1.2 Operations and Functions
5
Fig. 1.3 The rotation of the
function f (x,y)
counterclockwise by an angle
α generates a new function,
f ′ (x, y). The value of the
new function at P 2 is equal to
the value of the old function
at P 1 . Similarly, to find the
value of the new function at
P 1 ,wehavetoretrievethe
value of the old function at a
point P 0 , which is the point
that will be reached by the
clockwise rotation of P 1
in Fig. 1.3. In order to determine the mathematical form of the new transformed
function, we must be able to compare the value of the new function with the original function at the same point, i.e., we must be able to see how the property changes
at a given point. Thus, we would like to know what would be the value of ˆ
Rf in the
original point P 1 . Equation (1.8) cannot be used to determine this since the transformed function is as yet unknown and we thus do not know the rules for working
out the brackets in the left-hand side of the equation. However, this relationship
must be true for every point; thus, we may substitute ˆ
R −1 P 1 for P 1 on both the leftand right-hand sides of Eq. (1.8). The equation thus becomes
ˆ
Rf
ˆ
R
ˆ
R
−1 P 1
= ˆ
Rf ( ˆ
EP 1 ) = ˆ
Rf (P 1 ) = f
ˆ
R
−1 P 1
(1.9)
This result reads as follows: the transformed function attributes to the original point
P 1 the property that the original function attributed to the point ˆ
R −1 P 1 .InFig.1.3
this point from which the function value was retrieved is indicated as P 0 . Thus, the
function and the coordinates transform in opposite ways. 2 This connection transfers
the operation from the function to the coordinates, and, since the original function
is a known function, we can also use the toolbox of corresponding rules to work out
the bracket on the right-hand side of Eq. (1.9).
As an example, consider the familiar 2p orbitals in the xy plane: 2p x , 2p y . These
orbitals are usually represented by the iconic dumbell structure. 3 We can easily
find out what happens to these upon rotation, simply by inspection of Fig. 1.4,i n
which we performed the rotation of the 2p x orbital by an angle α around the z-axis.
Clearly, when the orbital rotates, the overlap with the 2p x function decreases, and
the 2p y orbital gradually appears. Now let us apply the formula to determine ˆ
R2p x .
The functional form of the 2p x orbital for a hydrogen atom, in polar coordinates,
reads: R 2p (r)Θ 2p|1| (θ )Φ x (φ), where R 2p (r) is the radial part, Θ 2p|1| (θ ) is the part
2 A more general expression for the transportation of a quantum state may also involve an additional
phase factor, which depends on the path. See, e.g., [1].
3 The electron distribution corresponding to the square of these orbitals is described by a lemniscate
of Bernoulli. The angular parts of the orbitals themselves are describable by osculating spheres.
5
Fig. 1.3 The rotation of the
function f (x,y)
counterclockwise by an angle
α generates a new function,
f ′ (x, y). The value of the
new function at P 2 is equal to
the value of the old function
at P 1 . Similarly, to find the
value of the new function at
P 1 ,wehavetoretrievethe
value of the old function at a
point P 0 , which is the point
that will be reached by the
clockwise rotation of P 1
in Fig. 1.3. In order to determine the mathematical form of the new transformed
function, we must be able to compare the value of the new function with the original function at the same point, i.e., we must be able to see how the property changes
at a given point. Thus, we would like to know what would be the value of ˆ
Rf in the
original point P 1 . Equation (1.8) cannot be used to determine this since the transformed function is as yet unknown and we thus do not know the rules for working
out the brackets in the left-hand side of the equation. However, this relationship
must be true for every point; thus, we may substitute ˆ
R −1 P 1 for P 1 on both the leftand right-hand sides of Eq. (1.8). The equation thus becomes
ˆ
Rf
ˆ
R
ˆ
R
−1 P 1
= ˆ
Rf ( ˆ
EP 1 ) = ˆ
Rf (P 1 ) = f
ˆ
R
−1 P 1
(1.9)
This result reads as follows: the transformed function attributes to the original point
P 1 the property that the original function attributed to the point ˆ
R −1 P 1 .InFig.1.3
this point from which the function value was retrieved is indicated as P 0 . Thus, the
function and the coordinates transform in opposite ways. 2 This connection transfers
the operation from the function to the coordinates, and, since the original function
is a known function, we can also use the toolbox of corresponding rules to work out
the bracket on the right-hand side of Eq. (1.9).
As an example, consider the familiar 2p orbitals in the xy plane: 2p x , 2p y . These
orbitals are usually represented by the iconic dumbell structure. 3 We can easily
find out what happens to these upon rotation, simply by inspection of Fig. 1.4,i n
which we performed the rotation of the 2p x orbital by an angle α around the z-axis.
Clearly, when the orbital rotates, the overlap with the 2p x function decreases, and
the 2p y orbital gradually appears. Now let us apply the formula to determine ˆ
R2p x .
The functional form of the 2p x orbital for a hydrogen atom, in polar coordinates,
reads: R 2p (r)Θ 2p|1| (θ )Φ x (φ), where R 2p (r) is the radial part, Θ 2p|1| (θ ) is the part
2 A more general expression for the transportation of a quantum state may also involve an additional
phase factor, which depends on the path. See, e.g., [1].
3 The electron distribution corresponding to the square of these orbitals is described by a lemniscate
of Bernoulli. The angular parts of the orbitals themselves are describable by osculating spheres.