ABSTRACT

The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We point out all properties of both of the original Mandal and the original Jaynes-Cummings Hamitonians. It is shown that these Hamiltonians are respectively pseudo-hermitian and hermitian [1] [2] . Like the direct approach to invariant vector spaces used in Refs. [3] [4] , we reveal the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator.

Keywords:

Pseudo-Hermiticity, Exact Solvability, Direct Method

1. Introduction

Several new theoretical aspects in quantum mechanics have been developed in last years. In the series of papers [5] [6] , it is shown that the traditional self adjointness requirement (i.e. the hermiticity property) of a Hamilton operator is not necessary condition to guarantee real eigenvalues and that the weaker condition PT-symmetry of the Hamiltonian is sufficient for the purpose. Following the theory developed in Refs. [5] [6] , let’s remind that a Hamiltonian is invariant under the action of the combined parity operator P and the time reversal operator T if the relation $H^{PT}=H$ is proved (i.e. PT-symmetry is said to be broken). As a consequence, the spectrum associated the previous Hamiltonian is entirely real.

An alternative property called pseudo-hermiticity for a Hamiltonian to be associated to a real spectrum is shown in details in the Refs. [1] [2] .

Referring the ideas of [1] [2] , we recall here that a Hamiltonian is said to be $\eta$ pseudo-hermitian if it satisfies the relation $\eta H{\eta }^{-1}=H^{+}$ , where $\eta$ denotes an invertible linear hermitian operator.

Another direction of quantum mechanics is the notions of quasi exact solvability and exact solvability [7] [8] [9] [10] .

In the last few years, a new class of operators has been discovered. This class is intermediate between exactly solvable operators and non solvable operators. Its name is the quasi-exactly solvable (QES) operators, for which a finite part of the spectrum can be computed algebraically.

This paper is organized as follows:

In Section 2, we briefly describe the general model which is expressed in terms of the creation and the annihilation operators. We show that the Hamiltonian describing the model is pseudo-hermitian if $\varphi =-1$ , or it is hermitian if $\varphi =+1$ .

In Section 3, we show in details the properties of the Mandal Hamiltonian namely the non-hermiticity, the non PT-symmetry, the pseudo-hermiticity and the exact solvability.

In Section 4, as in the previous section, it was pointed out that the original Jaynes-Cummings Hamiltonian is hermitian and exactly solvable.

2. The Model

In this section, we consider a Hamiltonian describing a system of a fermion in the external magnetic field, $B$ which couples the harmonic oscillator interaction (i.e. $\hslash \omega a^{+}a$ ) and the pseudo-hermitian interaction if $\varphi =-1$ , or the hermitian interaction if $\varphi =+1$ (i.e.) [1] [2] :

, (1)

where

$\sigma$ , ${\sigma }_{\pm }$ denote Pauli matrices,

$\rho$ , $\mu$ are real parameters,

$a^{+}$ , a refer the creation and annihilation operators respectively satisfying the usual bosonic commutation relation

$\left[a,a^{+}\right]=1$ , $\left[a,a\right]=\left[a^{+},a^{+}\right]=0$ and ${\sigma }_{\pm }\equiv \frac{1}{2}\left({\sigma }_x\pm i{\sigma }_y\right)$ .

Recall that the matrices ${\sigma }_{+},{\sigma }_{-},{\sigma }_x,{\sigma }_y$ and ${\sigma }_z$ can be expressed in the following matrix forms:

(2)

For the sake simplicity, one can choose the external field in the z-direction (i.e. $B=B_{0}z$ ) in order to reduce the Hamiltonian given by the Equation (1) and it becomes [1] [2] :

(3)

with $\epsilon =2\mu B_0$ and $\hslash =1$ .

3. Properties of the Original Mandal Hamiltonian

3.1. The Non-Hermiticity

In this section, we reveal that the Hamiltonian described by the Equation (3) is non- hermitian if $\varphi =-1$ . It is called Mandal Hamiltonian (i.e. $H_M$ ) and it takes the following form:

(4)

Taking account to the following identities:

(5)

let’s show that the Mandal Hamiltonian given by the above Equation (4) is non hermitian:

$H_M^{+}={\left(\frac{\epsilon }{2}{\sigma }_z\right)}^{+}+{\left(\omega a^{+}a\right)}^{+}+{\left[\rho \left({\sigma }_{+}a-{\sigma }_{-}{a}^{+}\right)\right]}^{+}$ ,

. (6)

Comparing the expressions given by the Equations (4) and (6), we see that they are different (i.e. $H_M^{+}\ne H_M$ ), as a consequence, we are allowed to conclude that the Mandal Hamiltonian $H_M$ is non-hermitian.

3.2. The Non PT-Symmetry of H_M

In this section, we prove that the Mandal Hamiltonian is non PT-symmetric [5] [6] . Recall that the parity operator is represented by the symbol P and the time-reversal operator is described by the symbol T.

The effect of the parity operator P implies the following changes [1] [2] :

(7)

Notice also the changes of the following quantities under the effect of the time reversal operator T:

(8)

Taking account to the relations (7) and (8), one can easily deduce the changes of the Mandal Hamiltonian under the effect of combined operators P et T as follows

,

, (9)

This above relation (9) can be written as follows

(10)

Comparing the relations (4) and (10), we see that they are different (i.e. $H_M^{PT}\ne H_M$ ), it means that the Mandal Hamiltonian $H_M$ is not invariant under the combined action of the parity operator P and the time-reversal operator T. In other words, the Mandal Hamiltonian $H_M$ is not PT-symmetric.

3.3. Pseudo-Hermiticity of H_M

In this section, we first prove that the non PT-symmetric Mandal Hamiltonian is pseudo-hermitian with respect to third Pauli matrix ${\sigma }_z$ [1] [2] :

(11)

with ${\sigma }_z{\sigma }_{\pm }{\sigma }_z^{-1}=-{\sigma }_{\mp }$ and $W_n={\text{e}}^{-\frac{\omega x^2}{2}}{\left(P_{n-1}\left(x\right),P_n\left(x\right)\right)}^t$ .

Comparing the Equations (6) and (11), it is seen that the following relation is satisfied:

$W_n={\text{e}}^{-\frac{\omega x^2}{2}}{\left(P_{n-1}\left(x\right),P_n\left(x\right)\right)}^t$ (12)

Taking account to this above relation, we are allowed to conclude that the Mandal Hamiltonian is pseudo-hermitian with respect to ${\sigma }_z$ .

Finally, we reveal a pseudo-hermiticity of $H_M$ with respect to the parity operator P:

(13)

Here we have used the relations (7) in order to obtain this above equation (13). As a consequence, one can conclude that the Mandal Hamiltonian is pseudo-hermitian with respect to the parity operator P.

Note that even if $H_M$ is non hermitian and non PT-symmetric, its eigenvalues are entirely real due to the pseudo-hermiticity property [1] .

3.4. Differential Form and Exact Solvability of H_M

In this step, our purpose is to change the Mandal Hamiltonian given by the Equation (4) in appropriate differential operator (i.e. $H_M$ is expressed in the position operator x and in the impulsion operator $H_{JC}^{+}=\frac{\epsilon }{2}{\sigma }_z+\omega a^{+}a+\left[\rho \left({\sigma }_{+}a+{\sigma }_{-}{a}^{+}\right)\right]$ ). Thus, referring to the ideas of exactly and quasi-exactly solvable operators studied in the Refs. [7] [8] [9] [10] , we reveal that $H_M$ preserves a family of vector spaces of polynomials in the variable x.

With this aim, we use the usual representation of the creation and annihilation operators of the harmonic oscillator respectively $a^{+}$ and a [1] [2] :

(14)

where $\omega$ is the oscillation frequency, m denotes the mass, x refers to the position operator and the impulsion operator is, $p^2=-\frac{{\text{d}}^2}{\text{d}{x}^2}$ .

Using appropriate units, we can assume $m=\hslash =1$ and the operators $a^{+}$ and a take the following forms:

(15)

Replacing the operators $a^{+}$ and a by their expressions given by this above Equation (15) in the Equation (4), the Mandal Hamiltonian $H_M$ takes the following form:

. (16)

In order to reveal the exact solvability of the above operator $H_M$ , we first perform the standard gauge transformation [2] :

(17)

After some algebraic manipulations, the new Hamiltonian ${\tilde{H}}_M$ (known as gauge Hamiltonian) is obtained

(18)

Replacing the Pauli matrices and by their respective expressions given by the relation (2), the final form of the gauge Hamiltonian is:

,

. (19)

Note that one can easily check if this above gauge Hamiltonian ${\tilde{H}}_M$ preserves the vector spaces of polynomials with $n\in {\rm N}$ . As the integer n doesn’t have to be fixed (i.e. it is arbitrary), ${\tilde{H}}_M$ is exactly solvable. Indeed, its all eigenvalues can be computed algebraically. Even if the gauge Mandal Hamiltonian ${\tilde{H}}_M$ is non-hermitian and non PT-symmetric, its spectrum energy is entirely real due to the property of the pseudo-hermiticity [1] [2] .

Thus, the vector spaces preserved by the operator $H_M$ have the following form

(20)

where $P_{n-1}\left(x\right)$ and $P_n\left(x\right)$ denote respectively the polynomials of degree n − 1 and n.

As the gauge Mandal Hamiltonian ${\tilde{H}}_M$ , it is obvious that the original Mandal Hamiltonian $H_M$ is exactly solvable. Due to this property of exact solvability, the whole spectrum of $H_M$ can be computed exactly (i.e. by the algebraic methods) [1] [2] [3] .

4. Properties of the Jaynes-Cummings Hamiltonian

4.1. The Hermiticity

In this section, considering $\varphi =+1$ , the Hamiltonian given by the Equation (3) leads to the standard Jaynes-Cummings Hamiltonian of the following form

$H_{JC}=\frac{\epsilon }{2}{\sigma }_z+\omega a^{+}a+\rho \left({\sigma }_{+}a+{\sigma }_{-}{a}^{+}\right)$ (21)

Our aim is now to prove that the above Hamiltonian $H_{JC}$ is hermitian.

Indeed, in order to reveal the hermiticity of the Jaynes-Cummings Hamiltonian given by the above relation (21), the following relation $H_{JC}^{+}=H_{JC}$ must be satisfied.

Consider now the following relation

$H_{JC}^{+}={\left(\frac{\epsilon }{2}{\sigma }_z\right)}^{+}+{\left(\omega a^{+}a\right)}^{+}+{\left[\rho \left({\sigma }_{+}a+{\sigma }_{-}{a}^{+}\right)\right]}^{+}$ , (22)

Taking account to the identities of the relation (5), this above equation leads the following expression:

$H_{JC}^{+}=\frac{\epsilon }{2}{\sigma }_z+\omega a^{+}a+\left[\rho \left({\sigma }_{+}a+{\sigma }_{-}{a}^{+}\right)\right]$ . (23)

Comparing the Equations (21) and (23), one can write that

$H_{JC}^{+}=H_{JC}$ . (24)

Referring to this equation (24), it is obvious that the standard Jaynes-Cummings Hamiltonian is hermitian. As a consequence, its eigenvalues are real due to the property of hermiticity.

4.2. Differential Form and Exact Solvability of H_JC

Along the same lines as in the above section 3.4, our purpose is to change the Jaynes-Cummings Hamiltonian given by the Equation (21) in appropriate differential operator (i.e. $H_{JC}$ is expressed in the position operator x and in the impulsion operator $p=-i\frac{\text{d}}{\text{d}x}$ ).

With this purpose, we use the usual expressions of the creation and annihilation operators of the harmonic oscillator respectively $a^{+}$ and a given by the Equation (15).

Substituting (15) in the Equation (21), the Jaynes-Cummings Hamiltonian $H_{JC}$ is written now as follows

$H_{JC}=\frac{\epsilon }{2}{\sigma }_z+\frac{p^2-\omega +{\omega }^2{x}^2}{2}+\rho \frac{\left[{\sigma }_{+}\left(p-i\omega x\right)+{\sigma }_{-}\left(p+i\omega x\right)\right]}{\sqrt{2\omega }}$ (25)

Operating on the above operator $H_{JC}$ the standard gauge transformation as

${\tilde{H}}_{JC}=R^{-1}{H}_{JC}R,R=\exp \left(-\frac{\omega x^2}{2}\right),$ (26)

after some algebraic manipulations, the new Hamiltonian ${\tilde{H}}_{JC}$ (known as gauge Hamiltonian) is obtained

$\begin{array}{c}{\tilde{H}}_M=\frac{\epsilon }{2}{\sigma }_z-\frac{1}{2}\frac{{\text{d}}^2}{\text{d}{x}^2}+\omega x\frac{\text{d}}{\text{d}x}+\rho \frac{\left[{\sigma }_{+}p+{\sigma }_{-}\left(p+2i\omega x\right)\right]}{\sqrt{2\omega }}\\ =\frac{\epsilon }{2}{\sigma }_z+\frac{p^2}{2}+i\omega xp+\rho \frac{\left[{\sigma }_{+}p+{\sigma }_{-}\left(p+2i\omega x\right)\right]}{\sqrt{2\omega }}\end{array}$ (27)

Replacing the Pauli matrices ${\sigma }_z,{\sigma }_{+}$ and ${\sigma }_{-}$ respectively by their matrix form given by the relation (2), the final form of the gauge Hamiltonian ${\tilde{H}}_{JC}$ is

${\tilde{H}}_M=\frac{\epsilon }{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)+\left(\begin{array}{cc}\frac{p^2}{2}+i\omega xp& 0\\ 0& \frac{p^2}{2}+i\omega xp\end{array}\right)+\rho \left(\begin{array}{cc}0& \frac{p}{\sqrt{2\omega }}\\ \frac{p+2i\omega x}{\sqrt{2\omega }}& 0\end{array}\right)$ ,

${\tilde{H}}_M=\left(\begin{array}{cc}\frac{p^2}{2}+i\omega xp+\frac{\epsilon }{2}& \rho \frac{p}{\sqrt{2\omega }}\\ \rho \frac{p+2i\omega x}{\sqrt{2\omega }}& \frac{p^2}{2}+i\omega xp-\frac{\epsilon }{2}\end{array}\right)$ . (28)

Note that one can easily check if this above gauge Hamiltonian ${\tilde{H}}_{JC}$ preserves the finite dimensional vector spaces of polynomials namely $V_n={\left(P_{n-1}\left(x\right),P_n\left(x\right)\right)}^t$ with $n\in {\rm N}$ . As the integer n is arbitrary, the gauge Jaynes-Cummings Hamiltonian ${\tilde{H}}_{JC}$ is exactly solvable.

As a consequence, its all eigenvalues can be computed algebraically. Indeed, the vector spaces preserved by the operator $H_{JC}$ have the following form

$W_n={\text{e}}^{-\frac{\omega x^2}{2}}{\left(P_{n-1}\left(x\right),P_n\left(x\right)\right)}^t$ (29)

where $P_{n-1}\left(x\right)$ and $P_n\left(x\right)$ denote respectively the polynomials of degree n − 1 and n.

As the gauge Jaynes-Cummings Hamiltonian ${\tilde{H}}_{JC}$ , it is obvious that the standard Jaynes-Cummings Hamiltonian $H_{JC}$ is exactly solvable. In other words, all eigenvalues associated to the Hamiltonian $H_{JC}$ can be calculated algebraically (i.e. by the algebraic methods) [1-3].

5. Conclusion

In this paper, we have put out all properties of the original Mandal Hamiltonian. We have shown that the Mandal Hamiltonian $H_M$ is non-hermitian and non-invariant under the combined action of the parity operator P and the time-reversal operator T. Even if the previous properties are not satisfied, it has been proved that the Mandal Hamiltonian $H_M$ is pseudo-hermitian with respect to P and with respect to ${\sigma }_3$ also. With the direct method, we have revealed that $H_M$ preserves the finite dimensional vector spaces of polynomials namely $V_n={\left(P_{n-1}\left(x\right),P_n\left(x\right)\right)}^t$ . Indeed, the Mandal Hamiltonian $H_M$ is said to be exactly solvable [1] [2] [3] [4] . Along the same lines used in Section 3, we have pointed out that the standard Jaynes-Cummings Hamiltonian $H_{JC}$ is hermitian and exactly solvable in Section 4.

Acknowledgements

I thank Pr. Yves Brihaye of useful discussions.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

Cite this paper

Nininahazwe, A. (2019) Pseudo-Hermitian Matrix Exactly Solvable Hamiltonian. Open Journal of Microphysics, 9, 1-9. https://doi.org/10.4236/ojm.2019.91001

References