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Extra info for Burgers turbulence
1 Commutation and anticommutation algebras 45 by condensing them into a single quantum state. In this circumstance, every single particle lies in the same one-particle state. If we time-evolve the system, we can think of the singleparticle wavefunction as a semiclassical variable describing the collective behavior of the condensate. Let us briefly recall one-particle quantum mechanics. 11) n so that |ψn (t)|2 = pn (t) gives the probability of being in state n. Applying Schrödinger’s ˆ equation, H|ψ = i ∂t |ψ , gives i ψ˙ n (t) = n|H|m ψm (t) m i ψ˙ n∗ (t) = − m|H|n ψm∗ (t).
3 Zero-dimensional ﬁeld theory H= 19 1 p2 + mω2 x2 , 2m 2 [x, p] = i . 33) quantum harmonic oscillator In the ground state, the particle in the harmonic potential undergoes zero-point motion, with uncertainties in position and momentum p and x which satisfy x p ∼ . 34) define the scale of zero-point motion. It is useful to define dimensionless position and momentum variables by factoring out the scale of zero-point motion: ξ= x , x pξ = p . 35) One quickly verifies that [ξ , pξ ] = i are still canonically conjugate, and that now H= ω 2 ξ + p2ξ .
A, vol. 112, p. 661, 1926.  P. Jordan and O. Klein, Zum Mehrkörperproblem der Quantentheorie (On the manybody problem of quantum theory), Z. , vol. 45, p. 751, 1927. The second quantization condition for bosons appears in equation (14) of this paper.  P. Jordan and E. Wigner, Uber das Paulische Aquivalenzverbot (On the Pauli exclusion principle), Z. , vol. 47, p. 631, 1928.  W. Heisenberg, Uber quanten theoretische Umdeutung kinematischer und mechanischer Beziehungen (Quantum theoretical reinterpretation of kinematic and mechanical relations), Z.
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