- #1

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It might be too much, but since this is my new hobby: are there any cool books that combine quantum mechanics and biology?

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- Thread starter micromass
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In summary, the essential mathematical knowledge needed to understand quantum mechanics includes finite dimensional linear algebra, functional analysis, general topology, measure theory, Banach spaces and algebras, Hilbert spaces, spectral theory, Lie groups and harmonic analysis. While this may seem like a lot, it is important to note that some of these concepts may already be familiar to mathematicians. Additionally, for a more rigorous understanding of the mathematical foundations, further knowledge in areas such as quantum field theory, gauge theory, and particle physics may be necessary. However, for a basic understanding of quantum mechanics and its applications, a solid understanding of finite dimensional linear algebra and some additional knowledge in areas such as calculus and

- #1

- 22,183

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It might be too much, but since this is my new hobby: are there any cool books that combine quantum mechanics and biology?

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- #2

atyy

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In my understanding, if one only has finite dimensional linear algebra, one has the complete essence of quantum mechanics.

- #3

aleazk

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If you are really interested in the mathematical foundations, then you need some good amount of stuff: basics of general topology, basics on measure theory (including its application to probabilty theory; the quantum logic formulation of QM is a generalization of this), Banach spaces and algebras (including spaces of operators), Hilbert spaces (the basic stuff like Riesz's theorem and bases, but also the general theory behind bounded operators and densely-defined unbounded operators); spectral theory (the spectral theorem for unbounded self-adjoint operators and the theory behind it); Lie groups and harmonic analysis (including the imprimitivity theorem; most of the foundational issues related to the study of basic quantum systems, like localizable and covariant systems, can be reduced to the study and application of this theorem, i.e., the classification of representations of different Lie groups).

Most mathematicians are already familiar with all this material, so I don't think you will find anything new (I say this because, if I remember well, you are a mathematician)

- #4

heatengine516

Gold Member

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Is this a trick question? Because isn't the correct answer than no one "really" understands quantum mechanics?

If not, then linear algebra, probability theory, partial differential equations, operator theory, spectral theory, combinatorics, group theory, and probably more.

- #5

Loststudent22

- 100

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Those suggestions might be interesting for you

- #6

Almeisan

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If you know the popular physics and you see the math of discrete problems like polarization of spin up/down, you kind of can fudge how it 'feels' for continuous stuff requiring complex functions and partial diff eqs.

I mean, how much mathematical knowledge of QM do you really need as a lay person?

Do you really want to solve Schrodinger's for real-life problems?

- #7

atyy

Science Advisor

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micromass said:It might be too much, but since this is my new hobby: are there any cool books that combine quantum mechanics and biology?

How about fake quantum biology?

https://www.quantamagazine.org/20141204-a-common-logic-to-seeing-cats-and-cosmos/

http://arxiv.org/abs/0905.1317

http://arxiv.org/abs/1210.2812 (read their concluding section for the link to renormalization)

http://arxiv.org/abs/1410.3831

- #8

Shobah

- 3

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micromass said:What math do I need toreallyunderstand quantum mechanics?

As esuna noted it depends on what you mean by 'really' understanding quantum mechanics. A full treatment would probably require some incursions into quantum field theory, gauge theory, particle physics... as well, for which some understand of Lie algebras is desirable, and all of the things listed above. Quantum

If your question had been

micromass said:What math do Ireallyneed to understand quantum mechanics?

I would have said that solid analysis skills (integration, Fourier transforms, solving DEs and PDEs, etc) will make you able to start learning QM effectively enough.

Example: I don't have any good textbooks in mind, but Prof. Tong's notes are available online and cover everything from introduction of the Dirac formalism through perturbation theory, all the way to quantum information, and recommend some books on the way.

http://www.damtp.cam.ac.uk/user/rrh/notes/pqm14_281014.pdf

- #9

atyy

Science Advisor

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However, mathematically, apart from the well-known complaints against Dirac's version of QM, there seem to be some interesting problems like Tsirelson's problem, which is to about how much the Bell inequalities can be violated.

http://arxiv.org/abs/1008.1142

http://arxiv.org/abs/0812.4305

The math needed for quantum mechanics is typically at the advanced undergraduate or graduate level. This includes knowledge of linear algebra, calculus, and differential equations. It is also helpful to have a strong understanding of complex numbers and vector spaces.

Linear algebra is a crucial tool in understanding quantum mechanics. It is used to describe the state of a quantum system and the evolution of that state over time. Many of the fundamental principles of quantum mechanics, such as superposition and entanglement, are best understood through the language of linear algebra.

Yes, a strong understanding of calculus is necessary for studying quantum mechanics. Many of the equations and concepts in quantum mechanics involve derivatives, integrals, and differential equations. It is important to have a solid foundation in calculus to fully grasp these concepts.

Yes, knowledge of differential equations is necessary for understanding quantum mechanics. The equations that govern quantum systems are often described using differential equations, and solving these equations is essential for predicting the behavior of quantum systems.

While it is possible to study some basic concepts of quantum mechanics without a strong math background, a deep understanding of the subject requires a solid grasp of advanced mathematics. Without this foundation, it may be difficult to fully comprehend the complex principles and equations involved in quantum mechanics.

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