Pseudo-Hermitian Matrix Exactly Solvable Hamiltonian

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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 ${\stackrel{\u02dc}{H}}_{M}$ (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 ${\stackrel{\u02dc}{H}}_{M}$ 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), ${\stackrel{\u02dc}{H}}_{M}$ is exactly solvable. Indeed, its all eigenvalues can be computed algebraically. Even if the gauge Mandal Hamiltonian ${\stackrel{\u02dc}{H}}_{M}$ 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 ${\stackrel{\u02dc}{H}}_{M}$ , 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

${\stackrel{\u02dc}{H}}_{JC}={R}^{-1}{H}_{JC}R,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}R=\mathrm{exp}\left(-\frac{\omega {x}^{2}}{2}\right),$ (26)

after some algebraic manipulations, the new Hamiltonian ${\stackrel{\u02dc}{H}}_{JC}$ (known as gauge Hamiltonian) is obtained

$\begin{array}{c}{\stackrel{\u02dc}{H}}_{M}=\frac{\epsilon}{2}{\sigma}_{z}-\frac{1}{2}\frac{{\text{d}}^{2}}{\text{d}{x}^{2}}+\omega x\frac{\text{d}}{\text{d}x}+\rho \frac{\left[{\sigma}_{+}p+{\sigma}_{-}\left(p+2i\omega x\right)\right]}{\sqrt{2\omega}}\\ =\frac{\epsilon}{2}{\sigma}_{z}+\frac{{p}^{2}}{2}+i\omega xp+\rho \frac{\left[{\sigma}_{+}p+{\sigma}_{-}\left(p+2i\omega x\right)\right]}{\sqrt{2\omega}}\end{array}$ (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 ${\stackrel{\u02dc}{H}}_{JC}$ is

${\stackrel{\u02dc}{H}}_{M}=\frac{\epsilon}{2}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)+\left(\begin{array}{cc}\frac{{p}^{2}}{2}+i\omega xp& 0\\ 0& \frac{{p}^{2}}{2}+i\omega xp\end{array}\right)+\rho \left(\begin{array}{cc}0& \frac{p}{\sqrt{2\omega}}\\ \frac{p+2i\omega x}{\sqrt{2\omega}}& 0\end{array}\right)$ ,

${\stackrel{\u02dc}{H}}_{M}=\left(\begin{array}{cc}\frac{{p}^{2}}{2}+i\omega xp+\frac{\epsilon}{2}& \rho \frac{p}{\sqrt{2\omega}}\\ \rho \frac{p+2i\omega x}{\sqrt{2\omega}}& \frac{{p}^{2}}{2}+i\omega xp-\frac{\epsilon}{2}\end{array}\right)$ . (28)

Note that one can easily check if this above gauge Hamiltonian ${\stackrel{\u02dc}{H}}_{JC}$ 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 ${\stackrel{\u02dc}{H}}_{JC}$ 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 ${\stackrel{\u02dc}{H}}_{JC}$ , 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.

References

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