Mathematics for Quantum Mechanics/Thermodynamics/Statistical Mechanics

In summary, the student is a rising sophomore with a major in microbiology and a deep fascination with atomic/quantum physics, relativity, thermodynamics, and statistical mechanics. They are seeking advice on what mathematical subjects they should know for their upcoming introductory courses in these areas, specifically whether they need to know proof-based, theoretical mathematics or if computational mathematics will suffice. The expert advises that for introductory courses, a working knowledge of calculus, linear algebra, and differential equations is necessary. They also suggest reading up on vector spaces, linear operators, inner-product spaces, and dual spaces for quantum mechanics and some mathematical probability theory for statistical mechanics. However, total rigor is not necessary unless the student plans on pursuing mathematical physics or applied math.
  • #1
bacte2013
398
47
Hello!

I am a rising sophomore with a major in microbiology. Although my main interest is in the microbiology and biochemistry, I am also deeply fascinated by the atomic/quantum physics, relativity, thermodynamics, and statistical mechanics. I will be taking those courses later on. I wrote this post to ask you what should I know in terms of mathematical subjects for those math-intensive courses; they are introductory courses. Do I need to know the proof-based, theoretical mathematics or am I fine with the computational mathematics? Do I need to know how to do the mathematical proofs for those courses?




MSK
 
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  • #2
For an introductory course, you certainly don't need the rigorous proof based mathematics. You should have a working knowledge of calculus (single and multi-variable), linear algebra, and differential equations. That should be about it.
 
  • #3
Thank you very much for the response! So I do not need to know the calculus levels of Spivak and Apostol for those introductory courses?
 
  • #4
Introductory courses require working knowledge. More advanced courses may require some more rigorous knowledge. But at no point in physics do you probably really ever need the total rigor of a mathematician, unless you plan on doing mathematical physics, or applied math. Not to say that it wouldn't help to be rigorous, but most physicists are not so concerned with rigor.

I am not familiar with those books, so I can't comment on their levels.
 
  • #5
bacte2013 said:
Thank you very much for the response! So I do not need to know the calculus levels of Spivak and Apostol for those introductory courses?

No, definitely not.

For QM, I would advise you to read up on vector spaces, linear operators, inner-product spaces and maybe even dual spaces. This will help a great deal.

A bit of mathematical probability theory will definitely help for Stat Mech since I've seen that Stat Mech books truly botch the job. But you definitely don't need to go in the nitty gritty details.
 

1. What is the relationship between mathematics and quantum mechanics?

Mathematics is the language of physics, and it plays a crucial role in understanding and describing the concepts and phenomena in quantum mechanics. The mathematical framework of quantum mechanics, known as Hilbert space, allows us to represent physical states, observables, and their corresponding operators in a precise and elegant manner.

2. How is mathematics used in thermodynamics?

In thermodynamics, mathematics is used to describe the behavior of macroscopic systems and their interactions with each other. It is used to define thermodynamic variables, such as temperature, pressure, and entropy, and to formulate the laws and principles that govern these systems. Additionally, mathematical techniques such as differential calculus and partial derivatives are used to solve thermodynamic equations and analyze the behavior of systems.

3. What role does mathematics play in statistical mechanics?

Mathematics is essential in statistical mechanics as it provides the tools and methods to analyze and understand the behavior of large systems consisting of a large number of particles. The principles of probability and statistics are used to describe the distribution of particles in a system, and mathematical techniques such as the Boltzmann distribution and partition function are used to calculate thermodynamic properties of the system.

4. How does mathematics help us understand quantum statistical mechanics?

Mathematics is crucial in understanding quantum statistical mechanics as it allows us to describe and analyze the behavior of quantum systems with many particles. The mathematical framework of quantum mechanics, along with statistical methods, is used to calculate the distribution of particles in a system and their corresponding thermodynamic properties. Mathematical techniques such as group theory and linear algebra are also used to solve and analyze quantum statistical mechanics problems.

5. What are the most important mathematical concepts for studying quantum mechanics, thermodynamics, and statistical mechanics?

The most important mathematical concepts for studying these fields include linear algebra, calculus, differential equations, probability and statistics, and group theory. Additionally, understanding concepts such as eigenvalues and eigenvectors, operators, and complex numbers is crucial in quantum mechanics, while knowledge of partial derivatives and integrals is essential in thermodynamics and statistical mechanics.

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