中国物理B ›› 2007, Vol. 16 ›› Issue (1): 16-22.doi: 10.1088/1009-1963/16/1/003

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Density matrix for an electron confined in quantum dots under uniform magnetic field and static electrical field

庞乾骏   

  1. Department of Physics, Shanghai Jiaotong University, Shanghai 200030, China
  • 收稿日期:2005-11-28 修回日期:2006-01-19 出版日期:2007-02-01 发布日期:2007-02-01
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos 60374037 and 60574036), the Program for New Century Excellent Talents of High Education of China(Grant No NCET 2005-290), The Special Research Fund for the Doctoral Program of High Education of China (Grant No 20050055013).

Density matrix for an electron confined in quantum dots under uniform magnetic field and static electrical field

Pang Qian-Jun(庞乾骏)   

  1. Department of Physics, Shanghai Jiaotong University, Shanghai 200030, China
  • Received:2005-11-28 Revised:2006-01-19 Online:2007-02-01 Published:2007-02-01
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos 60374037 and 60574036), the Program for New Century Excellent Talents of High Education of China(Grant No NCET 2005-290), The Special Research Fund for the Doctoral Program of High Education of China (Grant No 20050055013).

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摘要： Using unitary transformations, this paper obtains the eigenvalues and the common eigenvector of Hamiltonian and a new-defined generalized angular momentum (Lz) for an electron confined in quantum dots under a uniform magnetic field (UMF) and a static electric field (SEF). It finds that the eigenvalue of Lz just stands for the expectation value of a usual angular momentum lz in the eigen-state. It first obtains the matrix density for this system via directly calculating a transfer matrix element of operator \exp( -\beta H) in some representations with the technique of integral within an ordered products (IWOP) of operators, rather than via solving a Bloch equation. Because the quadratic homogeneity of potential energy is broken due to the existence of SEF, the virial theorem in statistical physics is not satisfactory for this system, which is confirmed through the calculation of thermal averages of physical quantities.

关键词: unitary transformation, IWOP technique, density matrix in ( x1, p2) representation, coherent rotational-entangled state

Abstract: Using unitary transformations, this paper obtains the eigenvalues and the common eigenvector of Hamiltonian and a new-defined generalized angular momentum (Lz) for an electron confined in quantum dots under a uniform magnetic field (UMF) and a static electric field (SEF). It finds that the eigenvalue of Lz  just stands for the expectation value of a usual angular momentum lz in the eigen-state. It first obtains the matrix density for this system via directly calculating a transfer matrix element of operator $\exp( -\beta H$) in some representations with the technique of integral within an ordered products (IWOP) of operators, rather than via solving a Bloch equation. Because the quadratic homogeneity of potential energy is broken due to the existence of SEF, the virial theorem in statistical physics is not satisfactory for this system, which is confirmed through the calculation of thermal averages of physical quantities.

Key words: unitary transformation, IWOP technique, density matrix in ($x_1, p_2$) representation, coherent rotational-entangled state

中图分类号:  (Entanglement and quantum nonlocality)

  • 03.65.Ud
03.65.Fd (Algebraic methods) 03.67.Lx (Quantum computation architectures and implementations)