–Klauder coherent states (GK-CSs) introduced in Refs. [4] and [5]

Consequently, the normalization function

Using the property (4) and the DOOT rules, it follows that

Particularly, we have

Consequently, the GK-CSs can be written as

In other words, the GK-CSs introduced in Ref. [4] can be obtained by applying the exponential operator #exp(−iγA₊A₋)# on the CSs depending on the real positive variable |J〉 and considering the rules of DOOT.

Let us examine some properties of the CSs |J〉.

Generally, the expectation value of an observable A is

For A = H we have 〈n|H|n′〉 = ħωe_nδ_nn′ and, consequently,

In conclusion, the CSs |J〉 fulfill all requirements imposed to a coherent state,^[4] even if they depend on the real variable J.

Using Eq. (7) and the DOOT rules, we can rewrite the GK-CSs (10) in an operatorial manner as

Let us demonstrate the unity operator resolution using the DOOT rules. We must have

Substituting Eqs. (32) and (19) into Eq. (31), we obtain

This result can be obtained also if we use Eq. (30), by performing the radial and angular integrals and taking into account the DOOT rules which require n′ = n. We have

Finally, bearing in mind the DOOT rules, the commutation of the ordered operator’s product #(A₊A₋)ⁿ#, we obtain

In order to extend the DOOT approach to the mixed states, let us now consider a quantum system which is in the equilibrium state with a field coupled to a reservoir at temperature T = (k_Bβ)⁻¹, where k_B is the Boltzmann constant. In this situation, the state of the examined system is a mixed state (thermal state) and the distribution function which describes a thermal state’s statistics is similar to the Maxwell–Boltzmann distribution. Then, the corresponding normalized density operator characteristic for this kind of state is (we consider the non-degenerated case)

Using the DOOT rules, the density operator can be written in an operatorial manner as follows:

For particular cases (quantum systems), the sum can be narrowed or compacted.

The Husimi’s Q-function is defined as the diagonal elements of the density operator ρ in the CSs representation.^[8] In our case, the GK-CSs |J, γ〉

It is straightforward to verify that the Husimi’s function is normalized to unity

The density operator can be expanded into a superposition of CSs projectors. This expansion is called the diagonal representation of the density operator, or Glauber–Sudarshan representation, or P representation^[9,10]

Substituting Eqs. (32) and (30) into Eq. (44), we have

In the last double sum, we must have n′ = n, due to the DOOT rules, in order to have the ordered product #(A₊A₋)ⁿ#. All terms with n′ ≠ n will be cancelled. In addition, the result of the angular integral is 1. Consequently, we have

Comparing with the expression (42) it is evident that we must have

If we perform the function change

This integral cannot be solved in a general case. The solutions are dependent on the concrete expressions of e_n and ρ(n). Ultimately, all calculations are reduced to a moment problem (Stieltjes or Hausdorff), as we will see in the last section dedicated to applications.

Let us apply the above-considered DOOT rules to two cases: pseudoharmonic oscillator and Morse oscillator. We have chosen these cases because in the first case it is a matter of an oscillator with an infinite number of bound states (and a linear energy spectrum with respect to the main quantum number n) and the second is an anharmonic oscillator with a finite number of bound states and with an energy spectrum expressed as a polynomial of the second degree in n.

5.1. Pseudoharmonic oscillator (PHO)

Let us consider a pseudoharmonic oscillator (PHO) with mass m and angular frequency ω = 2ω₀ (the corresponding one-dimensional harmonic oscillator has an angular frequency ω₀) and equilibrium distance r₀, with the potential (for simplicity, we consider that the PHO is in fundamental rotational state, with rotational quantum number J_rot = 0)^[11,12]

Their energy eigenvalues are

where k is the Bargmann index labeling the UIR of the SU(1,1) group characteristic for PHO. Consequently, we have

Then, the GK-CSs are^[12]

with

where the CSs depending on the real positive variable J are

Here Γ(x) is Euler’s gamma function, while is the Pochhammer symbol^[13] and also the normalization function, expressed as the hypergeometric function

Using Eqs. (3) and (4), the two arbitrary partner hermitic conjugate lowering A₋ and raising A₊ operators satisfy the following relations:

Consequently, the PHO Hamiltonian is

The following relations are valid, due to the recurrence procedure and hermitic conjugation:

Substituting the last two equations in the completeness relation for Fock vectors and using the DOOT rules, we obtain

where we obtain the DOOT ordered vacuum projector

as expected, according to Eq. (9).

Taking into account the above relations, the GK-CS |J, γ〉_PHO can be written in an operatorial manner as

Similarly, their conjugate counterpart _PHO〈J,γ|.

Consequently, the projector on the GK-CS |J, γ〉_PHO can be written, according to Eq. (30), as

In order to find the integration measure we appeal to Eqs. (21), (22), and (57) and we have

where

This is a Stieltjes moment problem and the function may be found using the substitution n = s − 1,^[5]

which leads to the solution

Finally, the integration measure is

Using this last expression, as well as the projector (68), and taking into account the DOOT rules and the value of angular integration after straightforward calculations, we prove the validity of the unity operator resolution identity

In Ref. [12] we have also demonstrated the validity of other Klauder’s requirements imposed to a coherent state.^[4]

With regard to the thermal states, the corresponding normalized density operator can be obtained using Eq. (42)

where we have used the expression of the Bose–Einstein population number (i.e., the average number of quanta)

as well as the expression of the partition function^[12]

The Husimi’s Q function is obtained using Eq. (43)

The P_PHO, quasi distribution function of the diagonal expansion of PHO normalized density operator ρ_PHO, i.e.,

may be obtained by inserting Eqs. (68) and (72).

The angular integral is easy to calculate, if we consider the definition of hypergeometric function ₁F₁ (a; b;x), as well as Eq. (72) and the DOOT rules,

Also, the appearing radial integral

can be brought to a Stieltjes moment problem by a suitable function, respectively exponent changes n + k = s − 1, similar to those of Eq. (68).

The final expression of P_PHO becomes

The diagonal expansion of PHO normalized density operator ρ_PHO is

This result was obtained earlier, by driving along an algebraic route (see Eq. (81) of Ref. [12]).

5.2. Morse oscillator (MO)

The one-dimensional non-rotational Morse Hamiltonian has the form of^[14]

where m_red is the reduced mass, D_e the dissociation energy, r_e the inter nuclear distance, and α the exponential parameter connected by the fundamental vibrational frequency ω through the relation

The Schrödinger equation for the Morse oscillator (MO) can be solved exactly, with the vibrational energy eigenvalues being

where ɛ ≡ ħωN/(N + 1), , and n = 0, 1, 2,…, [N/2] is a main quantum number. Note that the constant N is not an integer. Thus, the Morse potential has maximum [N/2] bound states.

Previously, using the Gazeau–Klauder formalism, Roy and Roy^[15] as well as Popov^[16] have constructed CSs corresponding to the Morse potential and examined some of their properties. Next we examine the same problem, but in the frame of DOOT. For this purpose, we use the shifted Hamiltonian H_MO, as well as the shifted dimensionless eigenvalues e_n

In order to find the two arbitrary partner hermitic conjugate lowering A₋ and raising A₊, we use Eqs. (3) and (4). These operators (which are not the group generators) satisfy the following relations:

In this manner, the MO shifted Hamiltonian can be written as

Applying the recurrence procedure and hermitic conjugation we find that

Using the property of the Euler gamma functions as well as the Pochhammer symbols^[17]

the positive constants ρ_MO(n) can be transformed as follows:

Consequently, the completeness relation for Fock vectors and the DOOT rules lead to the expression

where we have the DOOT ordered vacuum projector for MO

The function ₀F₁(;1 − N; − (N + 1)A₊A₋) is the truncated confluent hypergeometric function, in fact a confluent hypergeometric polynomial with the degree [N/2]. The result (95) is as expected, according to Eq. (9).

The corresponding expansion of the GK-CSs for the MO in terms of the energy eigenvectors |n,N〉 are^[16]

where the normalization function is

Using Eqs. (15), (16), and (90), the MO GK-CSs can be written as

where

and similar for their counterpart.

Using the DOOT rules, the projector is

according to Eq. (30). By putting J = 0 we recover the vacuum projector (95).

In order to find the integration measure from the diagonal expansion

according to the DOOT rules, we can take out the radial integral […] (it does not contain operators) from the symbol # #.

The result of the angular integration is unity, and in order to obtain the completeness relation (94), we must have n′ = n. We appeal to Eqs. (21) and (93)

where .

After the standard procedure (the exponent change n = s − 1), we obtain after solving the Stieltjes moment problem

and finally the integration measure

which is the same as that obtained in our previous paper (Eq. (32) in Ref. [16]).

The normalized density operator for a mixed (thermal) state of MO can be obtained using Eq. (42)

The exponential from this equation can be written as

Due to the fact that the Morse vibrational energy E_n is a polynomial of degree two in main quantum number n, in order to evaluate the density operator in a closed manner, we must apply another method. Taking into account that in the most cases (e.g., for diatomic molecules)

we can apply our previously introduced ansatz^[16] and expand the exponential exp(−βħωe_n) in the following manner:

Firstly, we apply this ansatz in order to calculate the partition function Z_MO(β)

which is suitable for the numerical calculations.

In a similar manner we can transform also the density operator (106) into

Starting from this expression, we will find the P-quasi distribution function P_MO(J) from the diagonal expansion of density operator

by equalizing the two manners of expressing the density operator, i.e., Eqs. (110) and (111).

After transforming Eq. (111) by substituting the expressions for the integration measure and GK-CSs projector, and equalizing the right-hand sides of Eqs. (110) and (111), it is obvious that we must have the following equality:

where [Int_J] is the radial integral

After developing the hypergeometric polynomials and using the DOOT rules, we see that in the products of raising and lowering operators we must have #(A₊)ⁿ(A₋)^n′ # = #(A₊A₋)ⁿ #δ_nn′, and the terms with n′ ≠ n are cancelled.

By inspecting the right-hand side of Eq. (112) it is evident that the P-quasi distribution function P_MO(J) must have the following structure:

Consequently, we have to solve the following Stieltjes moment problem:

where After the substitution we get the following result:

Finally, the P-quasi distribution function P_MO(J) becomes

which is identical to Eq. (60) in Ref. [16].

With these elements the properties of the GK-CSs of MO can be completely examined.

In the paper we have revisited the properties of the Gazeau–Klauder coherent states (GK-CSs) using a new operator ordering approach, the diagonal ordering operation technique (DOOT), introduced in our previous paper. In essence, this approach consists of applying the ordered product #A₊A₋ (or the operatorial functions of this ordered product) to different entities connected with the coherent states (in the present case the GK-CSs) in order to simplify respective algebraic calculations. The obtained results are the same as those obtained by using pure algebraic methods, which confirms the validity of the DOOT method.

We have defined two arbitrary partner hermitic conjugates densely defined and dimensionless lowering A₋ and raising A₊operators (which, generally, are not identical to the group generators of the corresponding quantum group), the only restriction being that they are connected with the Hamiltonian of the examined system by the relation H = ħωA₊A₋.

Regarding the GK-CSs introduced in Ref. [4], they can be obtained by applying the DOOT ordered exponential operator #exp(−iγA₊A₋)# on the CSs which depend on the real positive variable |J〉 and by considering the rules of DOOT.

In our opinion, the main conclusions of the present paper are as follows.