20世紀另一革命性進展是量子理論，源於馬克斯·普朗克（1856–1947）關於黑體輻射的開創性貢獻與愛因斯坦對光電效應的研究。1912年，數學家亨利·龐加萊發表了《量子理論研究》（Sur la théorie des quanta）。[17][18]他在這篇論文中首次提出了量子化的形式定義。早期量子物理的發展遵循阿諾爾德·索末菲（1868–1951）和尼爾斯·波耳（1885–1962）設計的啟發式框架，很快被馬克斯·玻恩（1882–1970）、維爾納·海森堡（1901–1976）、保羅·狄拉克（1902–1984）、埃爾溫·薛丁格（1887–1961）、薩特延德拉·納特·玻色（1894–1974）、沃夫岡·包立（1900–1958）發展的量子力學所取代。這一革命性理論框架基於對狀態、演化與測量的機率解釋，即無限維向量空間上的自伴算子。這空間稱作希爾伯特空間（數學家大衛·希爾伯特（1862–1943）、埃哈德·施密特（1876–1959）、里斯·弗里傑什（1880–1956）為尋求歐氏空間的推廣與研究積分方程式而引入）。約翰·馮·諾依曼在《量子力學的數學基礎》中嚴格定義了公理化的現代版本，並建立了希爾伯特空間現代泛函分析的相關部分——譜理論（大衛·希爾伯特引入，研究了無窮多變量的二次型。多年後，人們發現譜理論與氫原子光譜有關，他對這應用非常驚訝）。保羅·狄拉克用代數構造為電子建立了相對論模型，預言了電子的磁矩及其反粒子——正電子的存在。
Quote: " ... 理論家的負面定義是說他們不進行物理實驗，而正面... 是說他擁有百科全書式的物理知識，同時還有充分的數學武裝。根據這兩部分的比例，理論家可能接近實驗家，也可能接近數學家，後者我們一般視作數學物理專家。", Ya. Frenkel, as related in A.T. Filippov, The Versatile Soliton, pg 131. Birkhauser, 2000.
Irons, F. E. Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms. American Journal of Physics. August 2001, 69 (8): 879–84. Bibcode:2001AmJPh..69..879I. doi:10.1119/1.1356056.
Hassani, Sadri (2009), Mathematical Methods for Students of Physics and Related Fields, (2nd ed.), New York, Springer, eISBN 978-0-387-09504-2
Jeffreys, Harold; Swirles Jeffreys, Bertha, Methods of Mathematical Physics 3rd, Cambridge University Press, 1956
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研究生教材
Blanchard, Philippe; Brüning, Erwin, Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics 2nd, Springer, 2015, ISBN 978-3-319-14044-5
Cahill, Kevin, Physical Mathematics 2nd, Cambridge University Press, 2019, ISBN 978-1-108-47003-2
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古典物理專業書籍
Abraham, Ralph; Marsden, Jerrold E., Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics with an Introduction to the Qualitative Theory of Dynamical Systems 2nd, AMS Chelsea Publishing, 2008, ISBN 978-0-8218-4438-0
Adam, John A., Rays, Waves, and Scattering: Topics in Classical Mathematical Physics, Princeton University Press., 2017, ISBN 978-0-691-14837-3
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Boyer, Franck; Fabrie, Pierre, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, 2013, ISBN 978-1-4614-5974-3
Colton, David; Kress, Rainer, Integral Equation Methods in Scattering Theory, Society for Industrial and Applied Mathematics, 2013, ISBN 978-1-611973-15-0
Ciarlet, Philippe G., Mathematical Elasticity, Vol 1–3, Elsevier, 1988–2000
Galdi, Giovanni P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems 2nd, Springer, 2011, ISBN 978-0-387-09619-3
Hanson, George W.; Yakovlev, Alexander B., Operator Theory for Electromagnetics: An Introduction, Springer, 2002, ISBN 978-1-4419-2934-1
Kirsch, Andreas; Hettlich, Frank, The Mathematical Theory of Time-Harmonic Maxwell's Equations: Expansion-, Integral-, and Variational Methods, Springer, 2015, ISBN 978-3-319-11085-1
Knauf, Andreas, Mathematical Physics: Classical Mechanics, Springer, 2018, ISBN 978-3-662-55772-3
Lechner, Kurt, Classical Electrodynamics: A Modern Perspective, Springer, 2018, ISBN 978-3-319-91808-2
Marsden, Jerrold E.; Ratiu, Tudor S., Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems 2nd, Springer, 1999, ISBN 978-1-4419-3143-6
Müller, Claus, Foundations of the Mathematical Theory of Electromagnetic Waves, Springer-Verlag, 1969, ISBN 978-3-662-11775-0
Ramm, Alexander G., Scattering by Obstacles and Potentials, World Scientific, 2018, ISBN 9789813220966
Roach, Gary F.; Stratis, Ioannis G.; Yannacopoulos, Athanasios N., Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton University Press, 2012, ISBN 978-0-691-14217-3
