V. Papatheou

Exactly Soluble Harmonic Oscillator for a Particular Form of Time and Coordinates-Dependent Mass

An exact solution of Schrodinger equation is presented for the problem of a harmonic oscillator of mass varying with time and coordinates according to \begin{aligned} m(q,t)=m_{0}/\text{e}^{-\nu t}-k_{0}^{2}q^{2}. \end{aligned} This case, compared with known types of equivalent interactions, leads again to a new one that relates the damped harmonic oscillator with velocity-depentent interaction problem.

NON-HERMITIAN TUNNELING OF OPEN QUANTUM SYSTEMS

We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.

Effective Hamiltonian for Bloch-electrons in uniform electric and magnetic fields

Abstract It is possible to generalize the Peierls-Luttinger theorem for a Bloch electron in a uniform electric and magnetic field, by using an appropriate exponential transformation.

Statistical mechanics and the quantum friction

Abstract In this paper we calculate the density matrix for quantum-mechanical systems where the hamiltonian is similar to that of Caldirola-Kanai. For the case where the friction is a linear function of the velocity with a friction constant γ, we can calculate exactly the density matrix and the partition function of the harmonic oscillator and the oscillator in a uniform magnetic field.

Extension of the caldirola procedure in space

SummaryBased on results known until now of Caldirola’s chronon theory, we present an extension of this theory to the space co-ordinates with the use ofq-representation. First the new momentum operators are defined. These depend on a constant with dimension of length and their commutators with the position operator satisfy noncanonical commutation relations. The eigenfunctions and eigenvalues of the operators are found. We also study with this new extension the problems of free particles and the discrete time-dependent and time-independent Schrödinger equations are found. We next determine the coherent states of the new annihilation operators and finally calculate the Bloch density matrices for both cases&—Caldirola-Montaldi procedure and its new extension. These are noncanonical.RiassuntoBasandoci sui risultati conosciuti fino ad ora della teoria del tempo di Caldirola, si presenta un’estensione di questa teoria alle coordinate di spazio con l’uso della rappresentazioneq. Per prima cosa si definiscono i nuovi generatori d’impulso. Essi dipendono da una costante con dimensione di lunghezza e i loro commutatori con l’operatore di posizione soddisfano le relazioni di commutazione non canoniche. Si studiano anche con questa nuova estensione i problemi delle particelle libere e si trovano equazioni di Schrödinger dipendenti e indipendenti dal tempo. Si determinano poi gli stati coerenti dei nuovi operatori di annichilazione e infine si calcolano le matrici di densità di Bloch per entrambi i casi—la procedura di Caldirola-Montaldi e la sua nuova estensione. Queste sono non canoniche.РезюмеОсновываясь на известных результатах хрононной теории Калдиролы, мы предлагаем обобщение этой теории на пространственные координаты с использованиемq-представления. Сначала определяются новые операторы импульса. Они зависят от постоянной имеющей размерность длины и их коммутаторы с оператором положения удовлетворяют неканоническим коммутационным соотношениям. Определяются собственные функции и собственные значения. Мы также исследуем с помощью предложенного обобщения проблемы свободных частиц. Получаются уравнения Шредингера, дискретно зависящие от времени и не зависящие от времени. Затем мы определяем когерентные состояния новых операторов аннигиляции. В заключение, мы вычисляем блоховские матрицы плотности для двух случаев: процедуры Кардиролы-Монтальди и предложенного нового обобщения. Эти матрицы не являются каноническими.

Quantum friction in a periodic potential

In this paper we study the quantum friction problem using the Hamiltonian of Caldirola-Kanai for a periodic Mathieus type potential. In the sequel we study the lattice electron with friction we introduce a new effective Hamiltonian of the Caldirola-Kanai form for a Blochs band. Finally we study the cases of closed solutions of Schrodingers equation.

DEFORMED HARMONIC OSCILLATOR FOR NON-HERMITIAN OPERATOR AND THE BEHAVIOR OF PT AND CPT SYMMETRIES

In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.

Tight-binding model and Harper's equations are partial cases of the Caldirola-Montaldi procedure

SummaryIn the present paper the tight-binding model and Harper’s equations are derived directly from the Schrödinger equation by extending the Caldirola-Montaldi model in space. Then we apply approximation methods in order to determine the eigenvalues and eigenfunctions of Harper’s type equations which appear recently of significant physical interest (quantized Hall effect). Also, we calculate the Zener-tunnelling probability in the presence of electric and magnetic fields. Finally, we study the Caldirola-Montaldi model for free electrons in a uniform magnetic field.RiassuntoNel presente lavoro si deducono il modello di legame forte e le equazioni di Harper direttamente dall’equazione di Schrödinger estendendo il modello di Caldirola-Montaldi nello spazio. Poi si applicano metodi di approssimazione per determinare gli autovalori e le autofunzioni delle equazioni del tipo Harper che recentemente appaiono di notevole interesse fisico (effetto Hall quantizzato). Si calcola anche la probabilità del tunnelling Zener in presenza di campi elettrici e magnetici. Infine si studia il modello di Caldirola-Montaldi per elettroni liberi in un campo magnetico uniforme.РезюмеВ этой статье модель с ильной связи и уравнения Харпера вы водятся непосредственно из у равнения Шредингера, посредством обобщения модели Калдиролы-Монтальди. Затем мы применяем пр иближенные методы для определения собстве нных значений и собст венных функций уравнений ти па Харпера, которые представляю т значительный физич еский интерес (квантовый эф фект Холла). Мы также вычисл яем вероятность тунн елирования Зенера в присутствии электрического и маг нитного полей. В заклю чение, мы исследуем модель Калдиролы-Монтальди для свободных электр онов в однородном магнитно м поле.