20世纪另一革命性进展是量子理论，源于马克斯·普朗克（1856–1947）关于黑体辐射的开创性贡献与爱因斯坦对光电效应的研究。1912年，数学家亨利·庞加莱发表了《量子理论研究》（Sur la théorie des quanta）。[17][18]他在这篇论文中首次提出了量子化的形式定义。早期量子物理的发展遵循阿诺尔德·索末菲（1868–1951）和尼尔斯·玻尔（1885–1962）设计的启发式框架，很快被马克斯·玻恩（1882–1970）、维尔纳·海森堡（1901–1976）、保罗·狄拉克（1902–1984）、埃尔温·薛定谔（1887–1961）、萨特延德拉·纳特·玻色（1894–1974）、沃尔夫冈·泡利（1900–1958）发展的量子力学所取代。这一革命性理论框架基于对状态、演化与测量的概率解释，即无限维向量空间上的自伴算子。这空间称作希尔伯特空间（数学家大卫·希尔伯特（1862–1943）、埃哈德·施密特（1876–1959）、里斯·弗里杰什（1880–1956）为寻求欧氏空间的推广与研究积分方程而引入）。约翰·冯·诺依曼在《量子力学的数学基础》中严格定义了公理化的现代版本，并建立了希尔伯特空间现代泛函分析的相关部分——谱理论（大卫·希尔伯特引入，研究了无穷多变量的二次型。多年后，人们发现谱理论与氢原子光谱有关，他对这应用非常惊讶）。保罗·狄拉克用代数构造为电子建立了相对论模型，预言了电子的磁矩及其反粒子——正电子的存在。
Quote: " ... 理论家的负面定义是说他们不进行物理实验，而正面... 是说他拥有百科全书式的物理知识，同时还有充分的数学武装。根据这两部分的比例，理论家可能接近实验家，也可能接近数学家，后者我们一般视作数学物理专家。", 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
Marsh, Adam, Mathematics for Physics: An Illustrated Handbook, World Scientific, 2018, ISBN 978-981-3233-91-1
Mathews, Jon; Walker, Robert L., Mathematical Methods of Physics 2nd, W. A. Benjamin, 1970, ISBN 0-8053-7002-1
Menzel, Donald H., Mathematical Physics, Dover Publications, 1961, ISBN 0-486-60056-4
Riley, Ken F.; Hobson, Michael P.; Bence, Stephen J., Mathematical Methods for Physics and Engineering 3rd, Cambridge University Press, 2006, ISBN 978-0-521-86153-3
Stakgold, Ivar, Boundary Value Problems of Mathematical Physics, Vol 1-2., Society for Industrial and Applied Mathematics, 2000, ISBN 0-89871-456-7
Starkovich, Steven P., The Structures of Mathematical Physics: An Introduction, Springer, 2021, ISBN 978-3-030-73448-0
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
Geroch, Robert, Mathematical Physics, University of Chicago Press, 1985, ISBN 0-226-28862-5
Hassani, Sadri, Mathematical Physics: A Modern Introduction to its Foundations 2nd, Springer-Verlag, 2013, ISBN 978-3-319-01194-3
Marathe, Kishore, Topics in Physical Mathematics, Springer-Verlag, 2010, ISBN 978-1-84882-938-1
Milstein, Grigori N.; Tretyakov, Michael V., Stochastic Numerics for Mathematical Physics 2nd, Springer, 2021, ISBN 978-3-030-82039-8
Reed, Michael C.; Simon, Barry, Methods of Modern Mathematical Physics, Vol 1-4, Academic Press, 1972–1981
Richtmyer, Robert D., Principles of Advanced Mathematical Physics, Vol 1-2., Springer-Verlag, 1978–1981
Serov, Valery, Fourier Series, Fourier Transform and Their Applications to Mathematical Physics, Springer, 2017, ISBN 978-3-319-65261-0
Simon, Barry, A Comprehensive Course in Analysis, Vol 1-5, American Mathematical Society, 2015
Stakgold, Ivar; Holst, Michael, Green's Functions and Boundary Value Problems 3rd, Wiley, 2011, ISBN 978-0-470-60970-5
Stone, Michael; Goldbart, Paul, Mathematics for Physics: A Guided Tour for Graduate Students, Cambridge University Press, 2009, ISBN 978-0-521-85403-0
Szekeres, Peter, A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry, Cambridge University Press, 2004, ISBN 978-0-521-53645-5
Taylor, Michael E., Partial Differential Equations, Vol 1-3 2nd, Springer., 2011
Whittaker, Edmund T.; Watson, George N., A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions 4th, Cambridge University Press, 1950
经典物理专业书籍
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
Bloom, Frederick, Mathematical Problems of Classical Nonlinear Electromagnetic Theory, CRC Press, 1993, ISBN 0-582-21021-6
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
