Physical Aspects of PseudoHermitian and Symmetric Quantum Mechanics
Abstract
For a nonHermitian Hamiltonian possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of to a Hermitian Hamiltonian takes a simple form. We use this basis to construct the observables of the quantum mechanics based on . In particular, we introduce pseudoHermitian position and momentum operators and a pseudoHermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudoHermitian and in particular symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian , the physical observables , the localized states, and the conserved probability density for the nonHermitian symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the nonHermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of symmetric quantum mechanics and clarify its relationship with both the conventional quantum mechanics and the classical mechanics.
PACS number: 03.65.w
Contents
 1 Introduction
 2 Canonical Metric Basis
 3 Classical System and Its PseudoHermitian Canonical Quantization
 4 Localized States, Position Wave Functions, and the Probability Density

5 Application to the Symmetric Square Well
 5.1 Perturbative Calculation of and
 5.2 Construction of a Canonical Metric Basis
 5.3 Construction of the Hermitian Hamiltonian
 5.4 The Classical Hamiltonian
 5.5 Construction of the Observables
 5.6 Probability Density, Position Measurements, and Localized States
 5.7 Dynamical Consequences of NonHermiticity
 6 Discussion and Conclusion
 A Appendix
1 Introduction
Most of the recent publications on symmetric quantum mechanics focus on the study of the spectral properties of various (nonHermitian) symmetric Hamiltonians. The results reported in these publications are mainly mathematical. The purpose of the present paper is to address some of the most basic problems related to the physical aspects of symmetric and more generally pseudoHermitian quantum mechanics. In particular, we will offer a complete description of the nature and the construction of the physical observables and provide a method to compute various physical quantities in these theories. We will also elucidate the relationship between these theories and the conventional classical and quantum mechanics.
As our approach is motivated by the mathematical results obtained within the framework of the theory of pseudoHermitian operators [1, 2, 3], we begin our discussion by a brief review of the relevant developments.
A central question that arises in connection with the current interest in symmetric quantum mechanics [4, 5] is: “What are the necessary and sufficient conditions for the reality of the spectrum of a linear operator?” Ref. [2] provides the following answer to this question: If the operator acts in a Hilbert space and has a complete set of eigenvectors (i.e., it is diagonalizable) then its spectrum is real if and only if (one and consequently all of) the following equivalent conditions holds.

(C1) There exists a positivedefinite operator^{1}^{1}1An operator is called positivedefinite if it is Hermitian and has a strictly positive spectrum. that fulfils
(1) i.e., is pseudoHermitian [1] and the set^{2}^{2}2For a discussion of this set, see [6]. of all the metric operators satisfying includes a positivedefinite element.

(C2) is Hermitian with respect to some positivedefinite inner product on (which is generally different from its defining inner product .) A specific choice for is .
The framework provided in Refs. [1, 2] also explains the connection with symmetry. It turns out that, under the same conditions, pseudoHermiticity of is equivalent to the presence of an antilinear symmetry, symmetry being the primary example, [3, 9].
The condition that the Hamiltonian must have a complete set of eigenvectors may be relaxed by extending the analysis of [1, 2, 3] to blockdiagonalizable linear operators as discussed in [10, 11]. However, note that physically this condition is intertwined with the requirements of the quantum measurement theory. The failure to satisfy it is equivalent to allowing for the states that have zero overlap with all the energy eigenstates. As a result, the total probability of measuring any energy value for such a state is identically zero, i.e., one can never perform an energy measurement on such a state; it must not be possible to prepare it!
These physical considerations form the basis of a general framework, called pseudoHermitian quantum mechanics [12], that allows for formulating a quantum theory based on an eigenvalue problem for a linear operator acting in a (complex) vector space . A typical example is an eigenvalue (SturmLiouville) equation for a differential operator acting in a complex function space. Supposing that this eigenvalue problem has a solution, i.e., there are eigenvectors , one lets be the span of , endows with an arbitrary positivedefinite inner product, Cauchy completes [13] this inner product space to a Hilbert space , and views as a (possibly densely defined) linear operator acting in . Then, by construction, is a diagonalizable operator acting in , and the results of [1, 2, 3] apply.
As noted in [14], the equivalence of the reality of the spectrum of and the condition (C2) is the basic mathematical result underlying the construction of the socalled inner product for symmetric quantum systems [15]. Also as shown in [16], one can use the condition (C3) to map to a Hermitian Hamiltonian acting in . If one identifies the physical Hilbert space of the system with endowed with the positivedefinite inner product , then and are unitarily equivalent.
For models with a finitedimensional Hilbert space the construction of the Hermitian Hamiltonian is straightforward. In some cases has a much simpler form than , [16]. The situation is quite different for systems with an infinitedimensional Hilbert space, as almost nothing specific is known about the structure of . It is nevertheless expected to be a generally complicated nonlocal (nondifferential) operator [16].
The study of systems with an infinitedimensional Hilbert space is particularly important, because it is for such systems that one can seek for an underlying classical system and attempt to formulate an associated quantization scheme. Obviously, a proper treatment of these issues requires a careful study of the notion of a physical observable in pseudoHermitian and, in particular, symmetric quantum mechanics.
It has recently been shown [17] that the formulation of observables in symmetric quantum mechanics as originally proposed in [15] and reiterated in [18] is inconsistent with its dynamical aspects and that enforcing the rules of the standard measurement theory restricts the choice of the observables to linear operators that are Hermitian with respect to the inner product of the physical Hilbert space .^{3}^{3}3To resolve the inconsistency reported in [17], the authors of [15] have recently revised their definition of observables [19]. As noted in [20], it is not known if this corrected definition is generally compatible with the requirements of the quantum measurement theory. When the contour defining the boundary conditions of the problem is the real line, the definition reduces to ours (and consequently the above compatibility condition holds). But even in this case it is a more restrictive definition as it implies that the Hamiltonian must be (not only symmetric but also) symmetric, i.e., in representation it is a symmetric (infinite) matrix. This leads to some undesirable consequences [20]. Accepting this definition for the observables, one can easily show that the unitary mapping that maps to also maps the observables to the Hermitian operators acting in . This in turn means that a physical system described by the Hilbert space , the symmetric Hamiltonian , and the observables may be equivalently described by the Hilbert space , the Hermitian Hamiltonian , and the observables .
In this paper we will introduce a canonical basis in which the construction of the Hermitian Hamiltonian and the physical observables simplifies considerably. This allows us to determine the underlying classical system and develop a pseudoHermitian quantization scheme. We will also introduce and construct the pseudoHermitian position operator, the corresponding position wave functions, and the conserved probability density. As a concrete application of our general results we perform a thorough investigation of the symmetric square well Hamiltonian, computing the corresponding Hermitian Hamiltonian , the observables (in particular the pseudoHermitian position operator), the probability density, the position expectation values, and the localized states. We will also describe the effects of the nonHermiticity of the Hamiltonian on the latter quantities and discuss the underlying classical Hamiltonian.
Throughout this paper we will assume that is diagonalizable and has a nondegenerate, real, discrete spectrum. The extension of the results to degenerate spectra may be easily achieved following the approach of [1, 2, 3, 14]. The presence of a continuous part of the spectrum does not lead to any serious complications either. For example see [21, 22].
2 Canonical Metric Basis
Let be a Hilbert space and be a diagonalizable linear (Hamiltonian) operator having a real, nondegenerate, discrete spectrum. Following [1, 2, 3, 14], we shall label the eigenvalues of with and let denote a basis of consisting of the eigenvectors of ,
(2) 
Then one can construct another basis of that satisfies [3]
(3) 
In particular, form a biorthonormal system [23], and
(4) 
Here and throughout this paper, for any linear operator acting in , stands for the adjoint of , i.e., the unique linear operator satisfying .
A central result of [1] is that the operator
(5) 
satisfies (1). It is also manifestly positivedefinite, because it satisfies , where and is an orthonormal basis of , and that it is invertible, with the inverse given by
(6) 
We can use (5) to introduce the positivedefinite inner product:
(7) 
and identify the physical Hilbert space with the underlying vector space of endowed with this inner product. This means that as complex vector spaces and are identical, but as Hilbert spaces they are not.
In view of (1), the Hermitian Hamiltonian of condition (C3) has the form [16]:
(8) 
where is the unique positive(definite) square root of . The transformation corresponds to the linear mapping . It is a simple exercise to check that, for any pair of state vectors: Hence as a mapping of onto , is a unitary operator.^{4}^{4}4A linear map between two inner product (in particular Hilbert) spaces and with inner products and is said to be a unitary operator if for all , we have , [13]. is unitary if and only if it is invertible (onetoone and onto) and .
Now, consider a physical system that is described by the Hilbert space , the Hamiltonian , and the observables that are Hermitian operators acting in .^{5}^{5}5Being a Hermitian operator acting in , the Hamiltonian is also an observable. But as operators acting in neither nor are Hermitian. Because is a unitary transformation, is Hermitian if and only if is Hermitian. This, in particular, means that the observables may be constructed from the Hermitian operators according to [17]
(9) 
Consequently, we can also describe the physical system using the original Hilbert space , the Hermitian Hamiltonian , and the observables . The two descriptions are physically identical as there is a onetoone correspondence between the states and the observables used in these descriptions and more importantly the physical quantities such as the transition amplitudes or expectation values of the observables do not depend on the choice of the description.
The main ingredient of the above construction is the operator . It has three important properties:

As an operator mapping to , it is a unitary operator;

As an operator mapping to , it is a Hermitian operator;

As an operator mapping to , it is also a Hermitian operator.^{6}^{6}6This can be easily checked: . In particular, both and are physical observables.
Property P2 suggests that a natural method for computing the operators and is to use an orthonormal basis of that consists of the eigenvectors^{7}^{7}7Here we suppress the degeneracy labels for the eigenvectors for simplicity. Note also that in view of the nonuniqueness [24, 14] of one can assume without loss of generality that the eigenvalues of are nondegenerate. of . Denoting the eigenvalues of by , we have
(10) 
These in turn imply
(11) 
In the following we shall refer to as a canonical metric basis.
Let be a linear operator acting in , we can uniquely identify with its matrix representation in the basis , where
(12) 
Because is an orthonormal basis of , the matrix elements of are given by
(13) 
In particular, is Hermitian with respect to the defining inner product of if and only if is a Hermitian (possibly infinite) matrix, i.e., .
3 Classical System and Its PseudoHermitian Canonical Quantization
For , we can define the pseudoHermitian position () and momentum () operators according to
(20) 
where , , and and are the usual position and momentum operators acting in .
Clearly, the pseudoHermitian position and momentum operators satisfy the canonical commutation relation
(21) 
Indeed, together with the identity operator , they provide a unitary irreducible representation of the WeylHeisenberg algebra which has the physical Hilbert space as the representation space. The fact that by construction this representation is unitarily equivalent to the standard representation of the WeylHeisenberg algebra (that has as the representation space) is a manifestation of vonNeumann’s celebrated uniqueness theorem.^{8}^{8}8This theorem states that up to unitary equivalence the WeylHeisenberg algebra has a unique unitary irreducible (projective) representation [13].
Having introduced the pseudoHermitian position and momentum operators, we can also speak of the following pseudoHermitian canonical quantization of classical systems:
(22) 
where and stand for classical position, momentum, and Poisson bracket, respectively. For instance, pseudoHermitian quantization of the classical Hamiltonian for a free particle leads to the pseudoHermitian quantum Hamiltonian:
(23) 
which is a generally nonlocal (nondifferential) operator.
Note that in general the Hamiltonian operator , that is used to construct the metric operator and consequently define the above notion of pseudoHermitian quantization, does not have the standard form . For example, a symmetric Hamiltonian of the standard form [4] (with a complexvalued potential ),
(24) 
cannot generally be expressed in the form for any realvalued function . Nevertheless, because (in light of property P3) is also a physical observable, one can express and and consequently the Hamiltonian (24) as certain power series in and (modulo commutation relations (21).) This in turn implies that the classical Hamiltonian , whose pseudoHermitian quantization yields , is not generally of the standard (Kinetic+Potential) type. Rather it is a complicated (nonpolynomial) function of and .
The classical Hamiltonian may also be obtained using the Hermitian Hamiltonian which according to (8) and (24) takes the form
(25) 
Again this is a nonlocal operator which can be expressed as a power series in with dependent coefficients. This is because (according to property P2) and are Hermitian operators acting in . The classical Hamiltonian may be obtained by replacing and in the expression for by their classical counterparts and , respectively. Clearly the resulting is identical with the one obtained from .
Next, we wish to recall a simple procedure for associating a power series in and (i.e., a pseudodifferential operator) to a nonlocal linear operator . Suppose that may be expressed in terms of its kernel according to
(26) 
Then for real analytic wave functions, that form a dense subset of , we can expand appearing on the righthand side of (26) in Taylor series about . Substituting the result in (26), we find where
(27)  
(28) 
As a result, we have the following (densely defined) identity
(29) 
If the operator is Hermitian, we can express (29) in a manifestly Hermitian form, namely
(30) 
The classical counterpart of this operator is the following realvalued function of the phase space ().
(31) 
where means ‘Real part of’.
The results reported in this section clearly generalize to the Hilbert spaces where is or a topologically equivalent subset of . Together with the results of the preceding section, they lead to the following prescription for determining the classical Hamiltonian for a pseudoHermitian (particularly symmetric) quantum system:

Given the Hamiltonian , compute a metric operator ;

Diagonalize and construct the corresponding canonical metric basis ;

Compute the matrix elements of in this basis and use (18) to obtain the Hermitian Hamiltonian ;

Apply the above described method of associating a pseudodifferential operator to the operator , express the latter in a manifestly Hermitian form , and take and in the resulting expression. This yields a classical Hamiltonian for the theory.
The Hamiltonian obtained in this way generally involve . The strictly classical Hamiltonian will correspond to evaluating limit of . The latter is an admissible prescription only if this limit exists.
We end this section by making the last step of the above prescription more specific. Using (18), (28), (29) and (31), identifying the kernel of with , and denoting the normalized eigenfunctions of by , i.e.,
(32) 
we have
(33)  
(34) 
Admittedly, the computation of as outlined above is too complicated to be done exactly. In Sec. 5, we study its application to a simple symmetric model where a particularly useful approximation scheme allows for computing with any desired accuracy. Finally, we should like to add that the above prescription for computing may also be used to compute the pseudoHermitian observables such as the position operator .
4 Localized States, Position Wave Functions, and the Probability Density
Having introduced the pseudoHermitian position operator we can identify its (generalized [25]) eigenvectors with the localized states of the system. They are defined by
(35) 
In view of the identity , we have
(36) 
where are the usual position kets satisfying, for all ,
(37) 
Using these relations and the fact that is a unitary mapping, we can establish the orthonormality and completeness relations for the localized states :
(38) 
where denotes the projection operator defined by
(39) 
Next, consider a particle^{9}^{9}9Here by a particle we mean a quantum system having as its classical configuration space. whose state at a fixed time is described by the state vector . We can introduce the position wave function:
(40) 
and use (38) to expand the state vector in the position basis according to
(41) 
As seen from (40), the position wave function is generally different from . This is a direct consequence of the fact that as an operator acting in fails to be Hermitian. Furthermore, in view of (38) and (40),
Hence as a function mapping to , the wave function belongs to .^{10}^{10}10Here we view as an abstract vector belonging to . The state vector also belongs to the Hilbert space which coincides with . However, these two copies of should not be confused. The converse is also true in the sense that every squareintegrable function defines a state vector . Therefore, we may identify with the vector space of position wave functions for the system. It is also a straightforward exercise to show that the assignment of a wave function to each state vector , viewed as a map , is a unitary operator. In order to see this, let be arbitrary state vectors and and , then
In the following we will assume without loss of generality that is normalized with respect to the inner product , i.e., set .
According to the standard quantum measurement theory, the probability of finding the particle in a region at time is given by
(42) 
Hence
(43) 
is the probability density of the localization of the particle in space.
Unlike the naive “probability density” , defines a conserved total probability. This follows from the fact that is Hermitian with respect to the inner product of . It is instructive to demonstrate the conservation of total probability in the position representation. In order to do so, consider the timeevolution of the state vector as determined by the Schrödinger equation:
(44) 
Computing the inner product of both sides of this equation with (using the inner product ) and employing the completeness relation given in (38), we find
(45) 
where and is defined by
(46) 
Because, as an operator acting in , is Hermitian,
This is sufficient to conclude that is a Hermitian operator acting in . As a result, in the position representation the dynamics is determined by a Hermitian Hamiltonian; the timeevolution operator, , for the position wave functions is unitary; and the total probability
is conserved.
The Hamiltonian operator is directly related to the Hermitian Hamiltonian . Substituting (36) in (46) and using (7) and (8), we have
Hence, in light of (40), the Hamiltonian is the usual position representation of the Hermitian Hamiltonian , i.e.,
(47) 
This relationship between the Hamiltonian operators and extends to all the physical observables. Given an observable acting in and the corresponding operator acting in , we can define an associated Hermitian operator acting that realizes the action of on a state vector in terms of the corresponding position wave function according to
(48) 
The operator is the position representation of the abstract operator ;
(49) 
In view of (48), (49), (38), and (40), the expectation value of in a state described by the normalized state vector and position wave function is given by
(50) 
As shown in the preceding paragraphs, one can formulate both the dynamics and the kinematics of the theory using the position wave functions . In this formulation the observables and in particular the Hamiltonian are Hermitian operators acting in similarly to the conventional quantum mechanics. In order to use this formulation, however, one needs a more explicit expression for the wave function . We may derive such an expression using the canonical metric basis . In view of, (10), (11), (32), and (40), we have
(51) 
5 Application to the Symmetric Square Well
The symmetric square well potential, originally introduced by Znojil in [26], provides a simple model with generic properties of general symmetric potentials. Its Hamiltonian is given by
(52) 
where
(53) 
and . Usually one employs units in which , , and . This is equivalent to using the dimensionless variables
(54) 
and working with the dimensionless Hamiltonian:
(55) 
where
(56) 
In the representation, the eigenvalue problem for takes the form^{11}^{11}11The eigenvalues of the Hamiltonian (52) are given by .
(57) 
The Hilbert space to which the eigenvectors belong is^{12}^{12}12The Hilbert space associated with the unscaled Hamiltonian is obtained by changing in (58) to .
(58) 
Clearly, is not Hermitian with respect to the defining inner product of . This is an indication that is not the physical Hilbert space . In order to specify the latter we should determine an appropriate metric operator . This in turn requires the solution of the eigenvalue equation (57).
The eigenvalue problem for the symmetric square well admits an essentially explicit solution. A detailed discussion is provided in [26, 27]. If is below the critical value the Hamiltonian has a real spectrum [26].^{13}^{13}13 marks an exceptional point [28] where two real eigenvalues cross in such a way that the Hamiltonian becomes nondiagonalizable. Once exceeds , regains its diagonalizability, but a pair of complexconjugate eigenvalues appear in its spectrum [10, 6]. Increasing the value of indefinitely one encounters an infinite number of exceptional points passing each of which produces a complexconjugate pair of eigenvalues. For these values of the ‘nonHermiticity’ parameter , the Hilbert spaces and are identical as complex vector spaces, i.e., they are obtained by endowing their common vector space with different inner products.
In this paper we will only be concerned with the case . For these values of , one obtains the following complete set of eigenfunctions of :
(60) 
where are arbitrary nonzero real coefficients,
(61)  
(62) 
and with are the real solutions of the transcendental equation:
(63) 
The eigenvalues are given by
(64) 
Usually the coefficients are fixed arbitrarily [26] or kept as unimportant free coefficients [27]. We will fix them in such a way that in the limit , the eigenfunctions of (5) tend to the wellknown normalized eigenfunctions of the conventional (Hermitian) square well Hamiltonian (the case ):
(65) 
Because, by construction, are also the eigenfunctions of the operator, the continuity requirement (65) constrains to be invariant. The normalized and invariant eigenfunctions of the Hermitian infinite square well potential () are, up to a sign, given by
(66) 
where
(67) 
The eigenfunctions (66) form an orthonormal basis of the Hilbert space (58). We will denote the corresponding abstract basis vectors by , i.e.,
(68) 
The continuity requirement (65) together with Eq. (66) restrict the coefficients of the eigenfunctions (5). Specifically, if we only keep the leading order term in powers of and neglect the higher order terms, we find
(69) 
where stands for the integer part of . Both the eigenvalues and the eigenfunctions are therefore determined once we obtain the solutions of (63). As shown in [26], this equation may be easily solved numerically for various values of . In this paper, we will solve this equation perturbatively by expanding the relevant quantities in powers of .
5.1 Perturbative Calculation of and
Suppose that admits a power series expansion about :
(70) 
where are to be determined. Substituting (70) in (63), expanding both sides of the resulting equation in powers of , solving it term by term for , and using (70), (62) and (64), we find
(71)  
(72)  
(73) 
where
(74) 
and stands for terms of order and higher.
Eqs. (71) – (73) reveal the curious fact that the effective perturbation parameter is .^{14}^{14}14The condition that the above perturbative calculations would be unreliable for the ground state, i.e., , corresponds to which is slightly above the critical value . This is a clear indication that the nonHermiticity of the Hamiltonian only affects the low lying energy levels. This property of the symmetric square well Hamiltonian — which has been previously known [26] — is particularly significant, for as we explain below it implies that within the confines of the perturbation theory all the infinite sums appearing in the expressions (5), (18), and (51) for the metric operator , the Hermitian Hamiltonian , and the position wave functions may be safely truncated. For example, for , . Therefore, if we set and use as a measure of the contribution of the nonHermiticity of the Hamiltonian to the energy eigenvalues, we find for : and .^{15}^{15}15The value , say for an electron ( Kg) confined in a nanometer size well (), corresponds to an energy scale for the potential (53). This is comparable with the ground state energy () of the corresponding Hermitian infinite square well potential. More generally, we can ignore the effects of the nonHermiticity parameter for all the computations involving the levels with and still obtain results that are accurate at least up to three decimal places.
In the following, we will employ an approximation scheme that neglects the effects of the nonHermiticity parameter for all levels with greater than a given number . In view of the above discussion, the results obtained using this approximation will have an accuracy of the order of
(75) 
We will respectively refer to and as the ‘order’ and the ‘accuracy index’ of our approximation scheme.
5.2 Construction of a Canonical Metric Basis
Having obtained the expression for (71) and (72) for and , we can compute and use Eqs. (5), (60), and (69) to determine the eigenfunctions of the Hamiltonian as a power series in .
The computation of a metric operator