Mathematics for Quantum Mechanics/Thermodynamics/Statistical Mechanics

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Discussion Overview

The discussion revolves around the mathematical prerequisites for introductory courses in quantum mechanics, thermodynamics, and statistical mechanics. Participants explore the necessary level of mathematical understanding required for these subjects, considering both computational and proof-based approaches.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant, a rising sophomore, inquires about the mathematical knowledge needed for introductory courses in quantum mechanics and thermodynamics, questioning the necessity of proof-based versus computational mathematics.
  • Another participant suggests that for introductory courses, a working knowledge of calculus (single and multi-variable), linear algebra, and differential equations is sufficient, without the need for rigorous proof-based mathematics.
  • The original poster seeks clarification on whether advanced calculus levels, such as those presented in Spivak and Apostol, are necessary for these introductory courses.
  • A participant responds that while advanced courses may require more rigorous knowledge, most physicists do not prioritize mathematical rigor unless they are engaged in mathematical physics or applied math.
  • Further advice is given to familiarize oneself with vector spaces, linear operators, and inner-product spaces for quantum mechanics, as well as a basic understanding of probability theory for statistical mechanics, though detailed knowledge is not deemed essential.

Areas of Agreement / Disagreement

Participants generally agree that rigorous proof-based mathematics is not necessary for introductory courses, but there is some variation in opinions regarding the extent of mathematical knowledge required, particularly concerning specific texts and advanced topics.

Contextual Notes

There is a lack of consensus on the specific mathematical texts that may be beneficial, and participants express differing views on the necessity of rigorous mathematical training depending on future academic pursuits.

Who May Find This Useful

Students preparing for introductory courses in quantum mechanics, thermodynamics, and statistical mechanics, particularly those with a background in biology or related fields looking to understand the mathematical foundations needed for these subjects.

bacte2013
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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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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.
 
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?
 
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.
 
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.
 

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