| 时间相关的谐振子动力学研究 |
| Alternative Title | Dynamics of Harmonics Oscillator with Time-dependent Parameters
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| 张芳 |
| Thesis Advisor | 罗洪刚
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| 2011-05-27
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | 含时谐振子
不变算符法
高斯假设法
一般变换法
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| Abstract | 作为物理学中最基本的模型之一，谐振子模型在物理学中的很多领域都有着非常重要的应用，所以，对于各种不同类型的谐振子的研究一直是许多科研工作者非常关心的焦点。随着现代技术的进步，人们对含时系统的研究越来越深入，其目的是最终能够实现对这些系统的控制。时间相关的谐振子是目前人们关注的前沿问题之一，它广泛的应用在量子光学，等离子物理，分子物理，量子场论，量子信息和量子化学等领域。通过对含时谐振子的参数的调节，可以观察到许多先前时间无关的谐振子不能实现的现象。
本文主要包括两个方面的内容。在第一部分我们分别讨论了对不含时谐振子与含时谐振子的处理方法，尤其是含时谐振子，我们较详细的论述了两种目前普遍使用的的方法：含时不变算符方法和Gaussian 假设方法，并得到了各自的波函数解。在第二部分我们提出了一种新的处理含时谐振子的方法，即一般变换法，并用于精确求解时间相关的谐振子问题。我们计算了系统在各不同状态之间的跃迁概率随演化时间的变化。为了证明我们所用方法的正确性，将我们的结果与用其他方法得到的结果进行对比，发现经过一些变换后，最终都能转化为同一种形式。在我们所用的方法中，我们对含时Schrödinger 方程的求解问题转化为解一个微分方程组，这就使得我们的方法与在本文中提到的含时不变算符法相比，显得更为简便；而与Gaussian 假设方法相比，我们发现，在一些特定的条件下，我们所用的与Gaussian方法的结果可以重复，这样Gaussian 假设法是我们所用方法的一种特例，我们的方法更具有广泛性。 |
| Other Abstract | As one of the most fundamental models in physics, the harmonics oscillator plays an important role in various fields, for which many researchers become more and more interested in different types of harmonics oscillator. With the rapid development of modern techniques, in order to control the systems with time-dependent parameters successfully, researches about varied systems are more and more deeply. In present, one of the frontier issues is the problem of time-dependent harmonic oscillator, which has a wide range of applications throughout modern physics, such as quantum optics, plasma physics, molecular physics, quantum field theory, quantum information, quantum chemistry and so on. We can employ the time-dependent harmonic oscillator to explain many quantum-mechanical effects by means of the time-dependent parameters in the Hamiltonian of the harmonic oscillator.
This thesis is contributed to the study of two parts. Firstly, We discuss the time evolution of time-independent and time-dependent harmonic oscillator, separately. Because we are already familiar with the time-independent harmonic oscillator, in the following, We will focus on the time-dependent harmonic oscillator. There are various methods to determine the time evolution of time-dependent harmonic oscillator. In this thesis, we only introduce two approaches, which are famous and very simple for us. One is called the invariant operator method developed by Lewis and Riesenfeld in 1997, and another is the Gaussian type solution proposed by Husimi in 1953. Both of them can obtain the solution of the Schrödinger equation for the time-dependent harmonic oscillator. Secondly, we present a new treatment, named the general transformation method, for a kind of important quantal harmonic oscillator with a constant mass and a time-dependent frequency. We investigate the time evolution of this time-dependent oscillator system using the general transformation method and derive the exact wave function. In particular, we obtain a very general formula for the transition probabilities with the help of the solution for the time-dependent harmonic oscillator and show how it varies as a function of the transformation time between different initial quantum numbers and final quantum numbers. In order to verify our new treatment for the time-dependent harmonic oscillator, we compare our solution to others, which were derived by means of other methods, and apparently we find that they are the same via ... |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/229368
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| Collection | 物理科学与技术学院
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Recommended Citation
GB/T 7714 |
张芳. 时间相关的谐振子动力学研究[D]. 兰州. 兰州大学,2011.
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