Journal of Modern Physics, 2012, 3, 1856-1857
http://dx.doi.org/10.4236/jmp.2012.312233 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Steady Magnetohydrodynamic Equations for
Quantum Plasmas
Muhammad Asif, Umar Bashir
Department of Physics, COMSATS Institute of Information Technology, Lahore, Pakistan
Received September 30, 2012; revised October 30, 2012; accepted November 13, 2012
ABSTRACT
Steady Magnetohydrodynamic (MHD) Equations of force, density and energy for quantum plasmas have been derived.
These equations constitu te our Steady Magnetohydrodynamic model for quantum plasmas. All the quantum effects are
contained in the last term of quantum force equation and in the last three terms of quantum Energy Equation, so-called
Bohm potential and may be valuable for the description of quantum phenomena like tunneling.
Keywords: Steady Magnetohydrodynamic; Quantum Plasmas; Inhomogeneous Magnetic Field
1. Introduction
In classical plasma physics, fluid models are ubiquitous,
with their application ranging from astrophysics to con-
trolled nuclear fusion [1-3]. Magnetohydrodynamics, for
instance, provides one of the most useful fluid models,
focusing on the global properties of the plasma. The
purpose of work [4] is to produce a quantum counterpart
of magnetohydrodynamics, starting with the quantum
hydrodynamic model for charged particle systems. This
may provide yet another approach to the study of the
ways in which quantum physics can modify classical
plasma physics. However, it must be noted that the
quantum hydrodynamic model for charged particle sys-
tems was built for non-magnetized systems only [4]. In
order to arrive at a quantum modified magnetohydrody-
namics, work [4] sugg ests what the appropriate extension
would be of the quantum hydrodynamic model in cases
of nonzero magnetic field.
In this paper, we pr esent the Steady Magnetohydrody-
namic (MHD) Equations of force, density and energy for
quantum plasmas. These equations constitute our Steady
Magnetohydrodynamic model for quantum plasmas. All
the quantum effects are contained in the las t term of qu a n-
tum force equation and in th e last three terms of quantum
Energy Equation, so-called Bohm potential and may be
valuable for the description of quantum phenomena like
tunneling.
2. Steady Magnetohydrodynamic Equations
The Steady Magnetohydrodynamic (MHD) equations [5]
describe the plasma as a conducting fluid with conduc-
tivity which experiences electric and magnetic forces.
The fluid is specified by a mass density ρ, a flow velocity
V, and a pr essure P, which are all functions of space and
time. Here, we shall simply draw on the ordinary equa-
tions of hydrodynamics. The equation of motion has an
additional force per unit volume J due to the net mag-
netic force on the plasma particles. On fluid scales


D

 the plasma is overall charge neutral, so that the
net electric force is neglibly small. The MHD equations
represent the conservation of momentum, the conserva-
tion of mass and some equation of state (i.e., an energy
relationship). The simplified Steady Magnetohydrody-
namic Equations [5] are


VV PJB 

 (1)


0V 

 (2)


0PV

 (3 )
There are a number of assumptions behind the above
equations: the energy equation is the adiabatic equation
of state with

being the ratio of specific heats (5
3


for a monatomic gas); there are other possible choices for
an energy equation such as isothermal , etc. Oh-
mic heating J.E has been ignored, i.e., assumed to be less
important than the plasma’s thermal energy; radiative or
other heat losses are also neglected. We have also as-
sumed that the pressure is isotropic which is not gener-
ally true when the magnetic field can impose different
kinds of motion on the particles parallel and perpendicu-
lar to the field direction. Any possible heat flux has been
ignored, and there is no relative flow between the differ-

1  
C
opyright © 2012 SciRes. JMP

M. ASIF, U. BASHIR 1857
ent species in the plasma, so that the one fluid description
is valid. While the conductivity is usually large due to the
mobile electrons, the absence of collisions has the oppo-
site impact on other transport processes such as viscosity,
which w e h av e also therefore neglected.
3. Steady Quantum Force Balance Equation
The Steady force balance equation for quantum plasma is
obtaine d by putti ng the pressure P gi ven by [4,6] as
koQ
Pp p p

ko
pp pn
(4)
Since the kinetic and osmotic pressures measures the
kinetic and osmotic velocity dispersions, we can write
from [4,6]
(5)
Equation (5) depend only on the density, we obtain
from [4,6]

2
ie
mm
2
2
Ppn









 (6)
Using Equations (5) and (6), the Steady force balance
Equation (1) for quantum plasma is obtained as

2
ie
2
2
VV pJB
mm



  






0VV   
(7)
The derived Steady quantum force Equation (7) con-
stitute our quantu m hydrodynamic model for magnetized
systems. All the quantum effects are contained in the last
term of Equation (7), the so-called Bohm potential and
may be valuable for the description of quantum phe-
nomena like tunneling.
4. Steady Quantum Density Equation
The density Equation (2), for quantum plasmas is

 (8)
This is Steady density equation, which does not con-
tained the quantum effects.
5. Steady Quantum Energy Equation
Using Equations (5) and (6), the Steady energy Equation
(3), for quantum plasmas is obtained as


2 22
22
ie ie
Vp
V
pV
mm mm
 





  






2
0




 



(9)
The derived Steady Energy Equation (9) constitu te our
quantum hydrodynamic model for magnetized systems.
All the quantum effects are contained in the last three
terms of Equation (9), the so-called Bohm potential and
may be valuable for the description of quantum phe-
nomena like tunneling.
In comparison with standard fluid models for charged
particle systems, the Bohm potential is th e only quantum
contribution. In order to have a deeper understanding of
the importance of quantum effects, we compare Equa-
tions (7)-(9) with the rescaling of ideal quantum magne-
tohydrodynamic Equation (46) of [4], and obtained a non-
dimensional parameter 22
c
A
H
mV

, measuring the
relevance of quantum effects. Where A V is a Alfvén
velocity and c

is the cyclotron velocity. While for
ordinary plasmas H is n egligible, for dense astrophysical
plasmas, H can be of the order unity or more. Hence, in
dense astrophysical plasmas [7] such as the atmosphere
of neutron stars or the interior of massive white dwarfs,
quantum corrections to magnetohydrodynamics can be
experimentally important.
6. Conclusion
Steady Magnetohydrodynamic (MHD) Equations of force ,
density and energy for quantum plasmas have been de-
rived. The Steady force, density and Energy equations
constitute our reduced quan tum hydrodynamic model for
magnetized systems. All the quantum effects are con-
tained in the last term of force Equation (7) and in the
last two terms of Energy Equation (9), so-called Bohm
potential and may be valuable for the description of
quantum phenomena like tunneling.
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