Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential
Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential
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DOI：
10.1088/0305-4470/39/32/s18
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发表时间：
2005-08
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通讯作者：
A. Mostafazadeh
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作者：
A. Mostafazadeh
本文给出了一维准厄米哈密顿算子H = 12 p2 + v（x）的最一般度量算子（和正定内积）的系统微扰构造.我们证明了这个问题等价于求解（1 + 1）维迭代解耦双曲型偏微分方程的无穷系统。对于v（x）是纯虚数的情况，后者具有非齐次波动方程的形式，该方程有精确解。我们应用我们的一般方法得到了虚三次势的最一般的度量算子v（x）= i∈ x3。这揭示了一个无限类以前未知的CPT-以及非CPT-内积。我们计算了相应的幺正量子系统的物理观测量，并确定了其基础的经典系统。我们对虚立方势的结果表明，与量子系统不同，相应的经典系统对度规算符的选择不敏感。作为我们的方法的另一个应用，我们给出了一个完整的表征的伪厄米正则量子化的一个自由粒子在R中移动，这是符合通常选择的量子哈密顿量。最后，我们讨论了高维和具有固定长度尺度的系统的微妙之处。
We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, H = 1 2 p 2 + v(x), in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v(x} is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v(x) = i∈x 3 . This reveals an infinite class of previously unknown CPT- as well as non-CPT-inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in R that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.