To be used in future
 Section 3.3: Consequences for Quantum Mechanics Up Chapter 3: Symmetries Chapter 4: Charged Particles and Electromagnetic Fields 

3.4 Unitary operators

3.4.1 Exponent of generators

Once we know what the generators are, we can try and take many small steps to make one big step. Finding explicit expressions will rely on the basic formula from calculus The proof of this relations is simple; we use the binomial expansion and find

3.4.2 Unitarity

A very important class of operators (transformations) in quantum mechanics are those that preserve the norm of a wave function, or slightly more generally the scalar product (overlap) between any two wave functions. Let be one such operator. By definition the transformed ket becomes and thus if the scalar product is preserved. Since we require this to hold for any and , we find or
An operator that satisfies is called unitary

3.4.2.1 Unitarity of the transformation operators

We can show that any symmetry transformation that changes space in such a way as to leave the volume element invariant has an operator representation that is unitary: Consider the scalar product We find that if for any symmetry operation where the volume element satisfies (the small volume element gets transformed into a volume of equal size) we get We also know that Since and are arbitrary, we have

3.4.3 Translation in space

We start by taking dividing a translation into many smaller translations
It is easy to show that and thus , which shows that this is a unitary (norm preserving) operator. It is actually quite straightforward (and useful) to show that any operator of the form with a Hermitian operator is unitary [see example sheet].

3.4.4 Translation in time

It is even more useful to do this for time translations
The operator we have found over here is called the time evolution operator. Once again, it is trivially unitary.

3.4.5 Rotations

By now it should have become second nature to show that This obviously unitary.
For a problem with rotational symmetry, we normally usually use a basis of the form where and is a place holder for the remaining quantum numbers needed to specify the states. The choice of rotation angles we have made is not very well matched to this choice; we shall make an equivalent choice below.

3.4.5.1Euler angles

figure Figures/Eulerangles.png
Figure 3.3 The Euler angles
The axis/angle representation is not very well chosen for the basis (). Fortunately there is an alternative representation of rotations due to Euler that is commonly used because it has many useful properties, and also helps in this case. In this parametrisation we write a rotation as a product of three simple rotations. We first perform a rotation about the axis with angle ( ). This moves the and axes to point in a new direction in the plane. Then perform a rotation over an angle ( ) along the new axis. Finally, perform a rotation about an angle ( ) along the new axis. [J]  [J] These angles are called , and in many textbooks; we shall not use this notation, since it tends to lead to confusion with the polar angles. If we perform the transformation the operator implementing this transformation can clearly be written as Note the change in order! This is due to the fact that

3.4.5.2 Representation matrices

If we look at the matrix elements of in a basis of states (i.e., these are purely angular states, we suppress any additional labels. In polar angles we have ) we find that where we use the fact that cannot change the value of (since ). Clearly what remains now is the much simpler task to find the matrices that only depend on a single angle, see below for a simple example.
The matrices are called the Wigner rotation or Wigner matrices.

3.4.5.3 Angular momentum states and ladder operators

In order to work out some of the algebraic details of the angular momentum states it is convenient to change from and to the ladder operators These have the nice property We find that for the states satisfying we have Thus where is an as yet undetermined normalisation constant. If we write this can be used to show that (using the “obvious” relation ) and similar for . We cannot fix the phase of : A little thought shows this can be freely chosen, since it corresponds to a phase choice for the angular momentum eigenstates. We use Rewriting , we can now find the basic ingredients to evaluate the exponential of in the expression for .

3.4.5.4

It is instructive to look at the case . We quickly find that and thus, if we order the basis as we have the matrix representation for for Question: What must be the eigenvalues of the matrix? Check that this is indeed the case.
We can now calculate This could get complicated, but fortunately and Thus all odd powers and all even powers of the matrix are equal--apart from the “zeroth power”. We thus find that Thus

3.4.5.5 Spin

We have not yet looked at the thorny issue of intrinsic quantum numbers and their symmetries. One place where we can relatively straightforwardly do that is for angular momentum. We know that for a system with spin, the total angular momentum is conserved, and we can immediately generalise the transformation () to
Spin
What happens if we look at that case , , i.e., a pure spin particle? It is relatively easy to evaluate the values of using the Pauli matrix (details on example sheet); The most interesting fact is what happens under a rotation. For ease of calculation take it along the -axis, and we find that in other words, a spin state goes to minus itself under a space rotation!
 Section 3.3: Consequences for Quantum Mechanics Up Chapter 3: Symmetries Chapter 4: Charged Particles and Electromagnetic Fields