arXiv quant-ph/06120322006-12quant-phastro-phhep-thmath-phmath.MPphysics.atom-ph

谐振子的量子力学新论

A New Look at the Quantum Mechanics of the Harmonic Oscillator

H. A. Kastrup

备注：: 100 pages (Latex); v2: abstract simplified, remarks and references added
发表信息：: AnnalenPhys.16:439-528,2007
DOI：: 10.1002/andp.200610245

摘要：

经典的谐振子（HO）是在R mod 2 pi和I > 0中使用角度和作用变量phi的一般例子。但当（q，p）=（0，0）时，（q，p）到（q，p）的辛变换是奇异的.全局上{（q，p）}具有平面R^2的结构，但{（phi，I）}具有穿孔平面R^2 -（0，0）的结构。这意味着两个相空间的量子力学性质的不同：平面R^2的量子化群由（中心扩展的）平移由{q，p，1}生成，但{（phi，I）}的相应群是SO（1，2）= Sp（2，R）/Z_2，（Sp（2，R）：平面的辛群），具有李代数基{h_0 = I，h_1 = I cos phi，h_2 = - I sin phi}。在（phi，I）模型的QM中，三个h_j对应于SO（1，2）或其无穷多个覆盖群之一的不可约酉表示（正离散级数）的自伴生成元K_j，j= 0，1，2，其Bargmann指数k > 0决定了（phi，I）-Hamiltonian H（K）的基态能量E（k，n=0）= hbar Ω k。对于一个m重覆盖，最小的可能值是k=1/m，它可以任意小！算符Q和P，现在表示为K_j的函数，保持其通常的性质，但是当传递到更简单的Q，P模型时，HO的K_j量子模型的更丰富的结构被"擦除“！HO的（φ，I）变体意味着许多实验测试：同位素双原子分子的Mulliken型实验，原子，离子和BE凝聚体的谐波陷阱实验，带电粒子在磁场中的（Landau）能级实验，光在真空中的传播，通过电场或磁场。最后给出了场的量子真空能及其与宇宙学常数的关系的一个新的理论估计。

英文摘要：

Classically the Harmonic Oscillator (HO) is the generic example for the use of angle and action variables phi in R mod 2 pi and I > 0. But the symplectic transformation (ϕ,I) to (q,p) is singular for (q,p) = (0,0). Globally {(q,p)} has the structure of the plane R^2, but {(phi,I)} that of the punctured plane R^2 -(0,0). This implies qualitative differences for the QM of the two phase spaces: The quantizing group for the plane R^2 consists of the (centrally extended) translations generated by {q,p,1}, but the corresponding group for {(phi,I)} is SO(1,2) = Sp(2,R)/Z_2, (Sp(2,R): symplectic group of the plane), with Lie algebra basis {h_0 = I, h_1 = I cos phi, h_2 = - I sin phi}. In the QM for the (phi,I)-model the three h_j correspond to self-adjoint generators K_j, j=0,1,2, of irreducible unitary representations (positive discrete series) for SO(1,2) or one of its infinitely many covering groups, the Bargmann index k > 0 of which determines the ground state energy E (k, n=0) = hbar omega k of the (phi,I)-Hamiltonian H(K). For an m-fold covering the lowest possible value is k=1/m, which can be made arbitrarily small! The operators Q and P, now expressed as functions of the K_j, keep their usual properties, but the richer structure of the K_j quantum model of the HO is ``erased'' when passing to the simpler Q,P model! The (phi,I)-variant of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with the (Landau) levels of charged particles in magnetic fields, with the propagation of light in vacuum, passing through electric or magnetic fields. Finally it leads to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

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摘要：

英文摘要：

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