1. Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil

2. Quantum plasmas High density systems (e.g. white dwarfs) Small scale systems (e.g. ultra- small electronic devices) Low temperatures (e.g. ultra-cold dusty plasmas)

3. Some developments Dawson’s (multistream) model applied to quantum two-stream instabilities [Haas, Manfredi and Feix, PRE 62, 2763 (2000)] Quantum MHD equations [Haas, PoP 12, 062117 (2005)] Quantum modulational instabilities (modified Zakharov system) [Garcia, Haas, Oliveira and Goedert, PoP 12, 012302 (2005)] Quantum ion-acoustic waves [Haas, Garcia, Oliveira and Goedert, PoP 10, 3858 (2003)]

4. Modeling quantum plasmas Microscopic models: N-body wave-function  density operator  Wigner function Macroscopic models: hydrodynamic formulation

5. Wigner-Poisson system

6. Remarks In the formal classical limit ( ) the Wigner equation goes to the Vlasov equation The Wigner function can attain negative values (a pseudo-probability distribution only) The Wigner function can be used to compute all macroscopic quantities (density, current, energy and so on)

7. Hydrodynamic variables

8. Quantum hydrodynamic model (electrostatic plasma)

9. Bohm’s potential or quantum pressure term:

10. Application: quantum two-stream instability [Haas et al., PRE (2000)]

11. The quantum parameter (two-stream instability)

14. Magnetized quantum plasmas Electromagnetic Wigner equation: [Haas, PoP (2005)] This is an ugly looking equation so I will not try to show it! Sensible simplifications are needed  hydrodynamic models

15. Quantum hydrodynamics for (non- relativistic) magnetized plasma plus Maxwell’s equations and an equation of state.

16. Quantum magnetohydrodynamics Highly conducting two-fluid plasma  merging  QMHD [Haas, PoP (2005)] The quantum parameter (QMHD):

17. One-component magnetized quantum plasma: “1D” equilibrium

18. Vector potential

19. A pseudo-potential

20. Ampere's law  equivalent to a Hamiltonian system

21. Pressure balance equation It can be shown that

22. Remarks In general, the balance equation is an ODE for the density n Solving the Hamiltonian system for yields simultaneously and

23. Rewriting the balance equation

24. Free ingredients The pressure p = p(n) The pseudo-potential

25. Harris sheet solution In classical plasmas, the Harris solution more frequently is build using the energy invariant to solves Vlasov In quantum plasmas, in general a function of the energy is not a solution for Wigner This also poses difficulties for quantum BGK modes

26. Choice for Harris sheet magnetic field

27. Solving for and then for (using suitable BCs)

28. Balance equation for quantum Harris sheet solution Using a suitable rescaling:

29. Quantum parameter (quantum Harris sheet) It increases with 1/m, 1/L, and the ambient density.

30. Classical limit

31. Ultra-quantum limit

32. Numerical simulations (H=3) -15-10-5510 15 0.2 0.4 0.6 0.8 1 1.2

33. Numerical simulations (H=5)

34. Final remarks In the quantum case, a Harris-type magnetic field (together with ) is associated to an oscillating density The velocity field is also modified (it depends on the density) Stability questions were not addressed - what is the role of quantum correlations?