Vector and Tensor operators in quantum mechanics Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: November 24. 2015)
We discuss the properties of vector and tensor operators under rotations in quantum mechanics. We use the analogy from the classical physics, where vector and tensor of a quantity with three components that transforms by definition like
Vi ' ijV j Tij ' ik jlTikl j k ,l under rotation. First I will present a brief review in the case of classical physics. After that the properties of vector and tensor operators under rotation in quantum mechanics. The irreducible spherical tensor operators are introduced based on the rotation operators. The Cartesian tensors are decomposed into irreducible spherical tensors. The Wigner – Eckart theorem will be discussed elsewhere.
1. Definition of vector in classical physics (i) 2D rotation Suppose that the vector r is rotated through (counter-clock wise) around the z axis. The position vector r is changed into r' in the same orthogonal basis {e1, e2}.
y
r'
e 2 r e ' 2 x2
e1'
f x2 x1 f x x1 e1
In this Fig, we have
e e ' cos e e ' sin 1 1 , 2 1 e1 e2 ' sin e2 e2 ' cos
Spherical Harmonics as rotator matrices 1 9/3/2017
We define r and r' as
r' x 'e x 'e x e ' x e ' 1 1 2 2 1 1 2 2 ,
and
r x e x e 1 1 2 2 .
Using the relation
e r' e (x 'e x 'e ) e (x e 'x e ') 1 1 1 1 2 2 1 1 1 2 2 e2 r' e2 (x1'e1 x2 'e2 ) e2 (x1e1'x2e2 ')
we have
x ' e (x e 'x e ') x cos x sin 1 1 1 1 2 2 1 2 x2 ' e2 (x1e1'x2e2 ') x1 sin x2 cos or
x1 ' x1 11 12 x1 cos sin x1 () x2 ' x2 21 22 x2 sin cos x2
where
ij ei e j '
(ii) 3D rotation Next we discuss the three-dimensional (3D) case,
3 3 3 r x j e j , r' x j 'e j x j e j ' j1 j1 j1
Note that
ei (ei e j ')e j ' ij e j ' j j
ei ' (ei 'e j )e j (e j ei ')e j ji e j j j j
where
Spherical Harmonics as rotator matrices 2 9/3/2017
ij ei e j ' Then we get
3 3 3 r' xi 'ei x j e j ' x j ij ei ij x j ei i1 j1 ji1 i, j
or
xi ' ij x j j
or
x ' x 1 11 12 13 1 r' x2 ' z ()r 21 22 23 x2 x3 ' 31 32 33 x3 where
cos sin 0 z () sin cos 0 0 0 1
Using the relation r' z ()r , we can write down
3 3 3 r' z ()r z ()( x j e j ) x jz ()e j x j e j ' j1 j1 j1
where
z ()e j e j ' , ei z ()e j ei e j ' ij
We note that
ijik (ei e j ')(ei ek ') e j 'ek ' j,k i i
since
e j ' (ei e j ')ei (formula) i
Spherical Harmonics as rotator matrices 3 9/3/2017
______Now we consider more general case in order to get the definition of vector.
x3
P
A
Q
x2
x1
Suppose that
OP yi ei , OP' yi 'ei yiei ' i i i
OQ zi ei , OQ' zi 'ei zi ei ' i i i
Then we have
A PQ OQ OP (zi yi )ei i
A' P'Q' OQ' OP'
(zi ' yi ')ei i
(zi yi )ei ' i
Since
e j ' (ei e j ')ei ij ei , i j
Spherical Harmonics as rotator matrices 4 9/3/2017 the expression of A can be rewritten as
(zi ' yi ')ei (z j y j )e j ' (zi yi )ij ei . i j i, j
Therefore we get
zi ' yi ' ij (z j y j ) . j
Since the component of A is given by
Ai zi yi , and Ai ' zi 'yi ' in the old and new co-ordinate systems, we can write
Ai ' Rij Aj . j
We note that
Ai 'ei Aie j ' i i
In summary, under the rotation of the co-ordinate system, the components of the vector are transformed through
Ai ' ij Aj j where the vector is a tensor of rank 1. Similarly, the components of the tensor are transformed through
, for the tensor of rank 2 Ti j ' ik jlTkl j and
for the tensor of rank 3 Ti jk ' il jmknTlmn j
((Note)) The scalar (tensor of rank 0) is invariant under the rotation.
Spherical Harmonics as rotator matrices 5 9/3/2017 A'B' Ai B j (ei 'e j ') Ai B j i, j Ai Bi A B i, j i, j i ______2. Vector operators in quantum mechanics The operators corresponding to various physical quantities will be characterized by their behavior under rotation as scalars, vectors, and tensors.
Vi ijV j j
Note that is the rotation matrix. For the rotation by the angle around the z axis
cos sin 0 cos sin 0 z () sin cos 0 , , az ( ) z () sin cos 0 0 0 1 0 0 1
We assume that the state vector changes from the old state to the new state ' .
' Rˆ or
' Rˆ
A vector operator Vˆ for the system is defined as an operator whose expectation is a vector that rotates together with the physical system.
ˆ ˆ 'Vi ' ij Vj j or
ˆ ˆ ˆ ˆ R Vi R ijVj j or
ˆ ˆ ˆ ˆ Vi RijV j R j
((Note)) We show that
Spherical Harmonics as rotator matrices 6 9/3/2017 ˆ ˆ ˆ ˆ RV j R Viij i
From the relation
ˆ ˆ ˆ ˆ ˆ ˆ ˆ Vi RijV j R ij RV j R j j
we get
ˆ ˆ ˆ ˆ Viik ijik RV j R i i, j ˆ ˆ ˆ RV j R ijik j i ˆ ˆ ˆ jk RV j R j ˆ ˆ ˆ RVk R
or
ˆ ˆ ˆ ˆ RV j R Viij i
______We now consider a special case, infinitesimal rotation.
i Rˆ 1ˆ Jˆ n
i Rˆ 1ˆ Jˆ n
ˆ i ˆ ˆ ˆ i ˆ ˆ (1 J n)Vi (1 J n) ijV j j
ˆ i ˆ ˆ ˆ Vi [Vi ,J n] ijV j j where
ˆ ˆ ˆ ˆ ˆ ˆ V1 Vx , V2 Vy , V3 Vz
For n = ez,
Spherical Harmonics as rotator matrices 7 9/3/2017
1 0 z ( ) 1 0 0 0 1
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V1 [V1, J z ] 11V1 12V2 13V3 V1 V2
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V2 [V2 , J z ] 21V1 22V2 23V3 V1 V2
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ V3 [V3, J z ] 31V1 32V2 33V3 V3 or
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V1, J z ] iV2 , [V2 , J z ] iV1 , [V3 , J z ] 0
For n = ex,
1 0 0 x ( ) 0 1 0 1
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ V1 [V1, J x ] 11V1 12V2 13V3 V1
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V2 [V2 , J x ] 21V1 22V2 23V3 V2 V3
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V3 [V3, J x ] 31V1 32V2 33V3 V2 V3
or
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V1, J x ] 0 , [V2 , J x ] iV3 , [V3, J x ] iV2
For For n = ey,
Spherical Harmonics as rotator matrices 8 9/3/2017 1 0 y ( ) 0 1 0 0 1
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V1 [V1, J y ] 11V1 12V2 13V3 V1 V3
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ V2 [V2 , J y ] 21V1 22V2 23V3 V2
ˆ i ˆ ˆ ˆ ˆ ˆ ˆ ˆ V3 [V3, Jyx ] 31V1 32V2 33V3 V1 V3
or
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V1,J y ] iV3 , [V2, J y ] 0, [V3, J y ] iV1
Using the Levi-Civita symbol, we have
ˆ ˆ ˆ [Vi , J j ] iijkVk
We can use this expression as the defining property of a vector operator.
Levi-Civita symbol:ijk
123=231=312=1
132=213=321=-1
all other ijk. =0.
((Example))
When Vˆ Jˆ,
ˆ ˆ ˆ [Ji , J j ] iijk Jk
When Vˆ rˆ and Aˆ rˆ ,
ˆ [xˆi ,Lj ] iijk xˆk
Spherical Harmonics as rotator matrices 9 9/3/2017 When Aˆ pˆ,and Aˆ rˆ ,
ˆ [pˆi ,Lj ] iijk pˆk
((Mathematica))
Signature[i,j,k], which is equal to the Levi-Civita ijk .
((Note)) Speherical component We define the spherical components as
1 1 Vˆ (Vˆ iVˆ ) Vˆ , 1 2 x y 2
ˆ 1 ˆ ˆ 1 ˆ ˆ ˆ V (V iV ) V , V0 Vz 1 2 x y 2
We note that
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J x ] 0 , [V y , J x ] iVz , [Vz , J x ] iV y
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J y ] iVz , [Vy , J y ] 0, [Vz , J y ] iVx
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J z ] iVy , [Vy , J z ] iVx , [Vz , J z ] 0
We introduce the operators as
ˆ ˆ ˆ ˆ ˆ ˆ V Vx iVy , J J x iJ y
Using the above relation, we get
ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J x ] [Vx iVy , J x ] i[Vy , J x ] Vz
ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J x ] [Vx iVy , J x ] i[Vy , J x ] Vz
ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J y ] [Vx iVy , J y ] [Vx , J y ] iVz
ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J y ] [Vx iVy , J y ] [Vx , J y ] iVz
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J z ] [Vx iVy , J z ] [Vx , J z ] i[Vy , J z ] V
Spherical Harmonics as rotator matrices 10 9/3/2017
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [V , J z ] [Vx iVy , J z ] [Vx , J z ] i[Vy , J z ] V
These relations in turn can be shown to lead to
ˆ ˆ ˆ ˆ [V , J z ] V , [V , J z ] V
ˆ ˆ ˆ ˆ [V , J ] 0 , [V , J ] 0 ,
ˆ ˆ ˆ ˆ ˆ ˆ [V , J ] 2Vz , [V , J ] 2Vz ,
Using the above relations, we get
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [J z ,V1,] V1 , [J z ,V1] V1 , [J z ,V0 ] 0
ˆ ˆ ˆ ˆ [J ,V1] 0 , [J ,V1,] 0 ,
ˆ ˆ ˆ ˆ ˆ [J1,V1] 2V0 , [J ,V1] 2V0 ,
These satisfy the following relations
ˆ ˆ ˆ [J z ,Vq ] qVq
ˆ ˆ ˆ [J ,Vq ] 2 q(q 1)Vq1
where q = 0, 1.
3. Example for the spin 1/2 system We discuss the validity of the above formula for the spin 1/2 system.
The angular momentum operator
ˆ 0 1 ˆ 0 i ˆ 1 0 J x , J y , J z 2 1 0 2 i 0 2 0 1
The rotation operator
Spherical Harmonics as rotator matrices 11 9/3/2017 i cos i sin Rˆ ( ) exp( Jˆ ) 2 2 , x x i sin cos 2 2
i cos sin Rˆ ( ) exp( Jˆ ) 2 2 y y sin cos 2 2
i ei / 2 0 Rˆ ( ) exp( Jˆ ) z z i / 2 0 e
0 ei Rˆ ( )Jˆ Rˆ ( ) z x z i 2 e 0
0 iei Rˆ ( )Jˆ Rˆ ( ) z y z i 2 ie 0
1 0 ˆ ˆ ˆ Rz ( )J z Rz ( ) 2 0 1
The rotation matrix:
cos sin 0 z ( ) sin cos 0 , 0 0 1
0 ei Rˆ ( )Jˆ Rˆ ( ) Jˆ Jˆ Jˆ z x z 11 x 12 y 13 z i 2 e 0
0 iei Rˆ ( )Jˆ Rˆ ( ) Jˆ Jˆ Jˆ z y z 21 x 22 y 23 z i 2 ie 0
1 0 ˆ ˆ ˆ ˆ ˆ ˆ Rz ( )J z Rz ( ) 31J x 32 J y 33 J z 2 0 1
4. Example for the spin 1 system The angular momentum operator
Spherical Harmonics as rotator matrices 12 9/3/2017
0 1 0 0 i 0 1 0 0 ˆ ˆ ˆ J x 1 0 1 , J y i 0 i , J z 0 0 0 2 2 0 1 0 0 i 0 0 0 1
The rotation operator
1 cos isin 1 cos 2 2 2 i isin i sin Rˆ ( ) exp( Jˆ ) cos , x x 2 2 1 cos isin 1 cos 2 2 2
1 cos sin 1 cos 2 2 2 i sin sin Rˆ ( ) exp( Jˆ ) cos y y 2 2 1 cos sin 1 cos 2 2 2
ei 0 0 ˆ i ˆ Rz ( ) exp( J z ) 0 1 0 i 0 0 e
0 ei 0 ˆ ˆ ˆ i i Rz ( )J x Rz ( ) e 0 e 2 i 0 e 0
0 iei 0 ˆ ˆ ˆ i i Rz ( )J y Rz ( ) ie 0 ie 2 i 0 ie 0
1 0 0 ˆ ˆ ˆ Rz ( )J z Rz ( ) 0 0 0 0 0 1
The rotation matrix:
Spherical Harmonics as rotator matrices 13 9/3/2017 cos sin 0 z ( ) sin cos 0 0 0 1
0 ei 0 ˆ ˆ ˆ ˆ ˆ ˆ i i Rz ( )J x Rz ( ) 11J x 12 J y 13 J z e 0 e 2 i 0 e 0
0 iei 0 ˆ ˆ ˆ ˆ ˆ ˆ i i Rz ( )J y Rz ( ) 21J x 22 J y 23 J z ie 0 ie 2 i 0 ie 0
1 0 0 ˆ ˆ ˆ ˆ ˆ ˆ Rz ( )J z Rz ( ) 31J x 32 J y 33 J z 0 0 0 0 0 1
4 Scalar (tensor of rank zero)
The scalar is invariant under the rotation. In quantum mechanics, we may write
' Aˆ ' B' Aˆ .B where B denotes an arbitrary vector and also rotates with the system into the vector B'. This can be rewritten as
(Rˆ Aˆ Rˆ) B' Aˆ B or
Aˆ B' RˆAˆ Rˆ B
Then we have
ˆ ˆ ˆ ˆ Ai Bi ' (RAj R )Bj i j
Using the relation
Bi ' ij B j j
Spherical Harmonics as rotator matrices 14 9/3/2017 we get
ˆ ˆ ˆ ˆ ( Aiij )B j (RAj R )B j ji j or
ˆ ˆ ˆ ˆ RAj R Aiij i
5. Cartesian tensor operators The standard definition of a Cartesian tensor is that each of its suffix transforms under the rotation as do the components of an ordinary 3D vector. The Cartesian tensor operator is defined by
ˆ ' Tij ' ik jl Tkl k,l under the rotation specified by the 3x3 orthogonal matrix .
ˆ ˆ ˆ ˆ Tij Rik jlTkl R k,l
The simplest example of a Cartesian tensor of rank 2 is a dyadic formed out of two vectors Uˆ and Vˆ .
ˆ ˆ ˆ Tij UiVj
ˆ ˆ where Ui and Vi are the components of ordinary 3D vector operators. There are nine components: 1+3+5 = 9. This Cartesian tensor is reducible. It can be decomposed into the three parts.
Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ i j j i ( i j j i ) i j 3 ij 2 2 3 ij
The first term on the right-hand side, Uˆ Vˆ is a scalar product invariant under the rotation (corresponding to j = 0)
The second is an anti-symmetric tensor which can be written as
ˆ ˆ ijk (U V )k
There are 3 independent components (corresponding to j = 1).
Spherical Harmonics as rotator matrices 15 9/3/2017
0 A A 12 13 A12 0 A23 A13 A23 0 where
Uˆ Vˆ Uˆ Vˆ Aˆ i j j i ij 2
The third term is a 3x3 symmetric traceless tensor with 5 independent components (=6- 1), where 1 comes from the traceless condition (corresponding to j = 2).
S S S 11 12 13 S12 S22 S23 S13 S23 S33
with S11 S22 S33 0 , where
Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ S i j j i ij 2 3 ij
ˆ ˆ ˆ In conclusion, the Cartesian tensor Tij UiVj can be decomposed into irreducible spherical tensors of rank 0, 1, and 2. The Cartesian tensors are not very suitable for studying transformations under rotations, because they are reducible whenever their rank exceeds 1. It is therefore interesting to consider irreducible spherical tensors.
7. Irreducible spherical tensor Notice the numbers of elements of these irreducible subgroups: 1, 3, and 5. These are exactly the numbers of elements of angular momentum representations for j = 0, 1, and 2.
The first term is trivial: the scalar by definition is not affected by rotation, and neither is an j = 0 state.
To deal with the second and third terms, we introduce tensor operators having three and five components, such that under rotation these sets of components transform among themselves just as do the sets of eigenkets of angular momentum in the j = 1 and j = 2 representation, respectively.
m m Suppose we take a spherical harmonics Yl (,) Yl (n) , where the orientation of the unit vector n is characterized by and .
Spherical Harmonics as rotator matrices 16 9/3/2017 We now replace n by some vector Vˆ . Then we have a spherical tensor of rank k (in place of l) with magnetic quantum number (in place of m).
(k) mq ˆ Tq Ylk (V ) where
m mq ˆ q ˆ Yl (n) Yl k (V ) Tk (V )
The quantity
l m Pl,m (x, y, z) r Yl ( ,) is a homogeneous polynomial of order l.
4 The quantity P (x, y, z) rY q (,) is a first order homogeneous polynomial in x, y, 1,q 3 1 and z.
(i)
4 x iy P (x, y, z) rY 1(,) 1,1 3 1 2
Vˆ iVˆ T (1) x y 1 2
(ii)
4 P (x, y, z) rY 0 (,) z 1,0 3 1
(1) ˆ T0 Vz
(iii)
4 x iy P (x, y, z) rY 1(,) 1,1 3 1 2
Vˆ iVˆ T (1) x y 1 2
Spherical Harmonics as rotator matrices 17 9/3/2017
The above example is the simplest nontrivial example to illustrate the reduction of a Cartesian tensor into irreducible spherical tensors.
(k) mq ˆ Tq Ylk (V)
n' Rˆ n n
n' n Rˆ n
Using the closure relation,
Rˆ l,m l,m' l,m' Rˆ l,m m'
n' l, m n Rˆ l, m n l, m' l, m' Rˆ l, m m' or
m m' ˆ * Yl (n') Yl (n) l,m R l,m' m'
(l) ˆ ˆ Dm,m' (R) l,m R l,m'
m ˆ If there is an operator that acts like Yl (V ) , it is then reasonable to expect
ˆ m ˆ ˆ m' ˆ ˆ * m' ˆ (l) * ˆ R Yl (V )R Yl (V ) l,m R l,m' Yl (V )Dmm' (R) m' m' from the relation
m ˆ m ˆ ˆ m' m' ˆ Yl (n') R Yl (V )R , Yl (n) Yl (V )
ˆ (k ) We define a spherical tensor operator of rank k as a set of 2k+1, Tq , q = k, k-1,…, -k such that under rotation they transform like a set of angular momentum eigenkets,
k ˆ ˆ (k ) ˆ (k )* ˆ ˆ (k ) R Tq R Dqq' (R)Tq' , (1) q'k where
Spherical Harmonics as rotator matrices 18 9/3/2017 ˆ (k) mq ˆ Tq Yl k (V ) [q = k, k-1, ….., -k+1, -k; (2k+1) components)]
(k) ˆ ˆ Dqq' (R) k,q R k,q' or
(k )* ˆ ˆ * ˆ Dqq' (R) k,q R k,q' k,q' R k,q with q = k, k-1,…, -k. This can be rewritten as
k ˆ ˆ (k ) ˆ (k ) ˆ ˆ (k ) R Tq R Dq'q (R )Tq' q'k
The switching of Rˆ Rˆ leads to another expression
k ˆ ˆ (k ) ˆ (k ) ˆ ˆ (k ) RTq R Dq'q (R)Tq' (2) q'k
Considering the infinitesimal form of the expression (1), we have
ˆ ˆ k ˆ ˆ iJ n ˆ (k ) ˆ iJ n ˆ (k ) ˆ iJ n (1 )Tq (1 ) Tq' k,q'1 k,q q'k or
k ˆ (k ) i ˆ ˆ (k ) ˆ (k ) i ˆ (k ) ˆ Tq [J n,Tq ] Tq Tq' k, q' J n k, q q'k or
k ˆ (k ) ˆ (k ) ˆ [J n,Tq ] Tq' k, q' J n k, q q'k
In general, we have
k ˆ (k ) ˆ (k ) ˆ [J,Tq ] Tq' k,q' J k,q q'k
For n = ex, ey, ez
Spherical Harmonics as rotator matrices 19 9/3/2017 k ˆ (k ) ˆ (k ) ˆ ˆ (k ) [J z ,Tq ] Tq' k,q' J z k,q qTq (3) q'k
k ˆ (k ) ˆ (k ) ˆ [J x ,Tq ] Tq' k,q' J x k,q q'k
k ˆ (k ) ˆ (k ) ˆ [J y ,Tq ] Tq' k,q' J y k,q q'k
k ˆ (k ) ˆ (k ) ˆ ˆ (k ) [J ,Tq ] Tq' k,q' J k, q (k q)(k q 1)Tq1 (4) q'k where
ˆ J k,q (k q)(k q 1) k,q 1
These two commutation relations can also be taken as a definition of a spherical tensor of rank k.
We now consider
k ˆ ˆ (k ) ˆ (k ) ˆ ˆ (k ) RTq R Dq'q (R)Tq' q'k
This equation is rewritten as
k ˆ ˆ (k ) (k ) ˆ ˆ (k ) ˆ RTq Dq'q (R)Tq' R q'k
Then have
k ˆ ˆ (k ) (k ) ˆ ˆ (k ) ˆ RTq j,m Dq'q (R)Tq' j,m' j,m' R j,m qm''k
k (k ) ˆ ( j) ˆ ˆ (k ) Dq'q (R)Dm'm (R)Tq' j,m' q'k m'
(a) Spherical tensor operator of rank 1 The spherical tensor operator of rank 1 is related to the vector operator by the relation,
Spherical Harmonics as rotator matrices 20 9/3/2017 Vˆ iVˆ Vˆ iVˆ Tˆ (1) x y , Tˆ (1) Vˆ , Tˆ(1) x y 1 2 0 z 1 2
The vector operator Vˆ satisfies the commutation relation.
ˆ ˆ ˆ [Vi , J j ] iijkVk
Using this relation, we can show that
Vˆ iVˆ ˆ ˆ (1) ˆ x y 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ (1) [J z ,T1 ] [J z , ] [Vx iVy , J z ] (iVy Vx ) T1 2 2 2
Vˆ iVˆ ˆ ˆ (1) ˆ x y 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ (1) [J z ,T1 ] [J z , ] [Vx iVy , J z ] (iVy Vx ) T1 2 2 2
ˆ ˆ (1) ˆ ˆ [J z ,T0 ] [J z ,Vz ] 0 where
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Vx , J z ] i132Vy iVy , [Vy , J z ] i231Vx iVx .
((Note)) Using
Vˆ iVˆ Vˆ iVˆ Vˆ x y , Vˆ Vˆ , Vˆ x y 1 2 0 z 1 2 the vector operator Vˆ van be expressed as
1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ V Vx e x Vy e y Vz ez V1u1 V0u0 V1u1 (1) V u 1 where
e ie e ie u x y , u x y , u e . 1 2 1 2 0 z
The orthonormality relation of these new unit vectors is
u u' ,' (1)
Spherical Harmonics as rotator matrices 21 9/3/2017
Thus the ordinary scalar product of two vectors has the form
ˆ ˆ ' ˆ V W (1) VW ' (u u ' ) , ' ˆ (1) VW , ˆ ˆ ˆ ˆ ˆ ˆ V0W0 (V1W1 V1W1)
(b) Spherical tensor operator of rank 2
Spherical tensor of rank 2
(i)
2 0 P2,0 (x, y, z) r Y2 ( ,) 5 (3z 2 r 2 ) 16 (x iy)(x iy) (x iy)(x iy) [2z 2 ] 15 2 2 8 6 which corresponds to
2 15 Tˆ (1)Tˆ (1) 2Tˆ (1) Tˆ (1)Tˆ (1) Tˆ (2) 1 1 0 1 1 0 8 6
(ii)
15 x iy P (x, y, z) r 2Y 2 (,) ( )2 2,2 2 16 2 which corresponds to
ˆ (2) 15 ˆ (1) 2 T2 T1 16
(iii)
15 z(x iy) (x iy)z P (x, y, z) r 2Y 1(,) 2,1 2 8 2 2
Spherical Harmonics as rotator matrices 22 9/3/2017 which corresponds to
15 Tˆ (1)Tˆ (1) Tˆ (1)Tˆ (1) Tˆ (2) 0 1 1 0 1 8 2
(iv)
15 z(x iy) (x iy)z P (x, y, z) r 2Y 1(,) 2,1 2 8 2 2 which corresponds to
15 Tˆ (1)Tˆ (1) Tˆ (1)Tˆ (1) Tˆ (2) 0 1 1 0 1 8 2
8 Product of tensors (CG)
((Theorem)) (k ) (k ) Let Xˆ 1 and Zˆ 2 be irreducible spherical tensors of rank k1 and k2, respectively. q1 q2 Then
Tˆ (k ) k ,k ;q ,q k ,k ;k,q Xˆ (k1 )Zˆ (k2 ) q 1 2 1 2 1 2 q1 q2 q1 ,q2 is a spherical (irreducible) tensor of rank k, where
k1,k2;q1,q2 k1,k2;k,q is the Clebsch-Gordan (CG) coefficient.
((Proof))
Spherical Harmonics as rotator matrices 23 9/3/2017 RˆTˆ (k ) Rˆ k ,k ;q ,q k ,k ;k,q (RˆXˆ (k1) Rˆ )(RˆZˆ (k2 ) Rˆ ) q 1 2 1 2 1 2 q1 q2 q1,q2 k ,k ;q ,q k ,k ;k,q D(k1) (Rˆ)Xˆ (k1) D(k2 ) (Rˆ)Zˆ (k2 ) 1 2 1 2 1 2 q1q1' q1' q2q2 ' q2 ' q1,q2 ,q1',q2 ' k ,k ;q ,q k ,k ;k,q k ,k ;q ',q ' k ,k ;k",q' k ,k ;q ,q k ,k ;k",q" D(k") (Rˆ)Xˆ (k1)Zˆ (k2 ) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 q'q" q1' q2 ' q1,q2 ,q1',q2 ' k",q",q' k ,k ;q ',q ' k ,k ;k",q' D(k") (Rˆ)Xˆ (k1 )Zˆ (k2 ) k ,k" q,q" 1 2 1 2 1 2 q'q" q1' q2 ' q1',q2 ' k",q",q' k ,k ;q ',q ' k ,k ;k,q' Xˆ (k1)Zˆ (k2 ) D(k ) (Rˆ) 1 2 1 2 1 2 q1' q2 ' q'q q',q1',q2 ' (k ) ˆ (k ) Dq'q (R)Tq' q' where
ˆ ˆ (k1 ) ˆ (k1 ) ˆ ˆ (k1 ) RX q R D ' (R)X q ' 1 q1q1 1 q1 '
RˆZˆ (k2 ) Rˆ D(k2 ) (Rˆ)Zˆ (k2 ) q2 q2 q2 ' q2 ' q2 '
D(k1 ) (Rˆ)D(k2 ) (Rˆ) k ,k ;q ',q ' k ,k ;k",q' k ,k ;q ,q k ,k ;k",q' D(k") (Rˆ) q1q1 ' q2q2 ' 1 2 1 2 1 2 1 2 1 2 1 2 q'q" k"q"q'
(Clebsch-Gordan series) where
D D D D ... D , k1 k1 k1 k2 k1 k2 1 k1 k2 or
k" k1 k2 ,k1 k2 1,...,| k1 k2 |.
Orthogonality of the Clebsch-Gordan coefficient
j , j ;m ,m j , j ; j,m j , j ;m ',m ' j , j ; j,m 1 2 1 2 1 2 1 2 1 2 1 2 m1 ,m1 ' m2 ,m2 ' j,m and
Spherical Harmonics as rotator matrices 24 9/3/2017 j1, j2;m1,m2 j1, j2; j,m j1, j2;m1,m2 j1, j2; j',m' j, j' m,m' m1 ,m2
9 Tensor of rank 2 from CG In the case of
D1 D1 D2 D1 D0
we consider the case k = 2 for k1=1 (q1 = 1, 0, -1), and k2 = 1 (q2 = 1, 0, -1):
D1 D1 D2
We use the values of CG.
ˆ (2) ˆ (1) (1) T2 X1 Z1 ˆ ˆ U1V1 1 (Uˆ iUˆ )(Vˆ iVˆ ) 2 x y x y 1 i (Uˆ Vˆ Uˆ Vˆ ) (Uˆ Vˆ Uˆ Vˆ ) 2 x x y y 2 x y y x
Xˆ (1)Zˆ (1) Xˆ (1)Zˆ (1) Tˆ (2) 1 0 0 1 1 2 Uˆ Vˆ Uˆ Vˆ 1 0 0 1 2 Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ z x x z i y z z y 2 2
Spherical Harmonics as rotator matrices 25 9/3/2017 Xˆ (1)Zˆ (1) 2Xˆ (1)Zˆ (1) Xˆ (1)Zˆ (1) Tˆ (2) 1 1 0 0 1 1 0 6 Uˆ Vˆ 2Uˆ Vˆ Uˆ Vˆ 1 1 0 0 1 1 6
Uˆ Vˆ Uˆ Vˆ 2Uˆ Vˆ x x y y z z 6 1 (Uˆ Vˆ 3Uˆ Vˆ ) 6 z z
Xˆ (1)Zˆ (1) Xˆ (1)Zˆ (1) Tˆ (2) 0 1 1 0 1 2 Uˆ Vˆ Uˆ Vˆ 0 1 1 0 2 Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ Uˆ Vˆ z x x z i y z z y 2 2
ˆ (2) ˆ (1) ˆ (1) T2 X 1 Z1 ˆ ˆ U 1V1 1 (Uˆ iUˆ )(Vˆ iVˆ ) 2 x y x y
ˆ ˆ When Ui Vi xˆi (in a special case), we have
1 1 Tˆ ( 2 ) ( xˆ iyˆ )( xˆ iyˆ ) ( xˆ 2 yˆ 2 ) ixˆyˆ 2 2 2
ˆ ( 2 ) T1 xˆzˆ iyˆzˆ
xˆ 2 yˆ 2 2 zˆ 2 ˆ ( 2 ) T0 6
ˆ ( 2 ) T1 xˆzˆ iyˆzˆ
1 1 Tˆ ( 2 ) ( xˆ iyˆ )( xˆ iyˆ ) ( xˆ 2 yˆ 2 ) ixˆyˆ 2 2 2
Spherical Harmonics as rotator matrices 26 9/3/2017 Therefore we have the following relations.
2 2 ˆ (2) ˆ (2) xˆ yˆ T2 T2
Tˆ (2) Tˆ (2) xˆyˆ 2 2 2i
Tˆ (2) Tˆ (2) yˆzˆ 1 1 2i
Tˆ (2) Tˆ (2) xˆzˆ 1 1 2
xˆ2 yˆ 2 2zˆ2 ˆ (2) T0 6
((Note)) We use the commutation relations;
[xˆ, yˆ] 0 , [ yˆ, zˆ] 0 .
10 Tensor of rank 1 from CG We consider the case k = 1 for k1=1 (q1 = 1, 0, -1), and k2 = 1 (q2 = 1, 0, -1):
D1 D1 D1
Spherical Harmonics as rotator matrices 27 9/3/2017 X (1)Z (1) X (1)Z (1) T (1) 1 0 0 1 1 2 Uˆ Vˆ Uˆ Vˆ 1 0 0 1 2 1 (Uˆ Vˆ Uˆ Vˆ ) (Uˆ Vˆ Uˆ Vˆ ) i y z z y 2 z x x z 2i 1 1 (Uˆ Vˆ) (Uˆ Vˆ) 2 y 2i x
X (1)Z (1) X (1)Z (1) T (1) 1 1 1 1 0 2 Uˆ Vˆ Uˆ Vˆ 1 1 1 1 2
(Uˆ Vˆ Uˆ Vˆ ) x y y x 2i 1 (Uˆ Vˆ) 2i z
X (1)Z (1) X (1)Z (1) T (1) 0 1 1 0 1 2 Uˆ Vˆ Uˆ Vˆ 0 1 1 0 2 1 (Uˆ Vˆ Uˆ Vˆ ) (U V U V ) y z z y 2 z x x z 2i 1 1 (Uˆ Vˆ) (Uˆ Vˆ) 2 y 2i x
This tensors can be written in terms of its Cartesian components as
1 1 T (1) (C iC ) , T (1) (C iC ) 1 2 x y 1 2 x y
(1) T0 Cz
Then we get
T (1) T (1) (Uˆ Vˆ) 1 1 C x 2 x 2i
Spherical Harmonics as rotator matrices 28 9/3/2017 T (1) T (1) (Uˆ Vˆ) 1 1 C y 2i y 2i
(Uˆ Vˆ) T (1) C z 0 z 2i leading to
(Uˆ Vˆ) (Uˆ Vˆ) (Uˆ Vˆ) C i x , C i y , C i z x 2 y 2 z 2 or
(Uˆ Vˆ ) C i 2
This is, up to a factor, nothing but the ordinary vector product.
11 Tensor of rank 0 from CG We consider the case k = 0 for k1=1 (q1 = 1, 0, -1), and k2 = 1 (q2 = 1, 0, -1):
D1 D1 D0
X (1)Z (1) X (1)Z (1) X (1)Z (1) T (0) 1 1 0 0 1 1 0 3 U V U V U V 1 1 0 0 1 1 3 1 (U V ) 3
______REFERENCES
Spherical Harmonics as rotator matrices 29 9/3/2017 J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, second edition (Addison- Wesley, New York, 2011). Eugen Merzbacher, Quantum Mechanics, third edition (John Wiley & Sons, New York, 1998). Nouredine Zettili, Quantum Mechanics, Concepts and Applications, 2nd edition (John Wiley & Sons, New York, 2009). K. Gottfried and T-M. Yan, Quantum Mechanics: Fundamentals 2nd edition (Springer, 2003, New York) M. Tinkham Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964). D. Dubbers and H.-J. Stöckmann, Quantum Physics: The Bottom-Up Approach (Springer, 2013). ______APPENDIX We consider the rotation operator for the angular momentum J = ħ.
ˆ i ˆ Rz exp( J z)
ˆ ˆ ˆ ˆ We calculate R Ji R for i = 1 (x), 2 (y), and 3(z), and compare these with ij J j j
cos sin 0 z sin cos 0 0 0 1
Here we present the calculation for J = 3/2 using the Mathematica program.
((Mathematica))
Spherical Harmonics as rotator matrices 30 9/3/2017 Matrices j = 3/2
Clear "Global`" ;j 3 2; exp_ : exp . Complex re_, im_ Complex re, im ; — Jx j_, n_, m_ : j m j m 1 KroneckerDelta n, m 1 2 — j m j m 1 KroneckerDelta n, m 1 ; 2 — Jy j_, n_, m_ : j m j m 1 KroneckerDelta n, m 1 2 — j m j m 1 KroneckerDelta n, m 1 ; 2 Jz j_, n_, m_ : — m KroneckerDelta n, m ; Jx Table Jxj, n, m , n,j,j, 1 , m,j,j, 1 ; Jy Table Jy j, n, m , n,j,j, 1 , m,j,j, 1 ; Jz Table Jz j, n, m , n,j,j, 1 , m,j,j, 1 ; Jx MatrixForm
— 0 3 00 2 3 — 0 — 0 2 0 — 0 3 — 2 003 — 0 2
Spherical Harmonics as rotator matrices 31 9/3/2017 Jy MatrixForm
0 1 3 — 00 2 1 3 — 0 — 0 2 0 — 0 1 3 — 2 001 3 — 0 2
Jz MatrixForm 3 — 00 0 2 0 — 00 2 00 — 0 2 000 3 — 2
Rotation around the z axis
A1z MatrixExp Jz .Jx. MatrixExp Jz Simplify; — — A1z MatrixForm
0 1 3 — 00 2 1 3 — 0 — 0 2 0 — 0 1 3 — 2 001 3 — 0 2
Spherical Harmonics as rotator matrices 32 9/3/2017 A2z MatrixExp Jz .Jy.MatrixExp Jz Simplify; — — A2z MatrixForm
0 1 3 — 00 2 1 3 — 0 — 0 2 0 — 0 1 3 — 2 001 3 — 0 2
A3z MatrixExp Jz .Jz.MatrixExp Jz Simplify; — — A3z MatrixForm
3 — 00 0 2 0 — 00 2 00 — 0 2 000 3 — 2
Mz RotationMatrix , 0, 0, 1 Cos , Sin ,0 , Sin , Cos , 0 , 0, 0, 1 Mz MatrixForm Cos Sin 0 Sin Cos 0 001
Spherical Harmonics as rotator matrices 33 9/3/2017 B1z Mz 1, 1 Jx Mz 1, 2 Jy Mz 1, 3 Jz FullSimplify;
B2z Mz 2, 1 Jx Mz 2, 2 Jy Mz 2, 3 Jz FullSimplify; B3z Mz 3, 1 Jx Mz 3, 2 Jy Mz 3, 3 Jz FullSimplify; A1z B1z Simplify 0, 0, 0, 0 ,0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 A2z B2z FullSimplify 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 A3z B3z FullSimplify 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0
Rotation around the y axis A1y MatrixExp Jy .Jx. MatrixExp Jy Simplify; — — A1y MatrixForm
3 — Sin 1 3 — Cos 00 2 2 1 3 — Cos 1 — Sin — Cos 0 2 2 0 — Cos 1 — Sin 1 3 — Cos 2 2 00 1 3 — Cos 3 — Sin 2 2
Spherical Harmonics as rotator matrices 34 9/3/2017 A2y MatrixExp Jy .Jy.MatrixExp Jy Simplify; — — A2y MatrixForm
0 1 3 — 00 2 1 3 — 0 — 0 2 0 — 0 1 3 — 2 001 3 — 0 2
A3y MatrixExp Jy .Jz.MatrixExp Jy Simplify; — — A3y MatrixForm
3 — Cos 1 3 — Sin 00 2 2 1 3 — Sin 1 — Cos — Sin 0 2 2 0 — Sin 1 — Cos 1 3 — Sin 2 2 00 1 3 — Sin 3 — Cos 2 2 My RotationMatrix , 0, 1, 0 ; My MatrixForm Cos 0Sin 010 Sin 0Cos
Spherical Harmonics as rotator matrices 35 9/3/2017 B1y My 1, 1 Jx M y 1, 2 Jy My 1, 3 Jz FullSimplify;
B2y My 2, 1 Jx My 2, 2 Jy My 2, 3 Jz FullSimplify; B3y My 3, 1 Jx My 3, 2 Jy My 3, 3 Jz FullSimplify; A1y B1y Simplify 1 1 0, 3 — M y 1, 2 ,0,0 , 3 — M y 1, 2 ,0, — M y 1, 2 ,0 , 2 2 1 1 0, — M y 1, 2 ,0, 3 — M y 1, 2 , 0, 0, 3 — M y 1, 2 ,0 2 2 A2y B2y FullSimplify 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 A3y B3y FullSimplify 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0
Rotation around the x axis A1x MatrixExp Jx .Jx. MatrixExp Jx Simplify; — — A1x MatrixForm
— 0 3 00 2 3— 0 — 0 2 0 — 0 3 — 2 003 — 0 2
Spherical Harmonics as rotator matrices 36 9/3/2017 A2x MatrixExp Jx .Jy.MatrixExp Jx Simplify; — — A2x MatrixForm
3 — Sin 1 3 — Cos 00 2 2 1 3 — Cos 1 — Sin — Cos 0 2 2 0 — Cos 1 — Sin 1 3 — Cos 2 2 00 1 3 — Cos 3 — Sin 2 2 A3x MatrixExp Jx .Jz.MatrixExp Jx Simplify; — — A3x MatrixForm
3 — Cos 1 3 — Sin 00 2 2 1 3 — Sin 1 — Cos — Sin 0 2 2 0 — Sin 1 — Cos 1 3 — Sin 2 2 00 1 3 — Sin 3 — Cos 2 2 Mx RotationMatrix , 1, 0, 0 ;Mx MatrixForm 10 0 0 Cos Sin 0 Sin Cos B1x Mx 1, 1 Jx Mx 1, 2 Jy Mx 1, 3 Jz FullSimplify; B2x Mx 2, 1 Jx Mx 2, 2 Jy Mx 2, 3 Jz FullSimplify; B3x Mx 3, 1 Jx Mx 3, 2 Jy Mx 3, 3 Jz FullSimplify; A1x B1x Simplify 0, 0, 0, 0 ,0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 A2x B2x FullSimplify 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 A3x B3x FullSimplify 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0
Spherical Harmonics as rotator matrices 37 9/3/2017