Curriculum & Courses

Mathematics is a crucial part of a physics degree. It is not merely a constantly needed computational tool; rather, mathematical concepts are fundamental to how physics understands the world and makes relationships discussable. Accordingly, the histories of mathematics and physics are also closely intertwined.

Mathematics appears several times in the Bachelor of Science in Physics program:

Mathematics for Physics Students

In this two-part course, basic mathematical concepts and relationships that are crucial for physics are taught by (theoretical) physics instructors using examples closely tied to physics. Clarity of concepts is achieved primarily through the translation of fundamental mathematical relationships between different contexts—partly physical, partly various areas of mathematics. At the same time, this translation work allows students to learn and practice the computational skills used daily in physics.

Mathematics for physics students does not follow the path of traditional curricula, but instead begins with relatively new mathematical concepts of high generality and, from the very start, emphasizes the structural similarities between vectors and functions, or linear mappings and operators, with a focus on physical practice in quantum mechanics.

The material is divided between the two courses in such a way that nearly all the essential tools for the first semesters of a physics degree are covered in the very first lecture. These include, in particular, the fundamental concepts of differentiation and integration, vector calculus, the scalar product, the cross product, and differential equations. Accordingly, it is strongly recommended that students prioritize this course in their first semester; to facilitate this, Mathematics for Physics Students I is offered every semester, with a focus on modern mathematical concepts in the winter semester and on computational methods in the summer semester.

Mathematics for Physics Students II primarily covers linear algebra, vector calculus, coordinate systems, and transformations. Topics in function theory are also covered; this serves as the foundation for all courses in Theoretical Physics except Classical Mechanics. Therefore, this course can also be taken later; however, this will almost inevitably mean that (almost all of) Theoretical Physics cannot be studied until later as well.

Mathematics from the Mathematics Program

The mathematics taught in the mathematics program is significantly more rigorous than the mathematics taught to physics students. Even if examples are occasionally provided, the concepts are primarily explored by requiring students to prove mathematical theorems in accordance with the formal standards of mathematics. The formal, precise writing of a proof typically follows a different sequence than the development of the proof idea. By puzzling over the proof, turning it this way and that, and reorganizing the initially confused thoughts, students develop a unique way of thinking. This requires a high degree of willingness to engage with the logic and peculiarities of mathematics alongside one’s own field of study, even if the connection to physics often only becomes apparent many semesters later.

This way of thinking is essential for physics on the one hand, but on the other hand, it is not a strict prerequisite for following physics courses. Rather, it opens up a new perspective on the mathematical relationships that are worked with daily in physics. Accordingly, the mathematics courses in the program can be scheduled relatively freely, allowing students to focus initially on the parts of the program that suit them best during the often particularly challenging early stages of their studies.

Within the mathematics curriculum of the math program, Analysis I is a required course and should be taken first. There are two reasons for this: (1) It is also the first course in the math program and is therefore designed to introduce students to the mathematical way of thinking for the first time. (2) It covers sequences, series, continuity, differentiation, and integration. The idea behind all these concepts is to make complex relationships accessible by breaking them down into an infinite number of infinitely small, simple steps and to deal systematically with infinity. The development of calculus historically emerged from the interplay between mathematics and physics and was a decisive step during the Enlightenment toward the development of modern, quantitative natural sciences as we know them today.

After Analysis, students must take another course from the mathematics program. Here, they may also choose from advanced courses in the mathematics program: the necessary methodological foundations have already been laid by Analysis I, and the content-related foundations by Mathematics for Physics Students. The following courses may be selected:

im Sommersemester:

• Analysis II (Kann auch ohne die Mathematik für Studierende der Physik, allein mit Kenntnissen aus der Analysis I belegt werden)
• Funktionentheorie
• Numerische Mathematik

im Wintersemester:

• Analysis III
• Einführung in die Stochastik
• Elementare Differentialgeometrie

Weitere Veranstaltungen aus dem Mathematik-Studiengang können im Wahlbereich belegt werden. Diese Veranstaltungen dürfen sich inhaltlich nicht mit den anderweitig belegten Mathematik-Veranstaltungen überschneiden und sollten typischerweise erst nach der Mathematik für Studierende der Physik und der Analysis I belegt werden.

In the summer semester:

• Analysis II (Can also be taken without the Mathematics for Physics Students course, provided students have knowledge from Analysis I)
• Function Theory
• Numerical Mathematics

In the winter semester:

• Analysis III
• Introduction to Stochastics
• Elementary Differential Geometry

Additional courses from the mathematics program may be taken as electives. The content of these courses must not overlap with other mathematics courses taken elsewhere and should typically be taken only after Mathematics for Physics Students and Analysis I.

Since around the beginning of the 20th century, two distinct traditions have developed in physics, particularly in Germany: experimental physics and theoretical physics.

Experimental physics expands and systematizes physical knowledge based on experiments and uses experiments to test the validity of existing theories. Accordingly, each lecture and exercise session is followed by a laboratory practicum in which students learn to conduct experiments, handle data, and write scientific reports. Some of the experiments used are modern, while others are historical and were of great significance to the history of physics. Experimental Physics I and Experimental Physics II cover the various areas of mechanics & thermodynamics as well as optics & electricity in classical physics; they form the foundation for all advanced courses in experimental physics. (Further information on Lab A; on labs in atomic, solid-state, nuclear, and particle physics)

Theoretical physics uses mathematics to reorganize existing phenomenological knowledge. There are two main focuses here: On the one hand, it involves reconciling different theories; for example, at the end of the 19th century, the study of electricity, magnetism, and optics were separate but already highly developed branches of physics until Maxwell unified them into today’s coherent theory of electromagnetism. On the other hand, it involves applying existing physical knowledge to new questions. For example, Newtonian mechanics is ideally suited for identifying cause-and-effect relationships and, for instance, predicting whether a particular bridge will hold or collapse; the question in engineering, however, is how a bridge can be built using as little material as possible so that it does not collapse. It is not necessary, but recommended, to take the theoretical physics courses in the suggested order.

In the physics program at the University of Cologne, experimental physics and theoretical physics are taught on an equal footing and as distinct disciplines. The subject areas covered overlap to a large extent; for example, thermodynamics and statistical physics are addressed in both Experimental Physics I and Statistical Physics, albeit from entirely different perspectives.

Over the past few decades, computational physics has increasingly evolved from a practical tool into an independent approach to physics that combines both experimental and theoretical physics. Even newly developed fields in recent years, such as biological physics and astrophysics—which can be studied in the elective-required course in physics (see below)—do not follow the traditional division between experiment and theory.

In the required elective course in physics, students can study areas of physics taught and researched in Cologne that are not part of the mandatory curriculum. These currently include biological physics and astrophysics. The format of the courses is flexible and follows the established practices and current developments in each respective field.

A field not taken in the elective-compulsory course in Physics (see below) may still be taken in the elective section (see below).

The physics program includes two slots for courses outside of physics that students can choose freely. For students planning to spend a semester abroad, it makes sense to use these slots so that courses taken abroad can be easily transferred even if they do not otherwise fit well into their own degree program.

Elective Area

The goal of the elective area is to prepare students for interdisciplinary scientific work. Accordingly, it is recommended to take courses in this area from related natural sciences or mathematics. (Courses in fields not studied in the required elective course in physics may also be taken here.)

Courses from the elective area can be freely scheduled throughout the course of study and spread over several semesters. In particular: If the physics program is started with a focus on physics and the natural sciences, the elective area can be taken as early as the first semester and can also serve as a way to help students orient themselves if they are not yet entirely sure whether they really want to study physics or perhaps another natural science.

There is one exception: If mathematics courses are taken in the elective area, these must be advanced courses that go beyond the mathematics required for the physics program. Accordingly, it makes sense to take them only after completing the required physics mathematics courses.

Further information and recommendations regarding the elective track can be found on the corresponding subpage.

Integral Studies

Integral Studies is designed to encourage students to think outside the box. Students have almost complete freedom to choose what they want to do, but the idea is to take a course that is as far removed from physics as possible. Students planning a semester abroad who do not yet have (sufficient) proficiency in the language of their destination country should acquire knowledge of the respective foreign language in advance through the Studium Integrale.

Unlike the elective area, courses in the integral studies are ungraded.

Further information and recommendations regarding the integral studies can be found on the corresponding subpage.

The bachelor’s thesis is typically related to the research of a research group. Accordingly, the thesis is developed in close collaboration with members of the respective research group. It concludes with a colloquium, which is typically attended by the research group, fellow students, and other interested parties from the physics department.

Students must independently research the work of various faculty members, approach them, and agree on a topic. Experience shows that it is helpful to start early by actively approaching or writing to several faculty members or advanced students who have already written a bachelor’s thesis, asking them to explain what research is being conducted in their respective groups, etc. It makes sense to approach several different faculty members, allow time for consideration, and during these discussions also address the timeframe, talk about commitments outside of studies, potential second reviewers, etc.

The bachelor’s thesis can be registered as soon as a total of 148 credit points have been earned during the course of study.

As a rule, the two parts of Practicum A cannot be taken in the same semester because they both take place on Friday afternoons.

In contrast, the practicums in atomic, solid-state, nuclear, and particle physics can be taken at the same time and, in some cases, during the break between semesters.

• “Mathematics for Physics Students I” should definitely be taken in the first semester, as it is required for all other courses. “Experimental Physics I & II” can be taken concurrently, but all other courses should be taken only afterward. For this reason, “Mathematics for Physics Students I” is offered every semester. The courses in Theoretical Physics, with the exception of Classical Mechanics, also build upon Mathematics for Physics Students II and should only be taken after that.
• Experimental Physics I & II do not have to be taken “in sequence”; they can also be taken concurrently. Experimental Physics I is offered every semester, and Experimental Physics II every summer.
• As in the sample study plans, the lab courses build upon the corresponding Experimental Physics lectures. At the same time, the lab courses in Atomic / Solid-State / Nuclear & Particle Physics build upon the methods of Lab Course A and should be taken only after that. However, within both blocks, the order of the lab courses can be changed. The laboratory courses in atomic, solid-state, nuclear, and particle physics can also be taken simultaneously and, in some cases, during the lecture-free period.

Students can choose whether to take the entire Classical Field Theory course (9 credits) or to attend only the first two-thirds of the course (6 credits). Accordingly, the elective section is worth either 9 or 12 credits.

Submitting your thesis: Please send an email to the Registrar’s Office (physik-pa [at] uni-koeln.de) and provide the provisional title of your thesis, as well as the name of the professor supervising your work. We will then prepare the registration form and send it to you for your signature.

Submission of the thesis: Send the thesis to the reviewers via email as a PDF (PDF/A only) and put the Examination Office in CC (physik-pa [at] uni-koeln.de). Send a separate email to the Examination Office containing: the title page and the declaration of originality (all as PDFs and as separate files). We do not require a printed copy of the thesis! If desired, you may provide the printed versions to the examiners yourself.