행렬 역학

개요

• correspondence principle

• http://www.eolss.net/Sample-Chapters/C05/E6-06B-09-00.pdf

1

\(*_{mn}\) 은 transition \(E_{m}\to E_{n}\) 과 관계된 양들

\(Q=\left(q_{mn}e^{2\pi it\nu_{mn}}\right)\)

\(P=\left(p_{mn}e^{2\pi it\nu_{mn}}\right)\)

• 여기서 \(q_{mn},p_{mn}\) : amplitudes, \(\nu_{mn}\) : frequency 로 다음 조건을 만족시킴
  • \(q_{mn}=q_{nm}^{*}\)
  • \(p_{mn}=q_{nm}^{*}\)
  • \(\nu_{mn}=-\nu_{nm}\)
  • \(m \neq n\) 이면, \(\nu_{mn}\neq 0\)
  • \(\nu_{rs}+\nu_{st}=\nu_{rt}\)

2

• \([Q,P] = Q P - P Q = i \hbar\)
• Born-Jordan condition 이라고도 불리며 보어-좀머펠트 양자 조건에 해당

3

• \(H(P,Q)\) 해밀토니안

4

• 운동방정식
• \(\dot{Q}_i=\partial H/\partial P\)
• \(\dot{P}=-\partial H/\partial Q\)

\(H(P,Q)\) 는 대각행렬이며, 고유값은 \(E_n\)

\(E_{m}-E_{n}=\hbar \nu_{mn}\)

역사

• 1925 Heisenberg matrix mechanics
• 1926 Pauli hydrogen atom
• 1927 Heisenberg uncertainty principle
• 1930-31 Stone-von Neuman Theorem
• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

메모

On the other hand, matrix mechanics was invented by Heisenberg in June 1925, and presented in a fully developed form in Dirac’s first paper on quantum mechanics (received 7 November 1925) and also in the famous “three-men’s paper” of Born, Heisenberg and Jordan (received 16 November 1925).

• http://www.worldscibooks.com/etextbook/7271/7271_chap02.pdf
• A brief history of the mathematical equivalence between the two quantum mechanics
• Why were two theories (Matrix Mechanics and Wave Mechanics) deemed logically distinct, and yet equivalent, in Quantum Mechanics?
• Quantum Mechanics: Concepts and Applications
• Math Overflow http://mathoverflow.net/search?q=

관련된 항목들

• 푸리에 급수

수학용어번역

• 단어사전
  • http://translate.google.com/#en%7Cko%7C
  • http://ko.wiktionary.org/wiki/
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 한국통계학회 통계학 용어 온라인 대조표
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

매스매티카 파일 및 계산 리소스

• http://www.wolframalpha.com/input/?i=
• http://functions.wolfram.com/
• NIST Digital Library of Mathematical Functions
• Abramowitz and Stegun Handbook of mathematical functions
• The On-Line Encyclopedia of Integer Sequences
• Numbers, constants and computation
• 매스매티카 파일 목록

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

리뷰논문, 에세이, 강의노트

• B. L. van der Waerden, From Matrix Mechanics and Wave Mechanics to Unified Quantum Mechanics
• 임경순, 행렬역학의 전개 과정