Symmetry in quantum mechanics 


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SYMMETRYOPERATIONS GENERATE NEW SOLUTIONS from old when the underlying dynamics issymmetrical. a: Applying a rotation operator to the Earth’s orbit around theSun produce a different possible orbit. b: The same can be done in quantummechanics, here to a wavefunction of a hydrogen atom. The superposition of allpossible rotations of the wavefunction is itself a spherically symmetricwavefunction, in the single representation of the rotation group. From thisWigner saw the profound importance of irreducible representations in quantummechanics. FIGURE 2.

    Other superposition of rotated states will yield other irreduciblerepresentations are special: They cannot be further subdivided-any subset ofstates gets mixed by the symmetry group with all the other states of therepresentation. Furthermore any state can be written as a sum of statestransforming according to irreducible representations of the symmetry group.Wigner realized that these special states can be used to classify all thestates of a system possessing symmetries, and play a fundamental role in theanalysis of such systems. Consequently he understood the important role thatthe theory of representations of continuous and discrete groups would play inquantum mechanics and he proceeded to develop the requisite tools in the casesof physical interest.


量子力学中的对称 


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从过去基本动力学的对称性那里，对称方法得到了新的解。a：对地球围绕太阳的轨道应用一个旋转算符将得到一个不同的可能轨道。b：对量子力学同样也可以这么做，这里以氢原子的一个波函数为例。在这个旋转群的线性表象中，叠加这个波函数所有可能的旋转变换后将得到它自身一个球对称的波函数。由此维格纳看到了量子力学中最简表象的深远意义。图2

    其他旋转后的量子态叠加后会得到其他特殊的最简表象：对于由该表象其余所有量子态的变换群组合得来的量子态，它们不能再被分解成量子态的子集。此外，任何量子态可以写成根据对称群的最简表象而得的所有量子态变换之和的形式。维格纳意识到这些特殊的量子态可以用来区分一个对称性系统的所有量子态，并且在这样的系统分析中扮演着重要的角色。因此他认识到了连续和离散群的表象理论在量子力学中所扮演的角色的重要性，并继续完善在解决一些物理问题时所需的必要工具。