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On the description of rotational flow in quantum hydrodynamics
Volume 61A, number 2
PHYSICS LETTERS
18 April 1977
ON THE DESCRIPTION OF ROTATIONAL FLOW IN QUANTUM HYDRODYNAMICS R. SRIDHAR Department of Mathematics, P.S. G. Arts College, Coimbatore 641014, India Received 11 February 1977 3x VM(x) is a multiple of It is proved that the vanishing of the velocity-velocity commutator is possible only if f d the identity or a constant of motion; equivalence of Landau’s approach to that of Yamasaki et a!. is also discussed.
Since the quantization of classical hydrodynamics first performed by Landau [1] several attempts have been made to formulate a microscopic theory by constructing a velocity field as the canonical conjugate of the density fluctuation operator [2—41 It has been .
p(x). Defining j5(x) = p(x) (p) and taking the Fourier transform of (3) we have, —
17j I ~i
I U
=
—(
0~q,~ argued that the velocity operator as proposed by Landau is not suitable to describe rotational flow smce the velocity-velocity commutator is zero implying thereby a vanishing curl for the velocity operator [4, 51. Still, it is not clear whether the components of velocity operator commute [6, 7]. It is the purpose of this letter to establish a criterion for the vanishing of the velocity-velocity commutator; incidentally, we also demonstrate the equivalence of Landau’s definition of the velocity operator to that of Yamasaki et al.
[41.
X
~
)
I
+
1_( )1r+
~+
v
.i’1’1’
Q1Q2
Q,O
~
~
~
~
~
(p+q—Q
X
r1
~°
-Q —...—Q)
~
~
4
~l~’~oW2j P’~r with ~4(p) = fd3x exp(—ip-x)v,~(x) and -~
Consider the direct definition of
the velocity opera-
tor:
1(x)}, (1) v,2(x) p_1(x)J~(x)+J~(x)p where ~ = 1, 2, 3 denotes the spatial components of u(x) and p(x) and J~(x)are respectively the mass and current density operators satisfying the non-relativistic current algebra [8]. Using the commutation relation [9] ,
2(x)
a
~6(x—y)p(x)},(2)
~—
[p’(x),J~(y)] = ip~ it is easy to verify [v/2 (x) v (y)] ‘
=
/1
—i~(x y) V —
/2,j’
(x) p~(x)
‘
with V/2,0(x) =
au (x)
a~
av (x)
a° ~
Expand p~(x)about the equilibrium value (p) of
(3)
X~,0(p)= p~ v~(p) resulting in the integral equation (p) [~Y~(p),i~~(q)] = —X~,(p + q) —
~
—
[iyp
—
Q/2),ii~,(q— Q/2)] ~(Q).
(5)
QO
Thus the velocity-velocity commutator is zero only if X, low from non-relativistic current algebra. the fol2,~= the 0; though theof vanishing of X~,0pFrom does fact that right side eq. (4) involves andnot q only as (p + q), it follows that [~ 2(p),I~,(q)]will be zero only [I~(p),zero, iJjq)] will be zero if [17,jp + q), u0(0)jif equals a possibility thatonly exists if either ~Y0(0) is a multiple of the identity or a constant of motion, since the set {p(x), J~(x)}forms a complete set of current operators. It is readily seen that IY~(0)commutes with p(x) while, 117
Volume 61A, number 2
PHYSICS LETTERS
[J/2(x), ~(0)]= —iflV/2V(x).
(8)
18 April 1977
The author thanks the management and the P.5G. Arts College for encouragement.
Principal of
On using Schur’s lemma, it is observed that I~(0) is a multiple of the identity only if V/2 = 0; that is if (V X u) vanishes. Also,
References
[H, ~~(o)1
[1] L.D. Landau, J. Phys. (U.S.S.R) 5 (1941) 7.
~,
3 =
—ih
~
[2] R. Kronig and A. Thellung, Physica 18 (1952) 749;
fd3x{V/2,~(x)v/2(x)+ v/2(x)V/2,~(x)}, (9)
which in turn implies that i~~(0) is a constant of motion only if curl u is zero which happens to be the case for states involving purely potential flows. It may be noted that eq. (5) is the integral equation postulated by Yamasaki et a!. [4] and from the foregoing derivation it is clear that the convergence of this equation is related directly to the existence of the inverse p~(x).
118
A. 19 (1953) 217; H. Thellung, Ito, Prog. Physica Theor. Phys. 9(1953)117; J.M. Ziman, Proc. Roy. Soc. London A219 (1953) 257. [3] D.H. Kobe and G.C. Coomer, Phys. Rev. 7 (1973) 1312. [4] S. Yamasaki, T. Kebukawa and S. Sunakawa, Prog. Phys. [5] Theor. R. Fanelli and53(1975)1243. R.E. Struzynski, Phys. Rev. 182 (1969) 363 [6] D.D.H. Yee, Phys. Rev. 184 (1969) 196. [71 B.B. Varga and 5G. Eckstein, J. Low. Temp. Phys. 4 (1971) 563. [8] R.F. Dashen and D.H. Sharp, Phys. Rev. 165 (1968) 1857. [9] R. Vasudevan, R. Sridhar and N.R. Ranganathan, Phys. Lett. 29A (1969) 138.
PHYSICS LETTERS
18 April 1977
ON THE DESCRIPTION OF ROTATIONAL FLOW IN QUANTUM HYDRODYNAMICS R. SRIDHAR Department of Mathematics, P.S. G. Arts College, Coimbatore 641014, India Received 11 February 1977 3x VM(x) is a multiple of It is proved that the vanishing of the velocity-velocity commutator is possible only if f d the identity or a constant of motion; equivalence of Landau’s approach to that of Yamasaki et a!. is also discussed.
Since the quantization of classical hydrodynamics first performed by Landau [1] several attempts have been made to formulate a microscopic theory by constructing a velocity field as the canonical conjugate of the density fluctuation operator [2—41 It has been .
p(x). Defining j5(x) = p(x) (p) and taking the Fourier transform of (3) we have, —
17j I ~i
I U
=
—(
0~q,~ argued that the velocity operator as proposed by Landau is not suitable to describe rotational flow smce the velocity-velocity commutator is zero implying thereby a vanishing curl for the velocity operator [4, 51. Still, it is not clear whether the components of velocity operator commute [6, 7]. It is the purpose of this letter to establish a criterion for the vanishing of the velocity-velocity commutator; incidentally, we also demonstrate the equivalence of Landau’s definition of the velocity operator to that of Yamasaki et al.
[41.
X
~
)
I
+
1_( )1r+
~+
v
.i’1’1’
Q1Q2
Q,O
~
~
~
~
~
(p+q—Q
X
r1
~°
-Q —...—Q)
~
~
4
~l~’~oW2j P’~r with ~4(p) = fd3x exp(—ip-x)v,~(x) and -~
Consider the direct definition of
the velocity opera-
tor:
1(x)}, (1) v,2(x) p_1(x)J~(x)+J~(x)p where ~ = 1, 2, 3 denotes the spatial components of u(x) and p(x) and J~(x)are respectively the mass and current density operators satisfying the non-relativistic current algebra [8]. Using the commutation relation [9] ,
2(x)
a
~6(x—y)p(x)},(2)
~—
[p’(x),J~(y)] = ip~ it is easy to verify [v/2 (x) v (y)] ‘
=
/1
—i~(x y) V —
/2,j’
(x) p~(x)
‘
with V/2,0(x) =
au (x)
a~
av (x)
a° ~
Expand p~(x)about the equilibrium value (p) of
(3)
X~,0(p)= p~ v~(p) resulting in the integral equation (p) [~Y~(p),i~~(q)] = —X~,(p + q) —
~
—
[iyp
—
Q/2),ii~,(q— Q/2)] ~(Q).
(5)
QO
Thus the velocity-velocity commutator is zero only if X, low from non-relativistic current algebra. the fol2,~= the 0; though theof vanishing of X~,0pFrom does fact that right side eq. (4) involves andnot q only as (p + q), it follows that [~ 2(p),I~,(q)]will be zero only [I~(p),zero, iJjq)] will be zero if [17,jp + q), u0(0)jif equals a possibility thatonly exists if either ~Y0(0) is a multiple of the identity or a constant of motion, since the set {p(x), J~(x)}forms a complete set of current operators. It is readily seen that IY~(0)commutes with p(x) while, 117
Volume 61A, number 2
PHYSICS LETTERS
[J/2(x), ~(0)]= —iflV/2V(x).
(8)
18 April 1977
The author thanks the management and the P.5G. Arts College for encouragement.
Principal of
On using Schur’s lemma, it is observed that I~(0) is a multiple of the identity only if V/2 = 0; that is if (V X u) vanishes. Also,
References
[H, ~~(o)1
[1] L.D. Landau, J. Phys. (U.S.S.R) 5 (1941) 7.
~,
3 =
—ih
~
[2] R. Kronig and A. Thellung, Physica 18 (1952) 749;
fd3x{V/2,~(x)v/2(x)+ v/2(x)V/2,~(x)}, (9)
which in turn implies that i~~(0) is a constant of motion only if curl u is zero which happens to be the case for states involving purely potential flows. It may be noted that eq. (5) is the integral equation postulated by Yamasaki et a!. [4] and from the foregoing derivation it is clear that the convergence of this equation is related directly to the existence of the inverse p~(x).
118
A. 19 (1953) 217; H. Thellung, Ito, Prog. Physica Theor. Phys. 9(1953)117; J.M. Ziman, Proc. Roy. Soc. London A219 (1953) 257. [3] D.H. Kobe and G.C. Coomer, Phys. Rev. 7 (1973) 1312. [4] S. Yamasaki, T. Kebukawa and S. Sunakawa, Prog. Phys. [5] Theor. R. Fanelli and53(1975)1243. R.E. Struzynski, Phys. Rev. 182 (1969) 363 [6] D.D.H. Yee, Phys. Rev. 184 (1969) 196. [71 B.B. Varga and 5G. Eckstein, J. Low. Temp. Phys. 4 (1971) 563. [8] R.F. Dashen and D.H. Sharp, Phys. Rev. 165 (1968) 1857. [9] R. Vasudevan, R. Sridhar and N.R. Ranganathan, Phys. Lett. 29A (1969) 138.