Noether's Theorem: Translational Invariance and Momentum Conservation

Symmetries and Conservation Laws: The General Principle

Before we focus on translations, let's establish the general theorem that underpins all conservation laws in quantum mechanics. A symmetry of a physical system is a transformation that leaves the system's dynamics unchanged. In the Hamiltonian formulation, this means the Hamiltonian operator $H$ is invariant under the transformation.

If a unitary operator $U$ represents a symmetry, it must leave the Hamiltonian unchanged when performing a similarity transform, which, given its unitarity, is equivalent to commuting with the Hamiltonian:

Now, how does this lead to a conservation law? The answer lies in the time evolution of observables. An observable, represented by a Hermitian operator $Q$, is conserved if its expectation value is constant for any state, or equivalently, if the operator itself is constant in the Heisenberg picture.

Let's use the Heisenberg picture, where states are fixed and operators evolve in time according to $Q(t)=U_{time}^{†}(t)Q{U}_{time}(t)$, with $U_{time}(t)=e^{-iHt/\hslash }$. The time evolution of $Q(t)$ is governed by Heisenberg's equation of motion.

As the text shows, if an operator $Q$ (with no explicit time dependence) commutes with the Hamiltonian, $[Q,H]=0$, its time derivative in the Heisenberg picture is zero.

This means $Q$ is a conserved quantity, or a constant of motion.

Furthermore, for a continuous symmetry described by a unitary operator $U(\theta )=e^{-i\theta T}$ with generator $T$, the condition $[U(\theta ),H]=0$ for all $\theta$ is equivalent to the condition that the generator commutes with the Hamiltonian, $[T,H]=0$.

Thus, we have our central principle: The Hermitian generator of a continuous symmetry of the Hamiltonian is a conserved quantity.

From Translational Invariance to Momentum Conservation

We are now equipped with all the necessary tools. We simply need to apply the general principle to our specific case.

1. Symmetry: The symmetry is spatial translation.
2. Symmetry Operator: $U(\mathbf{a})=\exp (-\frac{i}{\hslash }\mathbf{a}\cdot \mathbf{P})$.
3. Generator: The momentum operator, $\mathbf{P}$.

According to our principle, if the Hamiltonian is invariant under spatial translations, then momentum must be conserved. Let's trace the simple steps of the proof.

Translational invariance of the Hamiltonian means:

This must be true for any $\mathbf{a}$, including an infinitesimal one, $\delta \mathbf{a}$. We can expand the translation operator to first order:

Substituting this into the commutator:

Since this must hold for any arbitrary infinitesimal displacement $\delta \mathbf{a}$, each component of the momentum operator must commute with the Hamiltonian:

We have just shown that translational invariance implies that the momentum operator commutes with the Hamiltonian. From our general principle established in the previous section, it immediately follows that momentum is a conserved quantity.

The following resource provides a self-contained derivation covering these exact steps.

Physical Interpretation and Examples

What does a translationally invariant Hamiltonian look like in practice?

• Free Particle: The Hamiltonian is $H=\frac{{\mathbf{P}}^2}{2m}$. Since any operator commutes with functions of itself, we have $[\mathbf{P},H]=0$ trivially. A free particle's momentum is conserved, which is what we expect.

• Particle in a Potential: The Hamiltonian is $H=\frac{{\mathbf{P}}^2}{2m}+V(\mathbf{X})$.
  Let's check the commutator:

  $$[\mathbf{P},H]=\left[\mathbf{P},\frac{{\mathbf{P}}^2}{2m}+V(\mathbf{X})\right]=\left[\mathbf{P},\frac{{\mathbf{P}}^2}{2m}\right]+[\mathbf{P},V(\mathbf{X})]=[\mathbf{P},V(\mathbf{X})]$$

  Momentum is conserved if and only if $[\mathbf{P},V(\mathbf{X})]=0$. This only happens if $V(\mathbf{X})$ is a constant (or zero), which brings us back to the free particle case. If the potential is not constant, it creates a force ($\mathbf{F}=-\nabla V$), and the particle's momentum is no longer conserved. This perfectly aligns with our classical intuition.

• System of Interacting Particles: Consider two particles interacting via a potential that depends only on their relative separation, $H=\frac{{\mathbf{P}}_1^2}{2m_1}+\frac{{\mathbf{P}}_2^2}{2m_2}+V({\mathbf{X}}_1-{\mathbf{X}}_2)$. This system is isolated, with no external forces.
  If we translate the entire system by a vector $\mathbf{a}$, so that ${\mathbf{X}}_1\to {\mathbf{X}}_1+\mathbf{a}$ and ${\mathbf{X}}_2\to {\mathbf{X}}_2+\mathbf{a}$, the potential term $V(({\mathbf{X}}_1+\mathbf{a})-({\mathbf{X}}_2+\mathbf{a}))=V({\mathbf{X}}_1-{\mathbf{X}}_2)$ is unchanged. The kinetic energy terms are also invariant. Therefore, the total Hamiltonian is translationally invariant.
  The generator for this global translation is the total momentum operator, ${\mathbf{P}}_{tot}={\mathbf{P}}_1+{\mathbf{P}}_2$. Following our theorem, this means:

  $$[{\mathbf{P}}_{tot},H]=0$$

  The total momentum of the isolated system is conserved. The individual momenta ${\mathbf{P}}_1$ and ${\mathbf{P}}_2$ do not commute with the interaction potential $V$, and thus are not conserved; the particles exchange momentum, but the total remains constant. This is a profound and satisfying result, directly mirroring the classical case.

This connection between symmetry and conservation is a recurring theme in physics. The following video gives a great overview of this "love story", framed within the classical Lagrangian formalism of Noether's theorem, which you will recognize as the direct predecessor to the quantum mechanical version we've just proved.

Conclusion

In this lesson, we have forged the final link in the chain connecting the geometry of space to a fundamental law of physics. By combining our knowledge of the translation operator with the general principle of symmetries in quantum mechanics, we have formally proven that translational invariance of the Hamiltonian leads directly to the conservation of momentum.

Key Takeaways:

• A symmetry in quantum mechanics is represented by a unitary operator $U$ that commutes with the Hamiltonian, $[U,H]=0$.
• The Hermitian generator $T$ of such a continuous symmetry is a conserved quantity, meaning it is constant in time. This follows from the Heisenberg equation of motion, as $[T,H]=0$.
• Applying this to spatial translations, where the generator is the momentum operator $\mathbf{P}$, we proved that if a Hamiltonian is invariant under translations (i.e., $[U(\mathbf{a}),H]=0$), then $[\mathbf{P},H]=0$, and momentum is conserved.
• This framework correctly predicts that momentum is conserved for free particles and for isolated multi-particle systems, but not for a single particle in a non-uniform potential.

In our next lesson, we will see this powerful logical structure at play once again. We will move from translations to rotations, construct the rotation operator, identify its generator as the angular momentum operator, and prove that rotational invariance of the Hamiltonian implies the conservation of angular momentum.