Quantum State And Physical Properties Of Electron In The Quantum Wire Under Time-dependent External Field

It is well known that getting the exact analytic solution of low-dimensional quantum system is still the very important and valuable work. Differing from the quantum perturbation method and the numerical simulation method, the exact analytic solution of quantum system for us present much exact physical information, such as squeezed coherent properties, Berry phase, quantum transiton. The quantum state and the physical properties of electron in the mesoscopic quantum system such as the quantum wire(nanowires) is still the interesting physical field. It is not only the important physical problem, but also are of important potential application in the future as the theoretical basis of tomorrow's nanocircuit and nanodevice. Without doubt,the mesoscopic system, such as quantum wire, will be very important parts of the quantum nanocircuit. In this paper, we investigate the quantum system of single electron in the quantum wire under the time-dependent linear external field;and we obtain the corresponding exact quantum state. By which we discuss some interesting physical properties of the quantum system, such as the coherent properties and quantum transition.The thesis consists of four chapters.In chapter one, we briefly introduce the research situation about low-dimensional quantum system and the case about quantum transition under the time-dependent external field.In chapter two, we study the quantum system of single electron trapped in the quantum wire under the linear time-dependent external field. By the test-function method and choose the suitable time-space variable, we obtain the exact quantum state of the corresponding quantum system. We study the squeezed coherent properties of the electron trpped. We find that the squeezed factor periodically vary with time and the instantaneous coherent state. We also obtain the expectation of the position and momentum, and we find the similar coherent properties, namely the expectation of the position and momentum agree with the corresponding classical results, which embody the classical-quantum correspondence. We calculate the expectation of system energy;and we find that the average energy is composed of classical and quantum parts, which is different from the energy of any known coherent state.For the time-dependent quantum state, we have discussed the quantum transition of electron under the external field in detail. For the weak field case, we find that our result agrees with the corresponding result obtained by the perturbation method. But with the increase of lasers strength, the quantum perturbation method can't give the reasonable physical result, however, our exact results haven't any limits to the external parameters. In the stronger field, our result well agree with the corresponding result obtained by the Green'function method. In particular, we find the resonance ladders of the expectation energy. To our surprise, when the initial state is the ground state, the energy platforms agree with the transiting peeks, which will help to realize the controlling of transiting process. At last, we exhibit the cotrolling to the wave-packet-train of electron under the external field. By adjusting the external field parameters, we can well realize the controlling of oscillating amplitudes, period and the centre position of the oscillation.In chapter three, we study the quantum system of single electron in the one-dimensional quantum wire under the linear time-dependent external field. By the test-function method, we get the Gaussian-type wave-packet-train solution, which is different from the plane-wave type solutions that have been reported. In our paper, we disscuss the relation between Gaussian-type solutions and the plane-wave type solutions in detail. To our interest, we find the coherent properties for the time-dependent linear quantum system, namely, the expectation of the position and momentum agree with the corresponding classical results, which embody the classical-quantum correspondence. We calculate the expectation of system energy;and we find that the average energy is composed of classical and quantum parts. We discuss the time-evolution of average energy for the time-dependent quantum system in detail. For the different strength of external field, we find the interesting energy gap and level overlaps.In the last part of the paper, we give a simple summary and discussion to the above-mentioned works. Here, our main works are involved in chapters two and three.