阿诺·索末菲Arnold Sommerfeld

阿诺·索末菲的母亲是Cäcile Matthias，父亲是Franz Sommerfeld。Franz是一名医生，来自柯尼斯堡的一个显赫家族。弗拉基米尔·阿诺尔德出生时，Franz 48岁，而他的妻子Cäcile 29岁；他们已结婚六年。阿诺尔德在1917年左右写下了自传笔记（他在晚年对其进行了更新），我们在本传记中引用了一些内容。特别地，让我们引用阿诺尔德关于他父母的描述（例如参见[1]，或索末菲的Collected Works，其中包含完整的自传文章）：-

我的父亲，一位执业医生，……是一位狂热的自然物品收藏家（琥珀、贝壳、矿物、甲虫等），也是自然科学的伟大朋友。……对于我精力充沛、智力活跃的母亲，我欠下了无限的恩情。

索末菲于1875年进入柯尼斯堡的Altstädtisches 文理中学（Gymnasium）。同一所学校里两个稍年长的学生是赫尔曼·闵可夫斯基和威廉·维恩。在这所高中，他在所有科目上都表现出色，但他不清楚应该进一步追求哪些科目：-

我对文学和历史的兴趣几乎超过了对精确科学的兴趣；我在所有科目上，包括古典语言，都同样出色。

他于1886年参加了中学毕业考试（高中毕业考试（Abitur）），并决定在大学专攻数学。索末菲在通过毕业考试后立即进入柯尼斯堡大学，尽管数学是他的主修科目，但他也追随自己广泛的兴趣，旁听了自然科学、哲学和政治经济学的讲座。当时柯尼斯堡的数学系因在那里任教的人才济济而引人注目。大卫·希尔伯特、阿道夫·赫维兹和费迪南德·冯·林德曼都为索末菲授课，在听了大卫·希尔伯特关于理想数理论的课程后，他感到抽象纯粹数学才是适合他的学科。

此时，柯尼斯堡大学以其由恩斯特·弗朗茨·诺伊曼创立的理论物理学派而闻名。然而索末菲的兴趣在数学而非物理。他也深深参与学生生活，加入了学生社团Burschenschaft，这被证明是一个主要的干扰[1]：-

积极参与[Burschenschaft]，加上其强制性的饮酒聚会和击剑决斗，妨碍了他在大学最初几年的系统而专注的学习。

事实上，他从那时起额头上就留下了一道长长的击剑疤痕。德国学生习惯于从一所大学转到另一所大学，但索末菲没有这样做，而是留在柯尼斯堡。晚年他后悔没有转学，因为最终在Burschenschaft上浪费了时间，但当时他从杰出的数学老师那里获得的灵感使他留在柯尼斯堡。

费迪南德·冯·林德曼是索末菲的博士论文导师，1891年，他因论文Die willkürlichen Functionen in der mathematischen Physik Ⓣ（数学物理中的任意函数）获得柯尼斯堡的博士学位。这是一篇在1891年夏天：-

……我在几周内构思并写成的论文。

在这篇学位论文中，他研究了用偏微分方程的特征函数以及其他给定函数组表示任意函数的问题。

完成博士学位后，索末菲留在柯尼斯堡，为取得能在文理中学教授数学和物理的教学文凭而工作。1892年通过教学文凭考试后，他开始了为期一年的兵役。许多学者认为兵役是一种不得不忍受的必要之恶，但索末菲并不这样看。他在柯尼斯堡的预备团服兵役，并在接下来的八年里继续自愿参加为期八周的军事训练——他真的很享受这种生活，而且在许多方面看起来也很像那么回事[1]：-

尽管他身材矮壮，但到了中年，借助向上翘起的蜡染胡须，他设法给人一种骠骑兵上校的印象。

1893年9月完成军事训练后，索末菲前往哥廷根，他知道那里是“数学高等文化的所在地”，并首先在矿物学研究所做了一年助手。然而，他很快就被菲利克斯·克莱因的魅力所折服：-

在讲座和讨论中，菲利克斯·克莱因的伟大个性给我留下了压倒性的印象。

1894年9月，他成为菲利克斯·克莱因的助手。他在自传笔记中写道：-

他有意识、有系统地菲利克斯·克莱因试图用数学物理的问题吸引我，并让我接受他在前几年讲座课程中形成的对这些问题的构想。我一直把菲利克斯·克莱因视为我真正的老师，不仅在数学方面，而且在数学物理以及力学方面也是如此。

他很快就非常熟悉菲利克斯·克莱因的讲座，因为他的一项职责是管理数学阅览室，并在那里提供菲利克斯·克莱因讲座的副本供学生使用（那时没有复印机！）。如前所述，索末菲的研究方向立即受到菲利克斯·克莱因的影响，后者当时正深入参与将复变函数论和其他纯数学应用于从天文学到动力学的一系列物理课题。索末菲在菲利克斯·克莱因指导下的第一项工作是一篇关于衍射数学理论的令人印象深刻的作品。他关于这一课题的工作包含重要的偏微分方程理论。他在哥廷根期间进行的其他重要工作包括研究电磁波在导线中的传播，以及研究运动电子产生的场。

1895年3月，索末菲在哥廷根提交了他的教授资格论文（Habilitation）学位论文The mathematical theory of diffraction，并成为数学privatdozent。他讲授的课题范围广泛，既讲授probability，也讲授物理学的偏微分方程。1895-96年，菲利克斯·克莱因关于陀螺的讲座促使菲利克斯·克莱因和索末菲开始了一个联合项目，撰写一部关于陀螺仪理论的四卷本著作。这部著作最终于1909-1910年出版，前两卷讨论数学理论，后两卷则讨论在地球物理学、天文学和技术中的应用。在哥廷根期间，他结识了Johanna Höpfner，她是大学管理员Ernst Höpfner的女儿，但当索末菲还是一名收入不足以养家的编外讲师时，他们无法结婚。

从1897年10月起，索末菲在克劳斯塔尔任教，并在那里的矿业学院成为数学教授。从教学角度看，这不是一份令人兴奋的工作，但它提供了足够的薪水让他得以结婚，而且这里离哥廷根足够近，使他能够继续与菲利克斯·克莱因及那里的其他人合作。索末菲与Johanna Höpfner的婚姻育有三子一女。

应菲利克斯·克莱因的要求，他承担了Encyklopädie der mathematischen WissenschaftenⓉ（数学科学百科全书）第五卷的编辑工作，该卷涉及数学物理。这是一项巨大的工程，在他离开克劳斯塔尔后很久仍占用了他大量时间。Reich在[11]中写道：-

1901年，索末菲向该卷的所有撰稿人发送了一份8页的指示。在他的说明中，他提出了一种表示向量、向量微积分和电磁量的特殊方式，这对所有撰稿人都成为强制性的。1903年9月，菲利克斯·克莱因成立了一个所谓的“向量委员会”，由索末菲、路德维希·普朗特和Rudolf Mehmke组成。其目标是创建统一的向量符号和微积分。

在克劳斯塔尔任职三年后，他成为亚琛工业大学的力学教授。这一任命与菲利克斯·克莱因有很大关系。正是他坚信工程学应建立在坚实的数学基础之上，因此他竭力推动任命索末菲担任这一教授职位。然而，索末菲在亚琛的新同事与菲利克斯·克莱因没有同样的远见，必须被说服：-

尽管我在亚琛的同事和学生起初对这位“纯数学家”持怀疑态度，但我很快就满意地发现自己不仅在教学中，而且在工程实践中都被接受为有用的成员；因此我被邀请提供专家意见并参加工程学会。

1906年，他成为慕尼黑的理论物理学教授。那里为他建立了一个研究所，设有讨论班教室、助手室和实验工作实验室。尽管他是一位数学物理学家，但他希望能够指导旨在检验其理论的实验工作。事实上，他在慕尼黑指导了大量学生，许多是理论物理学方向的，但对另一些学生，他指导他们进行实验研究项目。他最著名的学生包括彼得·德拜（1908年获得博士学位）、彼得·埃瓦尔德（1912年获得博士学位）、沃尔夫冈·泡利（1921年获得博士学位）、维尔纳·海森堡（1923年获得博士学位）和汉斯·贝特（1928年获得博士学位）。他在慕尼黑总共指导了近30名博士生。

马克斯·玻恩在[7]中，对索末菲指导学生的方​​式提供了洞见：-

理论物理学是一门吸引具有哲学头脑的年轻人的学科，他们会在没有足够基础的情况下思考最高原理。正是这类初学者，他知道如何应对，引导他们一步步认识到自己实际知识的缺乏，并为他们提供进行富有成果的研究所需的技能。……尽管有职责和科学工作，他仍有难得的能够为学生腾出时间的能力。……在这种友好而非正式的教学方式中，受邀参加从慕尼黑乘火车两小时可达的“苏德尔费尔德”滑雪聚会起到了很大作用。在那里，他和他的机械师……共同拥有一间滑雪小屋。晚上，当简单的饭菜做好、碗碟洗净、天气和积雪得到适当讨论后，谈话总是转向数学物理学，而这就是那些善于接受的学生了解大师内心想法的机会。

他在慕尼黑最早研究的课题之一是原子光谱。他研究了X射线是波的假说，并通过将晶体用作三维衍射光栅证明了这一点。从1911年起，他的主要兴趣领域变成了quantum theory。索末菲的工作使他用椭圆轨道取代了尼尔斯·玻尔原子的圆形轨道；他还在1916年引入了磁量子数，四年后又引入了内量子数。正是试图解释内量子数的理论工作导致了电子自旋的发现。

在职业生涯的后期，索末菲使用统计力学来解释金属的电子性质。这取代了亨德里克·洛伦兹在1905年基于经典物理学的早期理论。索末菲的方法是将金属中的电子视为简并电子气。他能够解释早期经典理论无法解释的特征。他的重要论著Atomic structure and spectral lines共出了六版，反映了1916年至1946年间理论光谱学的发展。

莫里斯·克莱因写到索末菲生命的最后阶段：-

索末菲的职业生涯末期因德国的事件而蒙上悲伤。反犹主义在那个国家一直存在，在希特勒时期变得恶毒，索末菲不得不目睹著名同事的流亡，其中包括阿尔伯特·爱因斯坦。

让我们在这段引文中插入另一段引文，这段引文出自希特勒上台约一年后索末菲写给阿尔伯特·爱因斯坦的一封信：-

我可以向你保证，我们的统治者对“民族的”一词的滥用，已经彻底使我戒除了在我身上曾如此明显的民族情感习惯。我现在愿意看到德国作为一个强国消失，并入一个和平的欧洲。

在这段插话之后，让我们继续莫里斯·克莱因的引文：-

他所能做的只是利用他在美国逗留一年和环球旅行一年期间建立的友谊来帮助安置难民。以这种方式失去如此多最优秀的人才，加上第二次世界大战，摧毁了德国的科学实力，索末菲感到有义务继续教学直到1947年，远远超过了通常65岁的退休年龄。他的生命以另一场悲剧告终。他年老时有些耳背，[与孙辈外出散步时]没有听到警告，于1951年春天被一辆卡车撞倒。两个月后，他因伤去世。

莫里斯·克莱因还这样称赞索末菲：-

[他]处于电磁理论、相对论和量子理论研究的前沿，他是伟大的系统化者和教师，在本世纪头三十年激励了许多最具创造力的物理学家。

最后让我们简要说明索末菲获得的众多荣誉。他获得了亨德里克·洛伦兹金奖、马克斯·普朗克奖章和奥斯特奖章。他当选为伦敦皇家学会、华盛顿的国家科学院、Berlin、慕尼黑、维也纳、Göttingen、Budapest、乌普萨拉和马德里的科学院院士，以及罗马的Academia dei Lincei、莫斯科的科学院 of the U.S.S.R和印度科学院的院士。他获得了许多大学的荣誉学位，包括罗斯托克、亚琛、加尔各答和雅典。

Arnold Sommerfeld's mother was Cäcile Matthias and his father was Franz Sommerfeld. Franz was a medical doctor who came from a leading Königsberg family. Franz was 48 years old when Arnold was born while his wife Cäcile was 29; they had been married for six years. Arnold wrote autobiographical notes in around 1917 (which he updated towards the end of his life) and we give some quotations from these throughout this biography. In particular let us quote what Arnold said about his parents (see for example [1], or Sommerfeld's Collected Works which contains the full autobiographical article):-

My father, the practicing physician, ... was a passionate collector of natural objects (amber, shells, minerals, beetles, etc.) and a great friend of the natural sciences. ... To my energetic and intellectually vigorous mother I owe an infinite debt.

Sommerfeld entered the Altstädtisches Gymnasium in Königsberg in 1875. Two slightly older pupils at the same school were Minkowski and Wien. At this High School, he excelled in all his subjects and it was not clear to him what subjects he should pursue further:-

I was almost more interested in literature and history than in the exact sciences; I was equally good in all subjects including the classical languages.

He took his final school examinations (the Abitur) in 1886 and decided that he would concentrate on mathematics at University. Sommerfeld entered the University of Königsberg immediately after passing the Abitur but although mathematics was his main subject, he followed his all round interests by attending lectures in natural sciences, philosophy, and political economy as well. The Mathematics Department at Königsberg at this time was remarkable for the talented staff who were teaching there. Hilbert, Hurwitz and Lindemann all lectured to Sommerfeld and, after attending a course by Hilbert on the theory of ideal numbers, he felt that abstract pure mathematics was the right subject for him.

At this time the University of Königsberg was famous for its school of Theoretical Physics which had been founded by Franz Neumann. However Sommerfeld's interests were in mathematics rather than physics. He was also much involved in student life and he joined the Burschenschaft, a student society, which proved a major distraction [1]:-

Active participation in [the Burschenschaft], with its compulsory drinking bouts and fencing duels, prevented systematic and concentrated study in his first few years at university.

In fact he received a long fencing scar on his forehead from this time. It was the custom of German students to move from one university to another, yet Sommerfeld did not do so, remaining in Königsberg. Later in life he regretted not moving because of the time he ended up wasting with the Burschenschaft, but at the time the inspiration he received from his remarkable mathematics teachers kept him in Königsberg.

Lindemann was Sommerfeld's doctoral thesis advisor and, in 1891, he was awarded his doctorate from Königsberg for his thesis Die willkürlichen Functionen in der mathematischen Physik Ⓣ. It was a thesis which, during the summer of 1891:-

... I conceived and wrote out in a few weeks.

In this thesis he studied the representation of arbitrary functions by the eigenfunctions of partial differential equations and other given sets of functions.

After completing his doctorate, Sommerfeld remained at Königsberg to work for the teaching diploma which would let him teach mathematics and physics at a Gymnasium. After passing his teaching diploma examinations in 1892, he then began a year of military service. Many academics find military service a necessary evil to be suffered but not so Sommerfeld. He did his military service in the reserve regiment in Königsberg and for the following eight years continued to undertake voluntary eight week military exercises - he really enjoyed the life and in many ways looked the part [1]:-

Despite his squat build, by middle age, with the aid of a turned up waxed moustache, he managed to give the impression of a colonel of the hussars.

In 1893, after completing his military training in the September, Sommerfeld went to Göttingen, knowing it to be "the seat of mathematical high culture", and he first spent a year as an assistant in the Mineralogical Institute. However he quickly fell under Klein's spell:-

Overwhelming was the impression which I received, in lectures and discussions, from Felix Klein's grand personality.

In September 1894 he became Klein's assistant. He wrote in his autobiographical notes:-

Consciously and systematically Klein sought to enthral me with the problems of mathematical physics, and to win me over to his conception of these problems as he had developed it in lecture courses in previous years. I have always regarded Klein as my real teacher, not only in things mathematical, but also in mathematical physics and in connection with mechanics.

He soon came to know Klein's lectures very well since one of his duties was to manage the Mathematical Reading Room, and to make copies of Klein's lectures available there for the use of students (no photocopiers in those days!). As indicated, the direction of Sommerfeld's research was immediately influenced by Klein who at this time was heavily involved in applying the theory of functions of a complex variable, and other pure mathematics, to a range of physical topics from astronomy to dynamics. Sommerfeld's first work under Klein's supervision was an impressive piece of work on the mathematical theory of diffraction. His work on this topic contains important theory of partial differential equations. Other important work which he undertook while at Göttingen included the study of the propagation of electromagnetic waves in wires, and the study of the field produced by a moving electron.

In March 1895 Sommerfeld presented his habilitation thesis The mathematical theory of diffraction to Göttingen and became a privatdozent in mathematics. He lectured on a wide range of topics, giving lectures on probability and also on the partial differential equations of physics. The lectures Klein gave in 1895-96 on the spinning top led to Klein and Sommerfeld starting a joint project to write a four volume text on the theory of gyroscopes. This would eventually be published in 1909-1910, the first two volumes dealing with the mathematical theory, while the final two volumes deal with applications to geophysics, astronomy and technology. While at Göttingen he had met Johanna Höpfner, the daughter of Ernst Höpfner who was the curator of university, but they could not marry while Sommerfeld was a privatdozent with insufficient income to support a wife.

From October 1897 Sommerfeld taught at Clausthal where he became professor of mathematics at the mining academy. This was not an exciting job from the teaching point of view, but it provided sufficient salary for him to marry, and it was also close enough to Göttingen to allow him to continue collaborating with Klein and others there. The marriage between Sommerfeld and Johanna Höpfner produced three sons and one daughter.

At Klein's request he took on the editorship of Volume V of the Encyklopädie der mathematischen Wissenschaften Ⓣ which dealt with mathematical physics. This was a big undertaking which occupied a lot of his time long after he had left Clausthal. Reich writes in [11]:-

In 1901 Sommerfeld sent a directive of 8 pages to all contributors of this volume. In his note he proposed a special way of denoting vectors, vector calculus and the electromagnetic magnitudes, which became obligatory for all contributors. In September 1903 Klein founded a so-called "vector commission" consisting of Sommerfeld, Ludwig Prandtl, and Rudolf Mehmke. Its aim was to create a unified vector symbolism and calculus.

Three years after taking up the appointment in Clausthal, he became professor of mechanics at the Technische Hochschule in Aachen. This appointment had much to do with Klein. It was he who strongly believed that engineering should be based on a strong mathematical base, so he had pressed hard for Sommerfeld to be appointed to this professorship. However Sommerfeld's new colleagues at Aachen did not have the same vision as Klein and had to be won over:-

Although my Aachen colleagues and students at first regarded the 'pure mathematician' with suspicion, I soon had the satisfaction of being accepted as a useful member not merely in teaching but also in engineering practice; thus I was requested to render expert opinions and to participate in the Engineering Society.

In 1906 he became professor of theoretical physics at Munich. There an institute was set up for him, with rooms for seminars, rooms for assistants, and laboratories for experimental work. Despite being a mathematical physicist, he wanted to be able to direct experimental work aimed at checking his theories. In fact he supervised a large number of students at Munich, many in theoretical physics, but for others he directed them in programmes of experimental research. His most famous students include Peter Debye (doctorate in 1908), Peter Ewald (doctorate in 1912), Wolfgang Pauli (doctorate in 1921), Werner Heisenberg (doctorate in 1923), and Hans Bethe (doctorate in 1928). In total he supervised nearly 30 doctoral students at Munich.

Max Born, in [7], gives an insight into Sommerfeld's way of supervising his students:-

Theoretical physics is a subject which attracts youngsters with a philosophical mind who speculate about the highest principles without sufficient foundations. It was just this type of beginner that he knew how to handle, leading them step by step to a realisation of their lack of actual knowledge and providing them with the skill necessary for fertile research. ... He had the rare ability to have time to spare for his pupils, in spite of his duties and scientific work. ... In this friendly and informal way of teaching a great part was played by invitations to join a skiing party on the 'Sudelfeld' two hours by rail from Munich. There he and his mechanic ... were joint owners of a ski hut. In the evenings, when the simple meal was cooked, the dishes were washed, the weather and snow properly discussed, the talk invariably turned to mathematical physics, and this was the occasion for the receptive students to learn the master's inner thoughts.

One of the first topics he worked on at Munich was atomic spectra. He studied the hypothesis that X-rays were waves and proved this by using crystals as three dimensional diffraction gratings. From 1911 his main area of interest became quantum theory. Sommerfeld's work led him to replace the circular orbits of the Niels Bohr atom with elliptical orbits; he also introduced the magnetic quantum number in 1916 and, four years later, the inner quantum number. It was theoretical work attempting to explain the inner quantum number that led to the discovery of electron spin.

In the later part of his career, Sommerfeld used statistical mechanics to explain the electronic properties of metals. This replaced an earlier theory due to Lorentz in 1905 based on classical physics. Sommerfeld's approach was to regard electrons in a metal as a degenerate electron gas. He was able to explain features which were unexplained by the earlier classical theory. His important treatise Atomic structure and spectral lines which ran to six editions, reflects the development of theoretical spectroscopy between 1916 and 1946.

Morris Kline writes about the last part of Sommerfeld's life:-

Sommerfeld's life was saddened toward the end of his career by events in Germany. Anti-Semitism, always present in that country, became virulent in the Hitler period and Sommerfeld was obliged to witness the emigration of famous colleagues, including Einstein.

Let us break into this quote to provide another quote, this one being in a letter Sommerfeld wrote to Einstein about a year after Hitler came to power:-

I can assure you that the misuse of the word 'national' by our rulers has thoroughly broken me of the habit of national feelings that was so pronounced in my case. I would now be willing to see Germany disappear as a power and merge into a pacified Europe.

After this interjection, let us continue with Morris Kline's quote:-

All he could do was use the friendships he had built up during a one-year stay in the United States and a one-year round-the-world trip to help place the refugees. The loss of so many of its best men in this way together with World War II, destroyed the scientific strength of Germany, and Sommerfeld felt obliged to continue teaching until 1947, long after the usual retirement age of 65. His life was ended by another tragedy. Somewhat deaf in his old age he failed to hear a warning [when out walking with his grandchildren] and was struck by a truck in the spring of 1951. He died of the injuries two months later.

Kline also gives this tribute to Sommerfeld:-

[He] was at the forefront of the work in electromagnetic theory, relativity and quantum theory and he was the great systematizer and teacher who inspired many of the most creative physicists in the first thirty years of this century.

Finally let us give a brief indication of the many honours Sommerfeld received. He won the Lorentz Gold Medal, the Planck Medal, and the Oersted Medal. He was elected to the Royal Society of London, The National Academy of Sciences in Washington, the Academies of Berlin, Munich, Vienna, Göttingen, Budapest, Uppsala, and Madrid, as well as the Academia dei Lincei of Rome, The Academy of the U.S.S.R in Moscow, and the Indian Academy of Sciences. He received honorary degrees from many universities including Rostock, Aachen, Calcutta, and Athens.