key: cord-0261795-scpejenc
authors: Kecita, F.; Bounames, A.; Maamache, M.
title: A real expectation value of the time-dependent non-Hermitian Hamiltonians
date: 2021-12-27
journal: nan
DOI: 10.1088/1402-4896/ac3dbd
sha: 6d7f723283dc86254198742fd8520d4f79aba484
doc_id: 261795
cord_uid: scpejenc

With the aim to solve the time-dependent Schr"{o}dinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent $mathcal{PT}$-symmetric one. Consequently, the solution of time-dependent Schr"{o}dinger equation becomes easily deduced and the evolution preserves the $mathcal{C(}tmathcal{)PT}$-inner product, where $mathcal{C(}tmathcal{)}$ is a obtained from the charge conjugation operator $mathcal{C}$ through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the $mathcal{C(}tmathcal{)PT}$ normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

It is commonly believed that the Hamiltonian must be Hermitian H = H + in order to ensure that the energy spectrum (the eigenvalues of the Hamiltonian) is real and that the time evolution of the theory is unitary (probability is conserved in time), where the symbol '+' denotes the usual Dirac hermitian conjugation; that is, transpose and complex conjugate. In 1998 this false impression has been challenged by Bender and Boettcher [1] who showed numerically that a few one-dimensional quantum potentials V (x) may generate bound states ψ(x) with real energies E even when the potentials themselves are not real. They show that because PTsymmetry is an alternative condition to Hermiticity. The central idea of PT -symmetric quantum theory is to replace the condition that the Hamiltonian of a quantum theory be Hermitian with the weaker condition: the invariance by space-time reflection. This allows one to construct and study many new Hamiltonians that would previously have been ignored.

These two important discrete symmetry operators are parity P and time reversal T . The operators P and T are defined by their effects on the dynamical variables x and p. The operator P is linear and has the effect of changing the sign of the momentum operator p and the position operator x : p → −p and x → −x. The operator T is antilinear and has the effect p → −p, x → x, and i → −i. It is crucial, of course, that when replacing the condition of Hermiticity by PT -symmetry, we preserve the key physical properties that a quantum theory must have. We see that if the PT -symmetry of the Hamiltonian is not broken, then the Hamiltonian exhibits all of the features of a quantum theory described by a Hermitian Hamiltonian.

In order to have a coherent and unitary theory, Bender et al. [2] have defined the PT inner-product associated to PT -symmetric Hamiltonians as follows

where PT f (x) = f * (−x). The advantage of this inner product is that the associated norm (f, f ), which independent of the global phase of f (x), is conserved in time. The application of this definition to the eigenfunctions of H and PT implies ψ m , ψ n PT = (−1) n δ mn ,

The situation here (that half of the the eigenfunctions of H and PT have positive norm and the other half have negative norm) is analogous to the problem that Dirac encountered in formulating the spinor wave equation in relativistic quantum theory. Following Dirac, Bender et al [2] constructed a linear operator denoted by C and represented in position space as a sum over the energy eigenstates of the Hamiltonian. The operator C is the observable that represents the measurement of the signature of the PT norm of a state. The properties of the new operator C resemble those of the charge conjugation operator in quantum field theory. Specifically, if the energy eigenstates satisfy (2), then we have Cψ n = (−1) n ψ n . C, called the charge conjugation symmetry with eigenvalues ±1, C 2 = 1, such that C commutes with the operator PT but not with the operators P and T separately, is the operator observable that represents the measurement of the signature of the PT norm of a state which determines its parity type. We can regard C as representing the operator that determines the C charge of the state. Quantum states having opposite C charge possess opposite parity type.

The introduction of operator C permits to formulate a positive CPT inner-product

thus Eq.(2) becomes ψ m , ψ n CPT = δ mn .

The non-Hermitian PT -symmetric models have been successfully used for describing several physical systems such the plasmons in nanoparticle systems [3] , the problems related to the quantum information theory [4] , nonclassical light [5] and the stability of hydrogen molecules [6] .

The generalization to time-dependent non-Hermitian case have been studied in . Note that the authors of Ref. [30] emphasize that in nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate t is a parameter and thus the time-reversal operator T does not actually reverse the sign of t. Some authors adopt the fact that the operator T changes also the sign of time t → −t [7, [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] , this case could lead sometimes to incorrect results.

In this work, we adopt the following strategy: we introduce a unitary transformation F (t) which commutes with the parity P and maps the solution |ψ(t) of the time-dependent Schrödinger equation involving a non-Hermitian Hamiltonian H(t) to the solution |χ(t) involving a non Hermitian Hamiltonian H required to be timeindependent and PT -symmetric. After performing this transformation, the problem becomes exactly solvable and the evolution preserve the CPT -scalar product

The other essential ingredient of this theory is the construction of a positive-definite inner product with respect to H(t) being non self-adjoint, so that its time-evolution operator is unitary and we obtain a consistent probabilistic interpretation so that the Hamiltonian under study exhibits real mean values. The most important step towards finding this positive-definite inner product is thus to find a new operator, which we call C(t) = F + (t)CF (t) such that, we obtain a conserved norm for our original system described by the solution |ψ(t) that is ψ(t) |ψ have been extensively studied in the literature in different ways [41] [42] [43] [44] [45] [46] [47] [48] . Finally, section 4 concludes our work.

Let us consider a non-hermitian time-dependent Hamiltonian H(t) where the quantum time evolution of the system is governed by the time-dependent Schrödinger equation (for simplicity we take = 1)

In order to study the evolution of the quantum systems associated to the time-dependent Hamiltonian H(t),

we seek that this Hamiltonian can be converted into a time-independent Hamiltonian by some time-dependent transformations. To this end, we initially perform a unitary transformation F (t) on |ψ(t)

by inserting (6) in Eq. (5), we obtain the time dependent Schrödinger equation for the state |χ(t)

such that the new Hamiltonian

is time-independent and PT -symmetric, i.e.;

its eigenstates |χ(t) preserve the CPT -inner product

and in this case the solution of the Schrödinger equation (7) can be written as

where |χ is an eigenstate of H PT 0 .

Knowing that our interest is the mean value of the non-Hermitian Hamiltonian H(t), for this aim we calculate firstly the expectation value of the Hamiltonian

from which we deduce that is

we note that the first term is nothing other than the expectation value of the Hamiltonian H(t) with a new C(t)PT -inner product

where [P, F (t)] = 0 and the new operator C(t) is defined as C(t) = F + (t)CF (t), which is similar to the operator C in the sense that verifies the property C 2 (t) = 1 since C 2 = 1.

Finally, We show that the exact solution of the time-dependent Schrödinger equation (5) can be found by introducing two consecutive unitary transformations. In order to solve the Schrödinger with the Hamiltonian specified by (16) , we first try to eliminate the time-dependent parameter α(t). This can be achieved by the transformation

The unitary operator F 1 (t) has the properties

In a representation x, the wave function is given by

Suppose that

Substituting (20) into (5)ruled by the Hamiltonian (16), we find the equation of motion for |φ(t)

where the Hamiltonian

look like the time-independent harmonic oscillators with variable mass m 0 subjected to a driving linear complex time-independent potential plus a time dependent (xp + px) terms. In order to obtain the usual time-dependent harmonic oscillator with a perturbative linear potential, we remove the cross term in (23) via the transformation

where its properties are

Thus, the following unitary transformation F (t) = F 2 (t)F 1 (t)

transforms the canonical operators x and p and their squares x 2 and p 2 as follows

therefore, the transformed Hamiltonian (8) reads

where

The central idea in this procedure is to require that the Hamiltonian (28) governing the evolution of |χ(t)

is time-independent. This is achieved by setting the global time-dependent frequency appearing in (28) equal to a real constant denoted by Ω 2 0 so that its time-derivatve leads to an auxiliary equation of the form

the resulting time independent non-Hermitian Hamiltonian

is PT -symmetric.

Note that when taking α(t) = 1 ρ 2 (t) , the above auxiliary equation (30) is transformed to the following new auxiliary equation

which admits the following solutions:

• for Ω 2 0 >ω 2 :

For an appropriate choice of the constants: A = B, we obtain the expression of α(t) as α(t) =

For an appropriate choice of the constants: A = B, we obtain the expression of α(t) as α(t) = 

The eigenequation of the PT -symmetric Hamiltonian H PT 0 has the form

and the solution of the corresponding Schrödinger equation (7) can be written as

Let us introduce a non unitary transformation of the form

such that

The action of U maps the PT -symmetric Hamiltonian H PT 0 to a Hermitian one as

where the eigenfunctions |ϕ n (x) of the Hermitian Hamiltonian h are

Then, the solutions |χ n (x, t) are obtained as

where the eigenvalues

are real and H n is the Hermite polynomial of order n.

A more general way to represent the C operator is to express it generically in terms of the fundamental dynamical operators x and p : C =e Q(x,p) . The exact formula of C associated to the theory described by the Hamiltonian (31) is given as a function of the parity operator P as

such that the operator C commute with PT and H PT 0 ,i.e., [C, PT ] = [C, H PT 0 ] = 0. We can easily verify that the CPT -inner product is conserved

Now it is not difficult to calculate the expectation value of the Hamiltonian H(t) C(t)PT defined previously

thus

where x 2 CPT = χ n (x)| CPx 2 |χ n (x) . By using the following relations

and

we get the expectation value of H(t) as

which is real for any positive real time-dependent function α(t) and more simple than the result given in Eq.

(28) in Ref. [7] with less constraints on the parameters of the problem.

Now, we calculate the expectation values x C(t)PT , x 2 C(t)PT , p C(t)PT and p 2 C(t)PT in the states ψ n (x, t) of H(t) defined in Eq. (16) . In the same way, using the CPT -inner product (42) and after straightforward calculation we obtain that

We calculate also the position and momentum uncertainties

Thus, the uncertainty product is given by

it is easy to check that the uncertainty product (57) is always real and greater than or equal to 1 2 and, consequently, it is physically acceptable for any value of n. The uncertainty product takes the minimal value ∆x∆p = 1 2 only for n = 0 and α(t) =constant, i.e., for time independent mass oscillators. Finally, the probability density of the wavefunction ψ n (x, t) of H(t) is in the form

thus

is the same as the probability density of the eigenstate χ n (x, t) of time independent H PT 0 which is also equal to the probability density of the eigenstate ϕ n (x) (38) of the standard harmonic oscillator (37) . Clearly, ϕ n (x) are elements from L 2 (R), and therefore the condition (59) yields that

under this observation, we deduce that the probability is finite.

The essential ingredient of quantum mechanical non Hermitian theory is the construction of a positive-definite inner product, so that its probability is conserved in time. The operator C(t) = F + (t)CF (t) confer to the norm its conservation. The main result of this paper is that the mean value of a time-dependent non-Hermitian

Hamiltonian H(t) is real in the new C(t)PT -inner product. For this, we introduced a unitary transformation F (t) that reduces the study of time-dependent non-Hermitian Hamiltonian H(t) to the study of time-independent PT -symmetric Hamiltonian H PT 0 , and derived the analytical solution of the Schrödinger equation of the initial system. Then, we defined a new C(t)PT -inner product and showed that the evolution preserves it. Furthermore, we proved that the expectation value of the time-dependent non-Hermitian Hamiltonian H(t) is real in the C(t)PT normed states since the transformation F (t) is unitary and [P, F (t)] = 0. As an illustration, we have investigated a class of quantum time-dependent mass oscillators with a complex linear driving force. The expectation value of the Hamiltonian, the uncertainty relation and probability density have been also calculated.

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