Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.

arXiv:quant-ph/0305140Fortschritte der PhysikFeb 26, 20032 citations

Non Hermitian Quantum Mechanics: Formalism and Applications

Edvardas Narevicius, Nimrod Moiseyev

Springer eBooksJan 1, 20030 citations

Quantum mechanics with a non-Hermitian Hamiltonian

Alexander N. Grigorenko

Physics Letters AJan 1, 19939 citations

Quantum mechanics in the real Pauli algebra.

Jiansu. Wei

In this thesis, real quantum mechanics (QM) is constructed over the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ in the Pauli algebra, where the trivector $i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3$ arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with $i\\sp2 = -1$ and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator ${\\cal T}$ is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since ${\\cal T}$ is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion $\\tilde\\Pi$ (an antilinear, anti-unitary transformation). Instead, the spatial reversal $\\bar\\Pi$ (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.In this thesis, real quantum mechanics (QM) is constructed over the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ in the Pauli algebra, where the trivector $i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3$ arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with $i\\sp2 = -1$ and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator ${\\cal T}$ is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since ${\\cal T}$ is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion $\\tilde\\Pi$ (an antilinear, anti-unitary transformation). Instead, the spatial reversal $\\bar\\Pi$ (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.

Jan 1, 19910 citations

Field quantization, photons and non-Hermitean modes

S. A. Brown, Bryan J. Dalton

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.

arXiv:quant-ph/0107039Journal of Modern OpticsJun 1, 20027 citations

Beyond-Hermitian Quantum Physics

Masahito Ueda

Bulletin of the American Physical SocietyMar 15, 20210 citations

In this thesis, real quantum mechanics (QM) is constructed over the field

{\\cal F}\\sb{P\\sb0\\oplus P\\sb3}

in the Pauli algebra, where the trivector

i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3

arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with

i\\sp2 = -1

and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field

{\\cal F}\\sb{P\\sb0\\oplus P\\sb3}

and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator

{\\cal T}

is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since

{\\cal T}

is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion

\\tilde\\Pi

(an antilinear, anti-unitary transformation). Instead, the spatial reversal

\\bar\\Pi

(an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.