The pursuit of exactness in quantum physics
Zapytaj Fizyka public lecture series
Warsaw, 15 April 2025Jean-Sébastien Caux
Beginnings
Astronomy
Quantum mechanics
The quantum hypothesis
Max Planck (1858-1947)
Quantum mechanics
'Matrix mechanics'
\[ \left[ q, p \right] = qp - pq = i \hbar \]
Uncertainty relation:
\[ \Delta q ~\Delta p \geq \frac{\hbar}{2} \]
Quantum mechanics
Erwin Schrödinger (1887-1961)
'Wave mechanics'
\[ i\hbar \frac{\partial}{\partial t} \Psi = H \Psi \]
Quantum mechanics
Max Born (1882-1970)
Probability interpretation
\[ p(x) = \left| \Psi(x) \right|^2 \]
Max Born (Nobel 1954 lecture)
I believe that ideas such as absolute certitude, absolute exactness, final truth, etc. are figments of the imagination which should not be admissible in any field of science.
On the other hand, any assertion of probability is either right or wrong from the standpoint of the theory on which it is based.
This loosening of thinking (Lockerung des Denkens) seems to me to be the greatest blessing which modern science has given to us.
For the belief in a single truth and in being the possessor thereof is the root cause of all evil in the world.
The Bohr atom
Niels Bohr (1885-1962)
The periodic table
The solvable, and the unsolvable
The two-body problem
aka the Kepler problem
Harmonic oscillator
The only thing we know how to solve?
The three-body problem
How complicated can it be?
In general: unsolvable!
Heisenberg model
\[ H = \sum_j S_j \cdot S_{j+1} \]
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Many-body physics
Unpredictable many-body physics
Superconductivity
Heike Kamerlingh Onnes (1853-1926)
Integrability
Bethe's solution of Heisenberg's model
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Exact solution (1931) using the 'Bethe Ansatz'
Energy levels exactly calculated
Hans A. Bethe (1906-2005)
Exact solutions versus strong correlations
John Scott Russell: legend or truth?
John Scott Russell
John Scott Russell (1808-1882)
Solitary wave of translation
Herriot-Watt University
Boussinesq, Korteweg & de Vries
\[ \partial_t \phi + \partial_x^3 \phi - 6\phi \partial_x \phi = 0 \]
Joseph Boussinesq (1842-1929)
Diederik Korteweg (1848-1941)
Gustav de Vries (1866-1934)
The baffling absence of ergodicity
Enrico Fermi (1901-1954)
© LANL
John Pasta (1918-1984)
© LANL
Stanisław Ulam (1909-1984)
Fermi-Pasta-Ulam-Tsingou
© LANL
Mary Tsingou (1928-)
The theory of solitons
Martin Kruskal (1925-2006)
Norman Zabusky (1929-2018)
Classical inverse scattering method
Symmetries and conservation laws
Emmy Noether (1882-1935)
Quantum integrability
Chen Ning Yang (1922-)
Rodney Baxter (1940-)
Ludvig Faddeev (1934-2017)
The power of integrability
Hans A. Bethe (1906-2005)
Experiments
Copper sulfate (CuSO4)
CuSO4: artist's view
Rogier Hiorns: Seizure
CuSO4: scientist's view
⤷ modeled by ⤵
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Neutron scattering
Applications of integrability
Quantum magnetism
Cold atoms
Quantum dots
Atomic nuclei
Integrability's recent history
A future out of equilibrium
Kapitsa's pendulum
Pyotr L. Kapitsa (1894-1984)
Kapitsa's pendulum
Pyotr L. Kapitsa (1894-1984)