Rotational Invariance and Angular Momentum
In the last chapter on symmetries, rotational invariance was not discussed, not because it is unimportant, but because it is all too important and deserves a chapter on its own. The reason is that most of the problems we discuss involve a single particle
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12.1. Translations in Two Dimensions Although we are concerned mainly with rotations, let us quickly review translations in two dimensions. By a straightforward extension of the arguments that led to Eq. (11.2.14) from Eq. (11.2.13), we may deduce that the generators of infinitesimal translations along the x andy directions are, respectively, Px
Py
coordinate basis
coordinate basis
-tli-
. a ox
(12.1.1)
. a oy
(12.1.2)
-z1i-
In terms of the vector operator P, which represents momentum, (12.1.3) Px and Py are the dot products of P with the unit vector (i or j) in the direction of the translation. Since there is nothing special about these two directions, we conclude
305
306
that in general,
CHAPTER 12
(12.1.4)
is the generator of translations in the direction of the unit vector fl. Finite translation operators are found by exponentiation. Thus T(a), which translates by a, is given by
where
a= a/a.
The ConsistenC}' Test. Let us now ask if the translation operators we have
constructed have the right laws of combination, i.e., if
(12.1.6)
T(b)T(a)=T(a+ b)
or equivalently if (12, 1.7)
This amounts to asking if P, and P, may be treated as c numbers in manipulating the exponentials. The answer is yes, since in view of Eqs. ( 12.1.1) and (12.1.2 ), the operators commute (12.1.8)
[P,,P,]=O
and their q number nature does not surface here. The commutativity of P, and P, reflects the commutativity of translations in the x and y directions. Exercise 12.1.1. * Verify that considering the relation
a· P is the generator of infinitesimal translations along a by
=
VJ(X ..
oa,, y- oa.. )
12.2. Rotations in Two Dimensions Classically, the effect of a rotation ¢ 0 k, i.e., by an angle c/>o about the ;; axis (counterclockwi se in the x y plane) has the following effect on the state of a particle: =[cos ci>o [ xj-+[·~J sm ci>o y y
[p,J [jJ'fi,. l p,
-+ ..
=
[cos rf>o sin cPo
-sin
cos
c/>oJ[xJ c/>o y
-sin r/>oJ. [Pxj cos cPo p, ..
(12.2.1)
(12.2.2)
Let us denote the operator that rotates these two-dimensional vectors by R( ¢ok). It is represented by the 2 x 2 matrix in Eqs. (12.2.1) and (12.2.2). Just as T(a) is the operator in Hilbert space associated with the translation a, let U[R(¢ok)] be the operator associated with the rotation R( ¢ok). In the active transformation picturet ilJI) -->llJIR) = U[R]ilJI) U(R]
( 12.2.3)
The rotated state llJI R) must be such that (X)R= (X) cos ¢o- ( Y) sin ¢o
(l2.2.4a)
( Y) R = (X) sin qJo + ( Y) cos qJo
(12.2.4b) ( l2.2.5a) ( l2.2.5b)
where
and (X)= ( lJIIX ilJI ), etc.
In analogy with the translation problem, we define the action of U[R] on position eigenkets: U[R]Ix, y) = lx cos ¢o- y sin ¢ 0 , x sin ¢o + y cos ¢ 0 )
(12.2.6)
As in the case of translations, this equation is guided by more than just Eq. (12.2.4), which specifies how (X) and ( Y) transform: in omitting a possible phase factor g(x, y), we are also ensuring that (Px) and (Py) transform as in Eq. (12.2.5). One way to show this is to keep the phase factor and use Eqs. (12.2.5a) and (12
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