Learning quantum mechanics between intuition and formalism

At ETH Zurich, BSc physics students make their first encounter with quantum mechanics (QM) in their second year of study: the Physics III course covers basic concepts in optics, statistical mechanics and QM. In their third year, Quantum Mechanics I provides students with deeper knowledge of the theory and of its formalism. Students interested in theoretical physics can then take the Quantum Mechanics II BSc course to learn more about quantum systems of many bosons or fermions. At MSc level theoretical physics students encounter quantum field theory, which is fundamental to theoretical particle physics, and can choose to find out more about quantum information, which opens the way to topics as disparate as quantum computing and quantum gravity.

First contacts

Severin Spillmann, MSc theoretical physics student, recalls the first impact with the formalism of QM. "At the start I had trouble with the concept of quantum state. I suppose I found wavefunctions a bit more intuitive," he says. He compares the transition from wave mechanics to the matrix formulation of QM (introduced in Quantum Mechanics I, in his time) to how classical Newtonian mechanics can be revisited with the introduction of the Lagrangian formalism.

The linear nature of the theory surprised Han-Miru Kim, MSc mathematics student who took most QM courses offered by the physics department. As he and his cohort delved into the formalism of QM, Kim – who generally found the lecture notes shared by all lecturers helpful – feels that more explicit links to and reminders of the linear algebra results used in quantum mechanical derivations would have afforded some extra clarity.

Karin Sim, third-year PhD student in theoretical physics, earned her BSc from a British university before coming to ETH Zurich for her Master's degree in Physics. She chose the theoretical physics curriculum and looked forward Quantum Field Theory I, which covered a subject that had so far remained a black box to her. Sim's impression is that ETH Zurich teaches QM with a highly rigorous approach placing a strong focus on the mathematical formalism of the theory. During her BSc she was introduced to quantum physics very gradually, in a way that didn't rely on mathematical formalism until late and instead aimed at building up some form of physical intuition for making sense of quantum mechanical phenomena. "There were frequent connections with classical physics," Sim recalls.

The role of intuition

Scientists spend years honing their intuition for how things work in their chosen discipline – seeing intuition not as a substitute for experiments or calculations, but rather as an additional input line that may sometimes guide them in their research activities. As QM is commonly presented as a subject that defies intuition, is it sensible to encourage students to rely on their physical intuition as they get to grips with this subject? If the answer is yes, what is the best way to develop some type of intuition in this context? Sim finds that the parallels between classical and quantum mechanics shaped her physical intuition in a helpful manner. When she worked as a teaching assistant on the Quantum Physics for Non-Physicists course, she saw students having few problems with the mathematical derivations – thanks to their robust knowledge of linear algebra – but noticed that they struggled a bit more to visualise some key concepts. "I think students need time to make sense of the probabilistic nature of measurement and the idea of superposition," she says.

Kim felt that it was harder for him to build his intuition in QM compared to his fellow physics students, who seemed somewhat quicker at making connections. Kim preferred to wait for the mathematical derivations – even when they featured "the occasional physicists' maths," he jokes.

For Spillmann, looking for parallels with classical physics doesn't feel all that helpful when learning QM. "I think it's better to have a clear cut – a point where you're exposed to a new way of doing things and cannot be tempted to try reconciling them with your classical intuition." In fact, Spillmann is convinced that QM courses should focus on the formalism of QM, which is an aspect that can be extremely daunting to tackle without a lecturer's experience to light up the way.

Beyond textbook quantum mechanics

QM stands out as a subject that continuously reveals itself in its depth and complexity to anyone studying it as well as actively working on it. "Learning quantum mechanics isn't like reading a story with a plot twist. I don't even think you need a 'wow' moment, you need to understand how things work and accept that quantum mechanics is a plot that crystallizes slowly as you progressively understand more of the subject," says Spillmann. He's used to allocating a sizable portion of his time to self-study on all his courses, sometimes starting to read on a subject before the lectures begin. With QM he felt this may not pay off as much, so he waited for the lecturer to guide him and his cohort through the learning process.

As a doctoral student, Sim now enjoys a fresh view on quantum mechanics. Her research into open quantum systems has a deep connection with non-Hermitian QM and quantum measurement: Sim likes how this research has led her to question some of the very foundations of the theory. "You learn that Hamiltonians must be Hermitian and all of a sudden you're invited to ask yourself, what if that isn't the case? What happens then?" This and many more questions are for the physics researchers of tomorrow to explore.