10
1 Operations
around y by a fourfold rotation axis along the z direction. In Chap. 3, we shall see
that this relation installs an equivalence between both twofold rotations whenever
such a ˆ
C
z
4 is present.
1.4 An Aide Mémoire
• Use a right-handed coordinate system.
• Always leave the Cartesian directions unchanged.
• Symmetry operations are defined in an active sense.
• Rotations through positive angles appear counterclockwise, when viewed from
the rotational pole; “counterclockwise is positive.”
• The transformation of the coordinates of a point is written in a column vector
notation.
• The transformation of a function space is written in a row vector notation.
• There is a dual relationship between the transformations of functions and of coordinates: ˆ
Rf (r) = f( ˆ
R −1 r).
• The transformation of an operator O is given by ˆ
RO ˆ
R −1 .
1.5 Problems
1.1 Use the stereographic representation of Fig. 1.1 to show that [ˆ ı, ˆ
C
z
2 ]=0.
1.2 The square of the radial distance of the point P 1 in the xy plane may be obtained
by multiplying the coordinate column by its transposed row:
r
2 = x
2
1 + y
2
1 =
x 1 y 1
x 1
y 1
(1.28)
Show that this scalar product is invariant under a rotation about the z-axis.
1.3 Derive the general form of a 2 × 2 matrix that leaves this radial distance invariant.
1.4 The translation operator T a displaces a point with x-coordinate x 1 t oan e w
position x 1 + a. Apply this operator to the wavefunction e ikx .
1.5 Construct a differential operator such that its action on the coordinate functions
x and y matches the matrix transformation in Eq. (1.16). What is the angular
derivative of this operator as the rotation angle tends to zero? Can you relate
this limit to the angular momentum operator L z ?
References
1. Berry, M.V.: In: Shapere, A., Wilczek, F. (eds.) Geometric Phases in Physics. Advanced Series
in Mathematical Physics, vol. 5, pp. 3–28. World Scientific, Singapore (1989)
1 Operations
around y by a fourfold rotation axis along the z direction. In Chap. 3, we shall see
that this relation installs an equivalence between both twofold rotations whenever
such a ˆ
C
z
4 is present.
1.4 An Aide Mémoire
• Use a right-handed coordinate system.
• Always leave the Cartesian directions unchanged.
• Symmetry operations are defined in an active sense.
• Rotations through positive angles appear counterclockwise, when viewed from
the rotational pole; “counterclockwise is positive.”
• The transformation of the coordinates of a point is written in a column vector
notation.
• The transformation of a function space is written in a row vector notation.
• There is a dual relationship between the transformations of functions and of coordinates: ˆ
Rf (r) = f( ˆ
R −1 r).
• The transformation of an operator O is given by ˆ
RO ˆ
R −1 .
1.5 Problems
1.1 Use the stereographic representation of Fig. 1.1 to show that [ˆ ı, ˆ
C
z
2 ]=0.
1.2 The square of the radial distance of the point P 1 in the xy plane may be obtained
by multiplying the coordinate column by its transposed row:
r
2 = x
2
1 + y
2
1 =
x 1 y 1
x 1
y 1
(1.28)
Show that this scalar product is invariant under a rotation about the z-axis.
1.3 Derive the general form of a 2 × 2 matrix that leaves this radial distance invariant.
1.4 The translation operator T a displaces a point with x-coordinate x 1 t oan e w
position x 1 + a. Apply this operator to the wavefunction e ikx .
1.5 Construct a differential operator such that its action on the coordinate functions
x and y matches the matrix transformation in Eq. (1.16). What is the angular
derivative of this operator as the rotation angle tends to zero? Can you relate
this limit to the angular momentum operator L z ?
References
1. Berry, M.V.: In: Shapere, A., Wilczek, F. (eds.) Geometric Phases in Physics. Advanced Series
in Mathematical Physics, vol. 5, pp. 3–28. World Scientific, Singapore (1989)