Noether's Theorem: Rotational Invariance and Angular Momentum Conservation

The General Connection: Symmetries and Conservation Laws

Before focusing specifically on rotations, let's establish the general principle that links any continuous symmetry to a conservation law. A symmetry transformation is one that leaves the Hamiltonian of a system invariant. As we've seen, such transformations are represented by unitary operators $U$. If a transformation is a symmetry, the Hamiltonian looks the same after the transformation:

This implies that the unitary operator commutes with the Hamiltonian, $[U,H]=0$.

Now, consider a continuous symmetry described by a parameter $\theta$, with $U(\theta )=\exp (-i\theta T/\hslash )$, where $T$ is the Hermitian generator. For the Hamiltonian to be invariant under this symmetry for any value of $\theta$, it must be invariant for an infinitesimal transformation, $U(\delta \theta )\approx I-i\delta \theta T/\hslash$. The condition $[U(\delta \theta ),H]=0$ then directly implies that the generator must commute with the Hamiltonian:

The following reading from David Tong's lecture notes on Principles of Quantum Mechanics formalizes this exact argument.

As the notes demonstrate, the condition $[T,H]=0$ is precisely the requirement for an operator $T$ (with no explicit time dependence) to be conserved. This can be seen from the Heisenberg equation of motion:

If $[T,H]=0$ and $T$ does not explicitly depend on time, then $\frac{dT(t)}{dt}=0$. Thus, we arrive at the central statement: the generator of a continuous symmetry of the Hamiltonian is a conserved quantity.

Rotational Invariance and Angular Momentum Conservation

We are now equipped to apply this general principle directly to rotations.

A system is rotationally invariant if its Hamiltonian is unchanged by any rotation. In the language of operator transformations, this means the Hamiltonian is a scalar operator. A scalar operator is, by definition, an operator that commutes with all rotation operators $U(R)$.

The following notes by Robert Littlejohn formalize this definition and its immediate consequence for the generators.

The logic is now complete and can be stated as a formal proof:

1. Premise: The system is rotationally invariant.
2. Definition: This means the Hamiltonian $H$ is a scalar operator.
3. Property of Scalar Operators: A scalar operator commutes with the generators of rotations. From our previous lesson, we know these generators are the components of the angular momentum operator, $\mathbf{J}$. Therefore, $[J_i,H]=0$ for $i=x,y,z$.
4. Conservation Law: Since each component of the angular momentum operator commutes with the Hamiltonian, each component $J_i$ is a conserved quantity. The time derivative of its expectation value is zero: $$\frac{d⟨J_i⟩}{dt}=\frac{i}{\hslash }⟨[H,J_i]⟩=0$$

We have thus shown that rotational invariance of the Hamiltonian implies the conservation of angular momentum.

Since $H$ commutes with $J_x$, $J_y$, and $J_z$, it must also commute with any function of them. In particular, it must commute with the total angular momentum squared operator, $J^2=J_x^2+J_y^2+J_z^2$.

Consequence of Conservation: Energy Degeneracy

The fact that angular momentum is conserved in rotationally symmetric systems has a profound and directly observable consequence: the degeneracy of energy levels.

Since $[H,J^2]=0$ and $[H,J_z]=0$, we know that we can find a basis of states that are simultaneous eigenstates of all three operators. These states are typically labeled by their eigenvalues: $|E,j,m_j⟩$.

However, the components of angular momentum do not commute with each other: $[J_x,J_y]=i\hslash J_z$. This non-commutativity is the key to degeneracy. Let's see how.

Consider an energy eigenstate $|E,j,m_j⟩$. What happens when we act on it with a raising operator, $J_{+}$, which as we know corresponds to an infinitesimal rotation? The resulting state is $J_{+}|E,j,m_j⟩\propto |E,j,m_j+1⟩$. What is the energy of this new state?

We can find out by acting on it with the Hamiltonian:

Since the Hamiltonian is rotationally invariant, it commutes with all components of $\mathbf{J}$, and therefore also with $J_{+}$.

This shows that the new state $J_{+}|E,j,m_j⟩$ is also an energy eigenstate with the exact same energy $E$.

Professor Susskind provides a very clear demonstration of this exact point.

This logic applies to all states within a given angular momentum multiplet. For a given total angular momentum quantum number $j$, there are $2j+1$ states, corresponding to $m_j=-j,-j+1,\dots ,j-1,j$. In a rotationally invariant system, these $2j+1$ states must all be degenerate in energy. This degeneracy is a direct and unavoidable consequence of rotational symmetry.

This is why, for example, in the hydrogen atom (neglecting smaller effects for now), the $2p$ orbitals ($l=1$)—$p_x,p_y,p_z$, or equivalently the $m_l=-1,0,1$ states—all have the same energy. The system possesses spherical symmetry, and these three states simply represent different orientations of the same angular momentum configuration, which can be rotated into one another.

Conclusion

In this lesson, we have solidified one of the most elegant and powerful ideas in quantum mechanics, connecting a fundamental symmetry of space to a fundamental conservation law.

Key Takeaways:

• A continuous symmetry of the Hamiltonian implies that its generator is a conserved quantity. This is the quantum mechanical statement of Noether's theorem.
• For a system with rotational invariance, the Hamiltonian is a scalar operator and commutes with the generators of rotation: $[H,J_i]=0$.
• This commutation relation guarantees that angular momentum is conserved, $\frac{d⟨\mathbf{J}⟩}{dt}=0$.
• The non-commuting nature of the rotation group generators ($[J_i,J_j]\ne 0$) combined with their conservation leads to a necessary degeneracy in the energy spectrum. For a given total angular momentum $j$, all $2j+1$ states corresponding to different $m_j$ values must have the same energy.

Having explored the consequences of continuous symmetries like translation and rotation, our next lesson will turn to a fundamentally different type: a discrete symmetry. We will analyze the parity operator, which corresponds to spatial inversion ($\mathbf{r}\to -\mathbf{r}$), and derive the important concept of parity selection rules.