3.5 Finite Rotations in 3D Euclidean Space and Angular Momentum in QM
An active rotation in 3D position space is defined as the rotation of a vector about some
point in a fixed coordinate system (a passive rotation being the rotation of the
coordinate system while the vector stays fixed). We claim that the rotation of a vector
about some axis in space is induced by a corresponding unitary and hermitian operator
expressed as a 3 by 3 matrix. These rotations are in direct correspondence to the
transformation of a state vector in an angular momentum basis (i.e. where the
eigenvalues are angular momentum) . That is, the rotation of a vector r , defining the
value of a function Ψ(r) at a point in position space, corresponds to a change of state of
the corresponding state vector |Ψ 〉 .
In order to develop the proper operators inducing finite rotations, one determines the
changes on a function (usually a vector function, say, f(r) ) under an infinitesimal rotation
which is induced by what we call generators of infinitesimal rotations. Then one
integrates the change in the function δf(r) to obtain the proper rotation operators U.
If the function f(r) is changed at r by an infinitesimal vector ε then we obtain the
function f(r’) = f(r-ε) (if you find this confusing, think of shifting, say, the sine function
by a certain amount just in 1D…) . Since ε is supposed to be a very small vector, we can
expand the new function in a Taylor series to first order:
And the corresponding change in the function is (to first order)
In general, a finite rotation R by an angle φ around an axis pointing along the unit vector
n results in the displacement of an entity or a change in property of some medium that
pervades all space (think of the electric field varying with position), located by the
vector r , is given by the vector (see figures below):
(See next page.)
Proof:
Assume we are rotating around the z-axis (we can always rotate the coordinate system to
satisfy this) and that the vector we want to rotate is given by:
The usual matrix rotation about z is:
With this, we have:
If the rotation angle is infinitesimal (δφ ) , then eqn. 3.5.1 becomes (via Taylor expansion
again)
Where
is a vector, pointing in the direction n of the axis of rotation,
defined by the right hand screw rule.
So, to first order in nδφ the change of the function f(r) under an infinitesimal rotation ε
by an angle δφ is,
Then, using
we find
Where α = nδφ is now defined to be the infinitesimal angle of rotation.
With this result, we see that the rotated function is given by
From here, we can go to a finite rotation φ around n by performing N such infinitesimal
rotations in succession, such that
If we decide to make all of these infinitesimal rotations the same magnitude, then the
total rotation is just
So, for N successive, infinitesimal rotations (it’s now time to start calling them
transformations), a finite rotation in eqn. 3.5.3 becomes
However, in the limit as N → ∞ , this becomes exact, and we find:
Accordingly, we write the rotation operator for finite rotations about n as
For any state |Ψ 〉 = ∑l∫Ψ(r) |r, l 〉 1 the rotation operator (eqn. 3.5.4) rotates all
probability amplitudes Ψ(r) into new amplitudes Ψ ’(r’) , describing a new state |Ψ ’ 〉.
Ψ ’(r’) = URΨ(r)
3.5.5
For a proper rotation, if the initial wave functions (and the state itself) are properly
normalized, then so are the transformed wave functions (i.e. a proper rotation can not
change the length of a vector !). For this to be the case, we must have
t
〈 Ψ ’ |Ψ ’ 〉 = 〈Ψ | UR UR |Ψ 〉 = 1
If the matrices (operators) L in the rotation operator are hermitian (which we will verify
t
-1
t
later), then this condition is satisfied, since then UR = UR or UR UR = I, as you can
easily verify from looking at the rotation operator.
1) This is a sum and integral over simultaneous discrete angular momentum eigenvectors and continuous position eigenvectors – this is
basically a state assembled from a complete set of solutions or eigenvectors of the Schrödinger equation. There is more detail in the
state as is indicated in this notation, but that is irrelevant at this point. We will find the full state, for all simultaneously
diagonalizable operators later on.
For any operator A , the definition of a rotationally transformed operator A ’ is given by
A’ Ψ ’(r’) = UR AΨ(r)
3.5.6
and inserting the identity operator expressed in terms of the rotation operators gives
t
A’ Ψ ’(r’) = UR A UR UR Ψ(r)
t
(A’ - UR A UR ) Ψ ’(r’) = 0
Which mean, that, since in general Ψ ’(r’) ≠ 0 , we must have
t
A’ = UR A UR
3.5.7
= e-iφn•L/ћ A eiφn•L/ћ
The so called generator of rotation L used here (corresponding to the symbol commonly
used to indicate orbital angular momentum) is just one example, serving to illustrate the
concept for all instances of angular momenta. In addition to angular momentum, we also
have spin angular momentum, denoted by S, which is just as special case of angular
momentum in general, as described in the introduction to this chapter. There is then also
total angular momentum J which is used to identify the total, added angular momentum
of a system and may include the addition of several orbital angular momenta, several spin
angular momenta, or the addition of spin and orbital angular momenta. All of these
represent some form of rotation in Euclidean space and the corresponding
transformation of a state, so that, in general we write
3.6 Angular Momentum Commutation Relations
We want to establish the quantum mechanical behavior of angular momentum, and the
first information we need for that is how the angular momentum operators commute with
each other as well as with other operators.
To this end, consider calculating the expectation value of some operator A . Assume that
A is in fact a three vector, such as another angular momentum operator.
Then,
〈A 〉 = 〈Ψ |A |Ψ 〉
〈A’ 〉 = 〈Ψ ’ |A |Ψ ’ 〉 = 〈Ψ | e-iφn•L/ћ A eiφn•L/ћ |Ψ 〉
This expectation value is a sum of vectors of eigenvalues 1 (of actual 3D vectors, as we
are used to, by virtue of the operator A being a vector operator). But for any individual
eigenstate of the hamiltonian with probability amplitude Ψ(r) the expectation value
corresponds to a single vector as written in eqn. 3.5.1 .
Where the last equality takes the infinitesimal rotation approximation with angle α.
Then, taking the form of the infinitesimal rotation operator from eqn. 3.5.3, we get
Where we only take terms on the left hand side to order α ( throwing away the term
with α2 ), so as to keep consistency with the first order expansion that lead to eqn. 3.5.3.
Therefore, we find that
Since J itself is a vector operator and this derivation made no reference to or any
restriction on any particular type, we can just plug in J itself again and see that this
leads to
Or, as we are more used to seeing
These are the commutation relations of the angular momentum components with
themselves.
1) It is a sum, because the general state is a superposition of eigenstates which all have (in general) different eigenvalues.
Properties of the Levi-Civita symbol: