Infinitesimal Rotations and the Angular Momentum Operator

Introduction

In our last lesson, we established a profound connection between spatial translation and momentum. By constructing the translation operator $U (a)$ , we rigorously identified the momentum operator $P$ as its infinitesimal generator. This led us to the principle that translational invariance implies momentum conservation, a cornerstone of physics.

We will now apply this same powerful line of reasoning to rotations. Our goal is to construct the operator that performs a rotation on a quantum state and, in doing so, to identify its generator. You will see that this generator is none other than the angular momentum operator. This will cement the idea that angular momentum is not just a classical analogy, but is fundamentally the agent of rotations in the quantum world, just as momentum is the agent of translations.

Rotations as Unitary Transformations

Before we construct the operator, let's revisit the general framework. Any symmetry transformation that preserves the physical content of a quantum system must preserve the normalization of states. This requires the transformation to be represented by a unitary operator. For a continuous symmetry parameterized by a vector $α$ , this unitary operator $U (α)$ can be built from an infinitesimal transformation.

The following lecture excerpt from Prof. James Binney provides an excellent overview of this logic, applying it directly to rotations.

010 Transformation of Kets, Continuous and Discrete Transformations and the Rotation Operator

This clip from an Oxford University lecture introduces the general framework for continuous transformations and then specializes it to rotations.

First, watch the segment on unitary transformations to review why these transformations must be unitary and how this leads to their generators being Hermitian operators. This is the abstract foundation. Then, watch the core section on rotations. The lecturer postulates that the rotation operator takes the form U(\mathbf{\alpha}) = \exp(-i \mathbf{\alpha} \cdot \mathbf{J}), identifying \mathbf{J} as a set of three Hermitian generators. He conjectures that this \mathbf{J} is the angular momentum operator. Our task in this lesson is to prove this conjecture by explicit construction.

As the video explains, we expect a rotation, specified by an axis $n^$ and an angle $θ$ (which can be combined into a vector $θ = θ n^$ ), to be represented by a unitary operator of the form:

U (θ) = exp (- \frac{i}{ℏ} θ \cdot J)

Here, $J$ is the vector of Hermitian generators. For an infinitesimal rotation by $δ θ$ , this operator becomes:

U (δ θ) \approx I - \frac{i}{ℏ} δ θ \cdot J

Our goal is to derive this form from first principles and thereby identify what $J$ is.

Constructing the Rotation Operator

Let's define how a rotation acts on a state. Consider an active rotation of a physical system by an angle $δ θ$ about an axis $n^$ . A state vector $∣ ψ ⟩$ is transformed into a new state $∣ ψ^{'} ⟩ = U (δ θ) ∣ ψ ⟩$ . The value of the new wavefunction $ψ^{'} (r)$ at a point $r$ must be equal to the value of the original wavefunction $ψ$ at the point that was rotated into $r$ . This point is $R^{- 1} r$ . Therefore:

ψ^{'} (r) = (U (δ θ) ψ) (r) = ψ (R^{- 1} r)

For an infinitesimal rotation, the inverse rotation matrix acts on $r$ as:

R^{- 1} r \approx r - δ θ \times r where δ θ = δ θ n^

So, the transformed wavefunction is $ψ^{'} (r) \approx ψ (r - δ θ \times r)$ . Notice the similarity to translation, where we had $ψ (r - a)$ . The displacement is now a position-dependent vector, $- δ θ \times r$ .

We can now use a Taylor series expansion for $ψ$ :

ψ (r - δ θ \times r) \approx ψ (r) - (δ θ \times r) \cdot \nabla ψ (r)

This gives us the operator's action:

U (δ θ) ψ (r) \approx [I - (δ θ \times r) \cdot \nabla] ψ (r)

Using the vector identity $(a \times b) \cdot c = a \cdot (b \times c)$ , we can rewrite the operator part:

U (δ θ) \approx I - δ θ \cdot (r \times \nabla)

Now, we bring in the quantum mechanical momentum operator, $P = - i ℏ\nabla$ , which implies $\nabla = \frac{i}{ℏ} P$ . Substituting this in:

U (δ θ) \approx I - δ θ \cdot (r \times \frac{i}{ℏ} P) = I - \frac{i}{ℏ} δ θ \cdot (r \times P)

Comparing this to the expected form $U (δ θ) \approx I - \frac{i}{ℏ} δ θ \cdot J$ , we have our result. The generator of rotations is:

J = r \times P

This is precisely the operator for orbital angular momentum. We have proven the conjecture: the angular momentum operator is the generator of rotations.

The following resource provides an alternative, very elegant derivation that explicitly builds the rotation operator from the translation operators we studied previously. Given your background, you will likely appreciate this constructive approach.

quantum mechanics - Complete derivation of generator of rotations - Physics Stack Exchange

This Stack Exchange answer provides a rigorous step-by-step construction of the rotation operator.

Please read the section titled "Group of Rotations". This derivation begins by expressing the argument of the rotated wavefunction, \psi(\mathbf{r} + d\theta \mathbf{K}\mathbf{r}), which corresponds to an infinitesimal rotation. The key insight is to decompose this infinitesimal change in position into a series of infinitesimal translations. Pay close attention to how the exponential form of the translation operator, which we know involves the momentum operator \mathbf{P}, is used to reconstruct the argument of the exponential for the rotation operator. The derivation beautifully culminates in showing that this exponential involves exactly the components of \mathbf{L} = \mathbf{r} \times \mathbf{P}, thereby identifying it as the generator. Read the whole section, from the definition to the final identification of the generator.

From Abstract to Concrete: $L_{z}$

The expression $L = r \times P$ is compact but abstract. To make it more concrete, let's look at the generator of rotations in the x-y plane, which is the z-component of angular momentum, $L_{z}$ .

Advanced Quantum Mechanics Lecture 2

In this excerpt from his "Advanced Quantum Mechanics" course, Prof. Leonard Susskind provides a direct, coordinate-based derivation of the generator for rotations in a 2D plane.

Please watch the segment from the beginning of this clip. The lecturer considers a particle in the xy-plane and calculates the change in its wavefunction under an infinitesimal rotation. By expressing the derivatives \partial/\partial x and \partial/\partial y in terms of the momentum operators P_x and P_y, he directly derives the familiar expression for the z-component of angular momentum: L_z = xP_y - yP_x. This provides a concrete example of the general result we found above.

By cyclic permutation of the coordinates in the expression for $L_{z}$ , we can write down all three components of the orbital angular momentum operator:

L_{x} L_{y} L_{z} = y P_{z} - z P_{y} = z P_{x} - x P_{z} = x P_{y} - y P_{x}

These three Hermitian operators are the generators of rotations about the x, y, and z axes, respectively.

The Mathematical Structure: Lie Algebra `so(3)`

The generators of any continuous symmetry group form a mathematical structure called a Lie algebra. The "product" in this algebra is defined by the commutator. For the group of 3D rotations, SO(3), the corresponding Lie algebra is denoted so(3). Its basis elements are precisely the generators $L_{x}, L_{y}, L_{z}$ .

The following resource delves into the formal mathematics of Lie groups and Lie algebras, providing the rigorous underpinnings for these physical concepts.

Understanding Quantum Mechanical Systems with Spherical ...

This document provides a mathematically-oriented discussion of the groups relevant to rotation in quantum mechanics.

Please read the beginning of Section 3.1 to find the definition of the special orthogonal group SO(n) as the group of rotations. Then, in the same section, focus on Proposition 3.8. It defines the Lie algebra so(n) as the set of real, anti-symmetric matrices (X^T = -X). It then shows the explicit form of a matrix in so(3). Start by reading the definition of SO(n). Then, skip to and read Proposition 3.8 in its entirety.

The anti-symmetric matrices described in Proposition 3.8 are the matrix representation of the generators of rotation. For example, the generator of rotations about the z-axis, which corresponds to the operator $L_{z}$ , can be represented in this basis by the matrix:

(J_{z})_{matrix rep.} = 010 - 1 00 000

(Note: Physicists often include a factor of $i$ to make the matrix representation Hermitian, but in pure mathematics so(3) is defined with real, anti-symmetric matrices). This provides a concrete matrix picture for the abstract operator $L_{z}$ .

Conclusion

In this lesson, we have mirrored the process we used for translations to uncover the deep connection between rotations and angular momentum. By explicitly constructing the operator for an infinitesimal rotation, we have proven that its generators are the components of the orbital angular momentum operator, $L = r \times P$ .

Key Takeaways:

The operator for a rotation by an angle $θ$ is given by $U (θ) = exp (- i θ \cdot J /ℏ)$ , where $J$ is the angular momentum operator.
For an infinitesimal rotation, $U (δ θ) \approx I - \frac{i}{ℏ} δ θ \cdot J$ .
We proved by construction that the orbital angular momentum operator $L = r \times P$ is the generator of these rotations.
The generators of the rotation group SO(3) form the Lie algebra so(3), whose elements can be represented by 3x3 real, anti-symmetric matrices.

Just as we showed that translational invariance implies momentum conservation, our next lesson will complete the parallel by proving that rotational invariance of a system's Hamiltonian implies the conservation of angular momentum.

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