12-2 - Probability current density

Prévia do material em texto

1 
 
Probability current density 
Masatsugu Sei Suzuki 
Department of Physics, SUNY at Binghamton 
(Date: November 22, 2013) 
 
In quantum mechanics, the probability current (sometimes called probability flux) 
is a mathematical quantity describing the flow of probability (i.e. probability per unit 
time per unit area). Intuitively, if one pictures the probability density as an 
inhomogeneous fluid, then the probability current is the rate of flow of this fluid. This is 
analogous to mass currents in hydrodynamics and electric currents in electromagnetism. 
It is a real vector, like electric current density. The notion of a probability current is 
useful in some of the formalism in quantum mechanics. 
Wikipedia: http://en.wikipedia.org/wiki/Probability_current 
 
1. Probability current density 
The probability density is defined by 
 
22
),()(),( ttt rrr   . 
 
The integral 
 
  rrr dtdtr
2
),(),(  , 
 
taken over some finite volume V, is the probability of finding the particle in this volume. 
Let us calculate the derivative of the probability with respect to time t. 
 











r
rrr
dHH
i
d
tt
dt
t
)]()[(
1
)(),(
***
*
*
2



 
 
Here 
 
 H
t
i 


 , 
 
where 
 
)(),( tt  rr  . 
 
Complex conjugate of this equation 
 
 
2 
 
***
*


HH
t
i 


  , 
 
where 
 
)(
2
2
2
* rV
m
HH 

, 
 
})](
2
{[})](
2
{[
)()()()(
2
2
**2
2
*****


rr V
m
V
m
HHHH


 
or 
 
)]()[(
2
)()( 2**2
2
***  
m
HH

, 
 
 

rrr d
mi
dt
t
)]()[(
2
),( 2**2
2   . 
 
Note that 
 
)()()( **2**2   . 
 
((Proof)) 
We use the formula of vector analysis. 
 
aaa   )( , 
 
*2**)(   . 
 
The complex conjugate of the above equation 
 
 2*** )(  . 
 
From the above equations, we get 
 
)()()( **2**2   . 
 
Then 
 
 

rrr d
mi
dt
t
)(
2
),( **
2   . 
 
 
3 
 
We note that 
 
 

aJrJr ddd
t
 , (Gauss’s theorem) 
 
or 
 
0


J
t
. (Equation of continuity) 
 
Then the probability current density can be defined as 
 
]Re[
1
)(
2
***  
immi

J , 
 
since 
 
)(
2
)(
2
1
]Re[
1
**
**
*






mi
iim
im



J
 
 
((Note-1)) What is the units of J? 
 
The unit of J should be [cm-2 s-1]. Since 1
2  rd , the unit of  * is [cm-4]. Then 
the unit of J is evaluated as 
 
 
2422
11
/ cmscmcmserg
serg




J 
 
The unit of  rr dJ )( is [cm/s] which is the unit of the velocity. 
 
((Note-2)) Comment on the formula 
((L.I. Schiff, Quantum Mechanics, McGraw-Hill, 1968)) 
 
)],(),(Re[
1
),( * t
i
t
m
t rrrJ    . 
 
 
“Although this interpretation of J is suggestive, it must be realized that J is not 
susceptible to direct measurement in the sense in which the probability P is. It would be 
 
4 
 
misleading, for example, to say that J(r, t) is the average measured particle flux at the 
point r and the time t, because a measurement of average local flux implies simultaneous 
high-precision measurements of position and velocity (which is equivalent to momentum) 
and is therefore inconsistent with the uncertainty relation. Nevertheless, it is sometimes 
helpful to think of J as a flux vector, especially when it depends only slightly or not at all 
on r, so that an accurate velocity determination can be made without impairing the 
usefulness of the concept of flux.” 
 
((Note-3)) The reason why J is called the probability current density. 
(J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd edition, Addison-
Wesley, 2011) 
 
What is the relation between the average of momentum and the probability current 
density? 
 
We may intuitively expect that the average value of the momentum operator at time t is 
related to the Probability current density ),( tJ r as 
 
 ),(
1
td
m t
rrJp 
 
where 
t
p is the average of the momentum operator and is defined by 
 
 pp ˆ
t
. 
 
This expectation value is in fact real since 
 
 ppp ˆˆ*   . 
 
We now calculate 
 
]ˆ
1
Re[  p
m
, 
 
which is in fact equal to  p̂1
m
. 
 
 
5 
 









rrJ
rrrrr
rrr
rprr
pp
dt
dtttt
im
dt
im
t
dtt
m
mm t
),(
)],(),(),(),([
2
]),(),(Re[
])(ˆ)(Re[
1
]ˆ
1
Re[
1
**
*






 
 
Thus we have 
 
)],(),(),(),([
2
),( ** tttt
im
t rrrrrJ    . 
 
((Note-4)) E. Merzbacher, Quantum Mechanics (John Wiley & Sons, 1998) 
 
By using the continuity equation, the time derivative of the average x coordinate can 
be written as 
 












r
raj
rrJ
rJ
r
dj
djdx
djdx
dx
d
t
xx
dt
d
x
x
x
)(
)(
)(

 
 
or 
 
 rJr ddt
d
, 
 
where the divergence term is removed under the assumption that  vanishes sufficiently 
fast at infinity. Using the expression of ),( trJ and integration by parts, we obtain 
 
  rpr didt
d
m  * . 
 
2. Equation of continuity 
The probability density is defined as 
 
6 
 
 
 * . 
 
We take the derivative of  with respect to t, 
 
tttt 









  *
*
* )( . 
 
Using the Schrödinger equation, we get 
 
 )
2
(
2
V
m
H
t
i 

 p
 , *
2
*
*
)
2
(  V
m
H
t
i 



p
 . 
 
and 
 

i

p , 
 
we have 
 
)(
2
)
2
(
1
)
2
(
1
)
2
(
1
)
2
(
1
])
2
[(
1
])
2
[(
1
*22*
2
2
**2
2
2
**
2
2
**
2










mi
mimi
mimi
V
mi
V
mit







pp
pp
 
 
We now calculate J , 
 
)(
2
)(
2
][
2
*22*
**22**
**






mi
mi
mi



J
 
 
where we use the vector formula 
 
AAA  fff )( . 
 
Then we have the equation of continuity 
 
7 
 
 
0


J
t

, 
 
or 
 
0)( 


 rJr ddt

. 
 
Using the Gauss’s theorem, this can be rewritten as 
 
0


 aJr ddt  , 
 
where da is the element of surface area vector whose direction is normal to the surface. 
 
3. Examples 
(a) The plane wave 
 
)(]exp[)](exp[),( rrp  EtiCEtiCtr

 . 
 
The probability current density is obtained as 
 
2
2
**
*
)]exp()exp()()exp()()exp([
2
)]()()()([
2


v
p
rprp
p
rp
p
rp
rrrrJ




C
m
i
C
ii
C
ii
C
i
C
mi
mi



 
 
 
 
Fig. Flux density J: particles number flowing per unit area per unit time. 
 
 
8 
 
avdtJadt
2 , 
 
or 
 
22 
m
p
vJ  . 
 
(b) The superposition of the plane waves 
 
Suppose that 
 
ikxikx BeAe  . 
 
Then we have 
 
)()(
)])(Re[
)]()Re[(
1
]Re[
1
22**
**
**
*
BA
m
k
BBAA
m
k
BeAeBeBeA
m
k
BeAe
xi
eBeA
m
xim
J
ikxikxikxikx
ikxikxikxikx
x














 
 
 
4. Conservation of the probability current density at the boundary for the one 
dimensional barrier problem 
We consider the boundary condition one dimensional barrier problem as shown 
below. 
 
The probability current density is defined by 
 
)(
2
** 
dx
d
dx
d
mi
J 

, 
 
 
9 
 
V0
a
2
-
a
2
a
a1
b1
a2
b2
a3
b3
x
Vx
 
 
Because of particle (flux) conservation, both  and 
dx
d
 must be continuous at the 
barrier border at x = -a/2 and x = a/2. In other words, when  and 
dx
d
 are continuous at 
the boundary, the flux conservation is valid. 
 
(a) Region-I 
 
ikxikx ebeax  11)( , for 2
a
x  
 
)()(' 11
ikxikx ebeaikx  . 
 
The probability current density is given by 
 
)(
2
1
2
1 bam
k
J I 

. 
 
(b) Region-II 
 
xx
II ebeax
 22)( 
 , for 
2
a
x  
 
)()(' 22
xx
II ebeax
   . 
 
 
10 
 
The probability current density 
 
)( *222
*
2 babam
iJ II 

. 
 
(c) Region III 
 
ikxikx
III ebeax
 33)( , for 2
a
x  
 
)()(' 33
ikxikx
IIIebeaikx
 . 
 
The probability current density 
 
)(
2
3
2
33 bam
k
J 

, 
 
where 
 
m
k
2
22
 , 
m
V
2
22
0
  . 
 
From the continuity of  and 
dx
d
, we get the relations between a1, b1, a2, b2, a3, and b3. 
 
(i) 












2
2
1
1 )
2
(
b
aa
M
b
a
, 
 
where 
 



















]
2
)(
exp[)(]
2
)(
exp[)(
]
2
)(
exp[)(]
2
)(
exp[)(
2
1
)
2
( 

ika
ik
ika
ik
ika
ik
ika
ik
k
a
M . 
 
(ii) 
 












2
2
3
3 )
2
(
b
aa
M
b
a
, 
 
where 
 
 
11 
 



















]
2
)(
exp[)(]
2
)(
exp[)(
]
2
)(
exp[)(]
2
)(
exp[)(
2
1
)
2
( 

ika
ik
ika
ik
ika
ik
ika
ik
k
a
M , 
 
We find that 
 
IIIIII JJJ  , 
 
or 
 
)()()(
2
3
2
3
*
222
*
2
2
1
2
1 bam
k
baba
m
iba
m
k

 
. 
 
5. Lagrangian of particles with mass m* and charge q* in the presence of 
magnetic field 
The Lagrangian L for the motion of a particle in the presence of magnetic field and 
electric field is given by 
 
)
1
(
2
1 *2* Avv 
c
qmL  , 
 
where m* and q* are the mass and charge of the particle. A is a vector potential and  is a 
scalar potential. 
 
The canonical momentum is defined as 
 
Av
v
p
c
q
m
L



 . 
 
The mechanical momentum (the measurable quantity) is given by 
 
Apvπ
c
q
m
*
*  . 
 
The Hamiltonian H is given by 
 
 *2
*
*
*2*
*
* )(
2
1
2
1
)( q
c
q
m
qmL
c
q
mLH  ApvvAvvp . 
 
 
The Hamiltonian formalism uses A and , and not E and B, directly. The result is that the 
description of the particle depends on the gauge chosen. 
 
 
12 
 
6. Current density for the superconductors 
We consider the current density for the superconductor.  is the order parameter of 
the superconductor and m* and q* are the mass and charge of the Cooper pairs. The 
current density is invariant under the gauge transformation. 
 
)](Re[
*
*
*
*
 AJ
c
q
im
q
s 

. 
 
This can be rewritten as 
 
A
AAJ
cm
q
im
q
c
q
ic
q
im
q
s
*
22*
**
*
*
*
*
**
*
*
*
*
)(
2
)]()[(
2







 
 
The density is also gauge independent. 
 
2
 rs . 
 
7. Probability current density (general case) 
Now we assume that 
 
)()()( rrr  ie . 
 
We note that 
 
)()(])([)(
)(])()()([)(
)(])([)()(
*
2
2
*
)()()(
2
*
)(*
*
*
rrArr
rArrrr
rArrAp
rrr
r














i
c
q
c
q
eei
i
e
c
q
e
ic
q
iii
i
. 
 
The last term is pure imaginary. Then the current density is obtained as 
 
ss qc
q
m
q
vAJ
2*
*
2
*
*
)(  


, 
 
or 
smc
q
vA *
*
 . 
 
 
13 
 
Since 
 
Apvπ
c
q
m s
*
*  , 
 
we have 
 
 smc
q
vAp *
*
. 
 
Note that Js (or vs) is gauge-invariant. Under the gauge transformation, the wave function 
is transformed as 
 
)()exp()('
*
rr 
c
iq

 . 
 
This implies that 
 
c
q


*
'  , 
 
Since  AA' , we have 
 
)(
)]()([
)''('
*
**
*
A
A
AJ






c
q
c
q
c
q
c
q
s






. 
 
So the current density is invariant under the gauge transformation. 
 
((Note)) 
ms

v 
 
Now we assume that 
 
)()()( rrr  ie . 
 
Noting that 
 
 
14 
 
)()()]([)(
])()()([)(
])([)(
2
)()()(
)(**
rrrr
rrrr
rrp
rrr
r












i
eei
i
e
e
i
iii
i
. 
 
The last term is pure imaginary. Then the probability current density is obtained 
as 
 
sm
vJ
22
)(    , 
 
or 

ms

v . 
 
8. Meissner effect: London equation 
We start with 
 
smc
q
vAp *
*
  . 
 
We assume that sn
2 is independent of r. Then we get 
 
ssss nqq vvJ
*2*   . 
 
Using the above two equations, we have 
 
s
snq
m
c
q
JAp
*
**
  . 
 
Suppose that p = 0. Then we have a London’s equation given by 
 
AJ
cm
nq s
s *
2*
 . 
 
From this equation, we have 
 
BAJ
cm
nq
cm
nq ss
s *
2*
*
2*
 . 
 
15 
 
 
Using the Maxwell’s equation 
 
sc
JB
4
 , and 0 B , 
 
we get 
 
BJB
2*
2*44
)(
cm
qn
c
s
s

 , 
 
where 
 
sn 
2 = constant (independent of r) 
 
2*
2*
2
4 qn
cm
s
L

  . (penetration depth). 
 
Then 
 
BBBB
2
2 1)()(
L
 , 
or 
 
BB
2
2 1
L
 . 
 
Inside the system, B become s zero, corresponding to the Meissner effect. 
 
9. Flux quantization 
We start with the current density 
 
ss qc
q
m
q
vAJ
2*
*
2
*
*
)(  


. 
 
Suppose that 
2sn =constant. Then we have 
 
AJ
 c
q
nq
m
s
s
*
*
*
 , 
 
or 
 
 
16 
 
  lAlJl dc
q
d
nq
m
d s
s 
*
**
*
 . 
 
The path of integration can be taken inside the penetration depth where sJ =0. 
 
   c
q
d
c
q
d
c
q
d
c
q
d
****
)( aBaAlAl , 
 
where  is the magnetic flux. Then we find that 
 

c
q
n
*
12 2 , 
 
where n is an integer. The phase  of the wave function must be unique, or differ by a 
multiple of 2 at each point, 
 
n
q
c
*
2 
 . 
 
The flux is quantized. When |q*| = 2|e|, we have a magnetic quantum fluxoid; 
 
e
ch
e
c
22
2
0 

 = 2.06783372 × 10-7 Gauss cm2. 
 
_____________________________________________________________________ 
REFERENCES 
1. J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, second edition 
(Addison-Wesley, New York, 2011). 
2. David. Bohm, Quantum Theory (Dover Publication, Inc, New York, 1979). 
3. Leonard Schiff, Quantum Mechanics (McGraw-Hill Book Company, Inc, New 
York, 1955). 
4. Eugen Merzbacher, Quantum Mechanics, third edition (John Wiley & Sons, New 
York, 1998). 
5. Albert Messiah, Quantum Mechanics, vol.I and vol.II (North-Holland, 1961).