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arXiv:hep-th/0412160v1 15 Dec 2004

Pseudo-hermitian interaction between an oscillator and a spin-1

particle in the external magnetic field

2

Bhabani Prasad Mandal∗

Department of Physics, Banaras Hindu University,

Varanasi-221005, India

Abstract

We consider a spin-1

oscillator through some pseudo-hermitian interaction. We find that the energy eigenvalues

for this system are real even though the interaction is not PT invariant.

2particle in the external magnetic field which couples to a harmonic

1Introduction

In the last few years the study of some nonhermitian Hamiltonian with real spectrum have

given rise to a growing interest in the literature. This was mainly initiated by Bender and

Boettcher’s observation that with properly defined boundary conditions the spectrum of

the Hamiltonian H = p2+ x2(ix)ν,

(ν ≥ 0) is real, positive and discrete. The reality of

the spectrum is a consequence of unbroken PT [ combined parity (P) and time reversal

(T) ] invariance of the Hamiltonian i.e. [H,PT] = 0 [1, 2]. However pairs of complex

conjugate eigenvalues appear when the PT symmetry is broken spontaneously. This is

also illustrated nicely with the help of a nonhermitian but PT invariant potential with

quasi-exactly solvable eigenvalues [3] .

This surprising result attracts lot of interest in last few years and many other such

nonhermitian but PT symmetric systems, mostly for one particle in one space dimension

have been investigated [4]- [17]. Validity of these results have also been tested for the

cases of nonhermitian extension of some exactly solvable many particle quantum systems

in one dimension [18] -[22]. Nonhermitian extension of some field theoretic models has

been considered in Refs. [5, 6].

However to develop a consistent quantum theory for these nonhermitian Hamiltoni-

ans one encounters serve difficulties [10, 13]. Firstly, the eigenstates of PT symmetric

nonhermitian Hamiltonians with real eigenvalues only do not satisfy standard complete-

ness relations. More importantly if one takes the natural inner product associated with

PT-symmetric system as

(f,g) =

d4x[PTf(x)]g(x),

?

∗e-mail address: bpm@bose.res.in

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then the half of the energy eigenstates have negative norms which makes it difficult to

maintain the familiar probabilistic interpretation of quantum theory. Recently Bender and

coworkers have found a new symmetry , C, inherent in all such Hamiltonian with unbroken

PT symmetry[2, 4]. This allows to introduce an inner product structure associated with

CPT conjugation for which the norms of the quantum states are positive definite and one

gets usual completeness relation. As a result the Hamiltonian and its eigenstates can be

extended to complex domain so that the associated eigenvalues are real and underlying

dynamics is unitary.

In an another approach Mostafazadeh [7, 8] has shown that the reality of spectrum

of nonhermitian Hamiltonian is due to so called pseudo-hermiticity properties of the

Hamiltonian. A Hamiltonian is called η pseudo-hermitian if it satisfies the relation

ηHη−1= H†,

(1.1)

where η is a linear hermitian operator. All PT symmetric nonhermitian Hamiltonian are

pseudo-hermitian and these consist a subclass of pseudo-hermitian Hamiltonian. All the

observations of PT symmetric nonhermitian Hamiltonian can be explained nicely in this

approach.

The purpose of this letter is to consider an example of nonhermitian Hamiltonian

which is not PT invariant but pseudo-hermitian and study the different properties of

such system. With this aim we consider a system consisting of a spin half particle in the

external magnetic field coupled to an oscillator via nonhermitian interaction. We find

that the spectrum is real even though the interaction term is not PT symmetric. The

explicit PT asymmetric system has also been considered recently [23].

Here is the plan of the paper. In section II we will discuss the Hamiltonian of the

system and its symmetries. We will find the energy eigenvalues and corresponding eigen-

fuctions explicitly for this system in section III. Section IV is kept for concluding remarks.

2The Model

We consider a system of a spin1

oscillator through some nonhermitian interaction described by the Hamiltonian

2particle in the external magnetic field,?B coupled to an

H = µ? σ ·?B + ¯ hωa†a + ρ(σ+a − σ−a†).

(2.1)

Here ? σ’s are Pauli matrices, ρ is some arbitrary real parameter and σ± ≡

are spin projection operators. a,a†are usual creation and annihilation operator for the

oscillator states and defined as

1

2[σx± iσy]

a =p − imωx

√2mω¯ h

,a†=p + imωx

√2mω¯ h

,

(2.2)

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with

a|n >=√n|n − 1 >, and a†|n >=√n + 1|n + 1 >,

where the notation |n > for number eigenvectors for the oscillator has been adopted.

For the sake of simplicity we can choose the external magnetic field in z-direction,

?B = B0ˆ z and the Hamiltonian for the system as given in Eq.(2.1) is reduced to ,

H =ǫ

2σz+ ¯ hωa†a + ρ(σ+a − σ−a†),

where ǫ = 2µB0. This system can also be thought of a two level system coupled to an

oscillator where ǫ is the splitting between the levels. Note that this Hamiltinian is not

hermitian as,

ǫ

2σz+ ¯ hωa†a − ρ(σ+a − σ−a†),

?= H,

(2.3)

(2.4)

H†

=

(2.5)

as σ†

not change sign as both are axial vectors but as it clear from the Eq (2.2) that both the

creation and annihilation operators change sign.

±= σ∓. Under parity transformation [i.e. x −→ −x; p −→ −p] both ? σ and?B do

P? σP−1= ? σ,

P?BP−1=?B,

PaP−1= −a,

Pa†P−1= −a†.

(2.6)

Note the interaction term of the Hamiltonian in Eq. (2.1) changes sign under parity

operation. The time reversal operator for the system of spin half particles is T = −iσyK

where K is complex conjugation operator. We note the changes of following quantities

under time reversal transformation as,

T? σT−1= −? σ,

T?BT−1=?B,

Tσ±T−1= −σ∓,

TaT−1= −a,

Ta†T−1= −a†.

(2.7)

We have considered the magnetic field as the external element in our system and it does

not change sign under time reversal operation. However one can consider magnetic field

in other way also, when it changes sign under time reversal as the current producing

magnetic field is reversed under time reversal. The results of this paper are same in both

cases. From Eqs. (2.6) and (2.7) we can see that the Hamiltonia in Eq. (2.1) is not PT

symmetric,

ǫ

2σz+ ¯ hωa†a + ρ(σ+a†− σ−a),

?= H.

PT H (PT)−1= −

(2.8)

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However this Hamiltonian is σz-pseudo-hermitian

σzHσ−1

z

=

ǫ

2σz+ ¯ hωa†a + ρ(σzσ+σza − σzσ−σza†),

ǫ

2σz+ ¯ hωa†a − ρ(σ+a − σ−a†),

= H†.

=

(2.9)

Here we have used the relations σzσ±σz= −σ±. In case of η− pseudo-hermitian Hamil-

tonian the choice of the operator η is not unique [7]. Therefore we look for whether our

Hamiltonian is pseudo-hermitian with respect to any other operator. Indeed it is also

pseudo-hermitian with respect to parity operator as,

PHP−1

=

ǫ

2PσzP−1+ ¯ hωPa†aP−1+ ρ(Pσ+aP−1− Pσ−a†P−1),

ǫ

2σz+ ¯ hωa†a − ρ(σ+a − σ−a†),

= H†.

=

(2.10)

Finally we found a symmetry of our Hamiltonian . It is invariant under the symmetry

generated by the combined operator, Pσzi.e.

[H,Pσz] = 0.

(2.11)

However it not surprising as it is shown in Ref.[7] that if a Hamiltonian is pseudo-hermitian

with respect to two different operator η1,η2 then the system is symmetric under the

transformation generated by η−1

2η1.

3The solutions

To find the energy eigenvalues and corresponding eigenvectors of the system described by

the Hamiltonian in the Eq. (2.4) we adopt the notation for the state as, |n,1

n is eigenvalue for the number operator a†a i.e. a†a|n >= n|n > and ms= ±1 are the

eigenvalues of the operator σzi.e. σz|1

is a ground state of the Hamiltonian with eigenvalue −ǫ

H|0,−1

2ms> where

2ms>= ms|1

2ms>. It is readily seen that |0,−1

2and it is non-degenerate.

2>

2>= −ǫ

2|0,−1

2> .

(3.1)

Note that the projection operators σ±have the following usual properties when they act

on the state |n,±1

σ+|n,1

σ−|n,−1

2>,

2>= 0;

σ+|n,−1

σ−|n,1

2>= |n,1

2>= |n,−1

2>,

2>= 0;

2> .

(3.2)

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We observe that the next possible states |0,1

2> is not a eigenstate of the Hamiltonian,

H|0,1

2>=ǫ

2|0,1

2> −ρ|1 −1

2> .

(3.3)

However this state along with the state |1,−1

and form a invariant subspace in the space of states as,

2> close under the action of the Hamiltonian

H|1,−1

2>= (¯ hω −ǫ

2)|1,−1

2> +ρ|0,1

2> .

(3.4)

First two excited states belong to this sector spanned by these two states, |0,1

|1,−1

2> and

2> wherein the Hamiltonian matrix is given by†

H1=

?

ǫ

2

ρ

−ρ −ǫ

2+ ¯ hω

?

.

The eigenvalues of this Hamiltonian matrix are given by λI,II

Note these eigenvalues are real provided |¯ hω − ǫ| ≥ 2ρ. Putting 2ρ = (¯ hω − ǫ)sinθ1we

find the eigenvectors corresponding to this doublet are

1

=1

2

?

¯ hω ±

?

(¯ hω − ǫ)2− 4ρ2?

.

|ΨI

1> = cosθ1

2|0,1

2|0,1

2> +sinθ1

2> +cosθ1

2|1,−1

2|1,−1

2>,

for λI

1=¯ hω

2(1 + cosθ1) −ǫ

2=¯ hω

2cosθ1,

|ΨII

1> = sinθ1

2>,

for λII

2(1 − cosθ1) +ǫ

2cosθ1.

(3.5)

It may be observed that these two states are not orthogonal to each other nor do they

have to be as H ?= H†. To find the next excited states we have to consider next in-

variant subspace. It can be easily checked that next invariant subspace is spanned by

the vectors,|1,1

obtained following the same method.

The result is easily generalized to the sector spanned by |n,1

wherein the Hamiltonian matrix is given by,

2>,|2,−1

2> and the eigenvalues and eigenvectors for this doublet can be

2> and |n + 1,−1

2>

Hn+1=

?

ǫ

2+ n¯ hω

−ρ√n + 1 −ǫ

ρ√n + 1

2+ (n + 1)¯ hω

?

.

Now we have the eigenvalues of this Hamiltonian matrix are given by

λI,II

n+1=1

2

?

(2n + 1)¯ hω ±

?

(¯ hω − ǫ)2− 4ρ2(n + 1)

?

.

(3.6)

†Similar two by two matrix Hamiltonian is also considered in ref. [24] for a completely different system.

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