Non-Hermitian Quantum Mechanics

In the1920s, based on the conclusion of lots of experiments and the old quantumtheory, new quantum mechanics was built to describe the principal of motion of micro-scopic particles. The microstructure of the matter was started to be researched deeply,and then the physical and chemical properties of the matter as well as its principal ofmotion were revealed further. Observables that describe the properties of the matterplay a crucial role, they are represented by operators in quantum mechanics. The ob-servables associated with the application in reality, thus, the observability is regardedas the prerequisite and necessary condition in the quantum mechanics. The Hermitianoperator is consistent with all the requirements mentioned above. So we take it forgranted that the obervables are characterized by Hermitian operators for a long time.In fact, the observable is only a necessary condition of the Hermiticity of the opera-tor. Many non-Hermitian operators also have real solutions, and they comply with aset of self-consistent quantum theory different from the one satisfied by the Hermitianoperator. These new ideas greatly expand the application of the Hermitian quantummechanics.This thesis mainly deals with the non-Hermitian P T symmetric theory and thepseudo-Hermitian quantum theory. Our work are organized as follows:By adding an imaginary interacting term proportional to ip1p2to the Hamilto-nian of a free anisotropic planar oscillator, we construct a new model which isdescribed by the P T-pseudo-Hermitian Hamiltonian with the permutation sym-metry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our P T-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also pointout the spontaneous breaking of permutation symmetry which plays a crucialrole in giving a real spectrum free of interchange of positive and negative energylevels in our model. Moreover, we find that the permutation symmetry of twodimensions in our Hamiltonian corresponds to the identity (not in magnitude butin attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, andreveal that the unequal-frequency condition imposed as a prerequisite upon thePais-Uhlenbeck oscillator can reasonably be explained as the spontaneous break-ing of this identity. An algebraic method for pseudo-Hermitian Hamiltonian systems is proposedthrough introducing the operator η+, defining new bra and ket vector states andredefining annihilation and creation operators to be η+-pseudo-Hermitian (notHermitian) adjoint of each other. As an example, a parity-pseudo-HermitianHamiltonian is constructed and analyzed in detail. Its real spectrum is obtainedby means of the algebraic method, where the corresponding operator η+is foundto be P V through a specific choice of V. The operator V is given in such a waythat on the one hand this P-pseudo-Hermitian Hamiltonian is also P V-pseudo-Hermitian self-adjoint and on the other hand P V ensures a real spectrum and apositive-definite inner product. Moreover, when the P-pseudo-Hermitian systemis extended to the canonical noncommutative space with noncommutative spatialcoordinates and noncommutative momenta as well, the first order noncommu-tative correction of energy levels is calculated, and in particular the reality ofenergy spectra and the positive-definiteness of inner products are found to be notaltered by the noncommutativity.Two non-Hermitian P T-symmetric Hamiltonian systems are reconsidered bymeans of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the P T-symmetric ones. Com-pared with the way converting a non-Hermitian Hamiltonian to its Hermitiancounterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian P T-symmetric Hamiltonian unchanged. In order to give the positivedefinite inner product for the P T-symmetric systems, a new operator V, insteadof C, can be introduced. The operator V has the similar function to the operatorC adopted normally in the P T-symmetric quantum mechanics, however, it canbe constructed, as an advantage, directly in terms of Hamiltonians. The spectraof the two non-Hermitian P T-symmetric systems are obtained, which coincidewith that given in literature, and in particular, the Hilbert spaces associated withpositive definite inner products are worked out.Non-Hermitian perturbation formulas with two independent small parameters intwo dimensions are deduced, where the usual positive inner products in Hermi-tian quantum mechanics are replaced by non-Hermtian positive inner products,such as the η+-pseudo-Hermitian inner products or the P T V inner products. Theformulas of eigenstates and the spectrum are given up to the first and secondorder of the two small parameters respectively. We propose a non-Hermitian but PT symmetric Hamiltonian by adding a non-Hermitian term proportional toixp to the free harmonic oscillator Hamiltonian, and obtain its real spectrum andeigenstates. Moreover, the model is generalized to noncommutative space withnoncommutative spatial coordinate operators and noncommutative momentumoperators. We get the eigenstates and real spectrum up to the first and second or-der of the two small parameters respectively in this case, and the eigenstates co-incide with the positive P TVinner products. The properties of the non-HermtianP T symmetric system keep unchange when it is generalized from the usual spaceto the noncommutative one.