Brownian Motion and Quantum Mechanics Two Sides of
Brownian Motion and Quantum Mechanics: Two Sides of the same Coin? Theo M. Nieuwenhuizen University of Amsterdam 100 Years of Brownian Motion Erice, 30 -7 -2005
Setup Generalities about foundations of Quantum Mechanics Results from Quantum Measurement Theory Quantum mechanics of the hydrogen atom Weak damping classical stochastic theories Phase space density for ground state + Yrast states Phase space forms for squares of spherical harmonics Conclusion A dream A random walk
Generalities Einstein, Feynman, ‘t Hooft, . . . I don’t understand Quantum Mechanics Some understanding should come from the solution of the quantum measurement problem Is QM complete? “Hidden variables”, sub-quantum mechanics ?
Realistic Model for a Quantum Measurement Allahverdyan, Balian, Th. M N. Europhys Lett 61, 452 (2003); cond-mat/0408316 quant-ph/. . . Tested system: spin ½ Apparatus=magnet+bath Ising magnet: starts out as metastable paramagnet ends up in stable ferromagnetic state: M > 0 if tested spin is up M < 0 if tested spin is down Excess energy is dumped in bath
Initial density matrix Tested system: arbitrary density matrix Apparatus: uncorrelated with apparatus After the measurement: - maximal correlation between S and A - Schrodinger cat-terms have decayed physical effect, no postulate
Look at magnetization of apparatus only: In practice: this is a classical distribution function It describes statistics of pointer values Probabilities = relative frequencies (von Mises) Derivation of the Born rule: via interpretation of result for macroscopic apparatus
Quantum Mechanics is a theory that describes the statistics of outcomes of experiments It cannot and should not describe individual experiments (otherwise than in a probablistic sense) Statistical physics Newton motion Quantum mechanics ? ?
In Nature: separate measurements occur There is no theory for single measurements Look for “subquantum mechanics”, “hidden variable theory” Stochastic Electrodynamics = SED attempts to present a physical cause of quantum behavior This talk: more general class of theories
Quantum mechanics of hydrogen atom: nucleus charge = -Ze Rest energy Spectrum: Rydberg energy Relativistic corrections Lamb shift Relativistic spectrum for m=c=1 Lamb shift: not from Schrodinger equation, but due to coupling to EM field weak effect weak coupling, weak Lorentz damping
Weak damping classical stochastic theories for hydrogen atom Phase space density Evolution Stationary distribution = function of conserved quantities Energy m=c=1 Angular momentum
The unsquared dance Define E(R) or R(E) by Related non-relativistic problem Effective angular momentum In QM: effective angular momentum
Bits and Pieces Go to cylindrical coords around r: Volume element in p-space , Consider Then Momentum integral and the ratio = non-relativistic groundstate density generates a factor r
Yrast states: l=n-1 (maximal) non-relativistic: 1 s, 2 p, 3 d, 4 f, … Phase space densities Momentum average gives square of relativistic wavefunctions: n=1: Ground state: P positive, so P differs from Wigner function W Reason: our p is instantanous; in Wigner function it is statistical Space average
Average energy not correct. . Doing the forbidden: Neglect correlations Approximate then quantum mechanical energy recovered at order Do this at all orders Exact quantum result regained for Yrast states:
2 p state: spherical harmonics In frame along r, cylindrical coordinates: L involves angles Search phase space forms Proposal: a) b)
Wigner(p) versus Phase space density(p) Test: scatter fast electrons on hydrogen atoms (Mott & Massey: Impulse approximation) Slow speeds: many revolutions during scattering: quantum cloud Fast speed: instantaneous position and speed of bound e is probed non-quantum result? Large but non-relativistic speads
Discussion Quantum measurement process Look for sub-quantum mechanics Considered class of theories includes Stochastic Electrodynamics Phase space densities proposed for Yrast states l=n-1 Integral over p gives QM density Integral over r does NOT give result from Wigner function Different method, same result: consistency Also 2 s state considered: works in the same approach (non-unique) l=1 phase space forms for squares of spherical harmonics proposed Ground state density positive; excited states partially negative Quantum energies recovered iff correlations neglected Physically: time scale separation : each new quantum operator corresponds to a classical average at a well separated time m
Theo’s dream 1) Schrodinger mechanics = SED or so de la Pena, Cetto, Cole, Khrennikov, . . 2) Particles, photons: solitons in electro-gravity Carter, Pereira, Arcos, Burinskii 3) Physical explanation for exclusion principle and QM-statistics timescales, 4) QM = statistics of stochastic soliton mechanics energetics This dream integrates basically all works of Albert Einstein.
Maybe this talk was a random walk Perhaps it was recurrent: going from nowhere to nowhere But there were magnificent views Now you may say I’m a dreamer But I’m not the only one I hope that one day you’ll join us And the world will live as one Imagine, John Lennon
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