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1
REGARDING HEISENBERG SHELLS
Dr Spyridon Koutandos
skoutandos1@gmail.com
skoutandos@yahoo.com
Abstract:
In this article the author examines the possibility of some hidden variables
existing behind quantum mechanics formulation, a phrase first used by Albert
Einstein. This time we derive new methods of explaining integrations used in
quantum mechanics.
In further calculations we will assume for ease as is the case usually that the angle
part of the wave function is already normalized and divided by 4 pi. For the behavior
of polarization regarding the signs the reader might want to refresh his knowledge
with the book of reference [9].
The result of the author [1] about the surface
polarization charge sigma is that it is possibility times the radius and the polarization
charge contained within equal psi surfaces is:
r
N
e
2
ψ
σ
−
=
(1)
enclosed
sphere
p
V
N
r
e
r
N
r
e
S
d
r
e
N
r
S
d
P
q
π
4
)
(
3
)
(

)
(
)
(
)
(
2
3
2
2
2
=
=
=
−
Ω
−
=
−
=
∫
∫
r
r
r
r
(2)
We should calculate the mean magnetic pressure on the surfaces of constant
probability. Coming back to the problem of calculating expression 14, we may notice
that , since the gradient of phase changes only along the equipotentials
of psi squared
and the magnetic pressure is proportional to this gradient times the vector potential A
which again is the magnetic induction times the vertical projection of the radius used
in expression 14 we have for a radial integration (where psi(r) is of course constant):
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( )
B
m
r
mN
e
d
rB
r
m
mN
e
d
r
A
mN
e
r
P
P
B
M
l
l
m
magnetic
frontiers
2
2
2
0
2
2
)
(
sin
)
(
2
sin
)
(
)
(
2
ψ
θ
π
φ
h
h
r
h
r
r
=
=
⋅
=
⋅
∇
=
=
=
⋅
∫
∫
(3)
We assume as is the case that the anfle parts of the wave function are already
normalized to 4 pi, so we have:
m
magnetic
l
sphere
m
E
N
r
E
N
r
B
m
r
mN
e
P
2
2
2
)
(
)
(
)
(
=
=
=
h
(4)
At this point the reader should be reminded that after psi comes the last letter of the
Greek alphabet, Omega=PV which is the big thermodynamic potential used by
Gibbs[7] along with some other symbols now used in quantum mechanics. At
conditions of equilibrium and no particle generations, therefore of zero chemical
potential, the free energy F is equal to Omega. The author has included pressure that
arises from the validity of the virial theorem. The polarization pressure is to be
associated with surface polarization charge and the normal pressure with the potential.
A
part
of
the
omega
should
include
the
magnetic
pressure.
m
p
l
m
enclosed
m
m
E
e
q
V
N
B
m
m
e
V
P
V
P
3
4
2
=
=
−
=
−
=
Ω
h
(5)
In equation 15d stands the Bohr magneton. We have made some assumptions such as
constant magnetic field but in general magnetic fields depend on angles and this rule
should apply. Specifically the magnetic field is formed by closed loops and may be
expressed as a gradient of the solid angle at which we look at it:
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 Spring '09
 SpyridonKoutandos

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