利用电荷量子比特促进矩阵力学的教学

Using charge qubits to facilitate the teaching of matrix mechanics

在量子力学教学中,矩阵力学的表示和计算是一个非常重要的内容.二能级体系作为典型而又简单的例子,可以加深学生们对矩阵力学的理解.然而目前教材中对二能级体系的重视比较有限.本文引入量子信息前沿研究中的一个体系:电荷量子比特作为教学案例.在教学中,首先利用哈密顿量对体系建模,描述单电子在两个量子点中的占据和隧穿过程.然后,通过能量本征函数表示、求解含时薛定谔方程和求解力学量平均值的运动方程等3种方法,得到体系状态和测量概率随时间变化的关系.最后与相干实验结果比较,发现理论结果能很好的描述实验现象,而且有清晰的物理图像.这样的教学方式有助于学生更加直观和深入地掌握矩阵力学表示下的量子力学规律与计算方法.

The representation and calculation of matrix mechanics is a very important element in the quantum mechanics course. The two-level system serves as a simple example to deepen students’ understanding of matrix mechanics. However, the current textbooks give rather limited attention to the two-level system. In this paper, we introduce a system in the research frontier of quantum information: the charge qubit. In the teaching, the system is first modeled using a Hamiltonian operator to describe the occupation and tunneling processes of a single electron in two quantum dots. Then, the state and the time evolution of the system are obtained by three methods such as energy eigenfunctions representation, solving the time-dependent Schr9 dinger equation and solving the equation of motion for the expectation value of an operator. Finally, by comparing the theoretical analysis and experimental results, students can find that the theory is consistent with the experiment. Thus these teaching contents can help students to grasp more intuitively and deeply the quantum mechanical laws and calculations with the matrix mechanics representation.