Noether's Theorem in Quantum Mechanics

Hello! Welcome to the first lesson of our course on Intermediate to Advanced Quantum Mechanics.

Given your strong background in both theoretical and computational physics, particularly your experience with field theory and condensed matter, we're going to dive right into one of the most profound principles in modern physics: the connection between symmetry and conservation laws.

This lesson addresses the first learning outcome of our initial module: to apply the theory of continuous symmetries and derive conserved quantities using Noether's theorem in quantum mechanics. We will explore this principle from two complementary viewpoints: the Lagrangian formulation, which you'll recognize from classical field theory, and the Hamiltonian/operator formalism, which is the native language of quantum mechanics. By the end of this lesson, you'll be able to derive conservation laws from the symmetries of a system.

1. The Core Idea: Symmetry and the Invariance of Action

At its most fundamental level, a symmetry is a transformation that leaves the physical laws describing a system unchanged. In the Lagrangian formulation of physics, this translates to the action $S$ being invariant under the transformation. Emmy Noether's celebrated theorem provides the rigorous link between such an invariance and a conservation law.

To see how this works in a general context you are familiar with from your QFT studies, let's start with the classical field theory derivation.

As the video demonstrates, for any continuous global symmetry of the action, there exists a corresponding conserved Noether current, $J^{\mu }$. The invariance of the action under an infinitesimal transformation forces this current to satisfy a continuity equation:

$${\partial }_{\mu }{J}^{\mu }=\frac{\partial J^0}{\partial t}+\vec{\nabla }\cdot \vec{J}=0$$

Here, $J^0$ is interpreted as a charge density ($\rho$), and $\vec{J}$ is the current density.

By integrating the charge density over all of space, we define the total charge $Q$:

$$Q=\int \rho d^{3}x=\int J^0{d}^{3}x$$

If we assume the current $\vec{J}$ vanishes sufficiently quickly at infinity (a standard physical assumption), the divergence theorem shows that this total charge is conserved in time:

$$\frac{dQ}{dt}=0$$

This is the essence of Noether's theorem in the language of field theory: a continuous global symmetry implies a conserved charge.

2. The Quantum Mechanical Picture: Symmetry and Generators

In quantum mechanics, we typically work with states in a Hilbert space and operators, rather than Lagrangians and actions. So, how does Noether's beautiful idea translate into this framework?

The key is to connect the conserved charge from the classical theorem to the generator of the symmetry transformation in quantum mechanics. A symmetry operation, like a translation or a rotation, is represented by a unitary operator, $U$, that acts on the quantum states. A continuous symmetry is one that can be built up from infinitesimal transformations, and these transformations are "generated" by a Hermitian operator.

The following video provides an excellent conceptual bridge, building up the idea of generators from first principles.

The video highlights the central pairings:

• Spatial Translation (parameter: distance) is generated by Momentum. Conservation of momentum follows from the invariance of physics under translation.
• Rotation (parameter: angle) is generated by Angular Momentum. Conservation of angular momentum follows from the invariance of physics under rotation.
• Time Translation (parameter: time) is generated by the Hamiltonian. Conservation of energy follows from the invariance of physics under time translation.

Our task now is to formalize this connection using the operator language of quantum mechanics.

3. Noether's Theorem in Operator Formalism

Let's derive Noether's theorem directly within the Hamiltonian framework. The key reading below formalizes the ideas we've just discussed.

Let's summarize the logical steps of the proof, which are crucial:

1. Symmetry as a Unitary Operator: A symmetry transformation is represented by a unitary operator $U$. If this is a symmetry of the dynamics governed by the Hamiltonian $H$, then $H$ must be unchanged by the transformation. This means $H$ transformed by $U$ is equal to $H$:

   $$U^{†}HU=H$$

   This is equivalent to the statement that the unitary operator commutes with the Hamiltonian:

   $$[U,H]=0$$

2. Continuous Symmetries and Generators: A continuous symmetry depends on a continuous parameter $ϵ$. The unitary operator can be written in terms of a Hermitian operator $G$, called the generator:

   $$U(ϵ)=e^{-iϵG/\hslash }$$

   For an infinitesimal transformation ($ϵ\ll 1$), we can expand this to first order:

   $$U(ϵ)\approx 1-\frac{iϵ}{\hslash }G$$

3. Deriving the Conservation Law: We now substitute this infinitesimal form back into the commutation relation $[U,H]=0$:

   $$\left[1-\frac{iϵ}{\hslash }G,H\right]=0$$

   Since the identity operator commutes with everything, this simplifies to:

   $$-\frac{iϵ}{\hslash }[G,H]=0⟹[G,H]=0$$

   The generator of the symmetry must commute with the Hamiltonian.

4. Connecting to Conservation: The Heisenberg equation of motion for the expectation value of an operator $G$ that does not explicitly depend on time is:

   $$\frac{d⟨G⟩}{dt}=\frac{1}{i\hslash }⟨[G,H]⟩$$

   Since we just proved that $[G,H]=0$, it immediately follows that:

   $$\frac{d⟨G⟩}{dt}=0$$

   The expectation value of the generator is conserved. The generator $G$ is the conserved observable.

This completes the proof: For every continuous symmetry of the Hamiltonian, the corresponding Hermitian generator is a conserved quantity.

Test your understanding!

Briefly explain, in your own words, the conceptual difference between the Lagrangian derivation of Noether's theorem (which yields a conserved current $J^{\mu }$) and the Hamiltonian derivation (which yields a conserved generator $G$). How are the two results related?

Show answer

The Lagrangian derivation is based on the invariance of the action, a functional of fields over spacetime. It's a statement about the entire history of the system. The result is a local conservation law expressed as a continuity equation, ${\partial }_{\mu }{J}^{\mu }=0$. This tells us how a conserved quantity flows in space and time.

The Hamiltonian derivation is based on the invariance of the Hamiltonian, an operator acting on states at a single instant in time. The result is a global conservation law, $[G,H]=0$, which implies that the expectation value of the generator $G$ is constant in time.

The two are related because the conserved charge $Q=\int J^0{d}^{3}x$ from the Lagrangian formalism corresponds to the conserved operator $G$ in the Hamiltonian formalism. The time-component of the Noether current, $J^0$, is the density of the conserved observable $G$.

4. An Advanced Application: Symmetries of the Schrödinger Field

The standard examples of momentum, energy, and angular momentum are fundamental. However, since you're aiming for an advanced understanding, let's explore a more subtle application. We can treat the Schrödinger wavefunction $\psi (\mathbf{x},t)$ itself as a classical field and write down a Lagrangian for it. Applying Noether's theorem to this Lagrangian can reveal some less obvious conserved quantities.

The following paper explores exactly this, providing a taste of how field-theoretic tools can be applied directly to non-relativistic quantum mechanics.

This analysis reveals two interesting continuous symmetries for the free particle Schrödinger Lagrangian:

1. A constant shift in the field: $\psi \to {\psi }^{\prime }=\psi +ϵ$, where $ϵ$ is a constant complex number. This leads to the conserved quantity: $$Q=\int \psi (\mathbf{x},t)d^{3}x$$
2. A spatially linear shift in the field: $\psi \to {\psi }^{\prime }=\psi +\overrightarrow{ϵ}\cdot \mathbf{x}$. This leads to the conserved quantity: $$\vec{Q}=\int \mathbf{x}\psi (\mathbf{x},t)d^{3}x$$

These are valid conserved quantities arising from Noether's theorem, yet they are rarely discussed. Why? Unlike probability ($\int |\psi {|}^2{d}^{3}x$), these quantities are not gauge-invariant (they change under a phase transformation $\psi \to e^{i\alpha }\psi$), and their conservation depends more sensitively on the boundary conditions of $\psi$ at infinity. This serves as a great example of the power and subtlety of the theorem.

Conclusion

In this lesson, we have established the fundamental connection between continuous symmetries and conservation laws, a cornerstone of modern physics.

Here are the key takeaways:

• Noether's Theorem states that for every continuous global symmetry of a physical system, there is a corresponding conserved quantity.
• In the Lagrangian formalism, a symmetry is a transformation that leaves the action invariant, leading to a conserved current $J^{\mu }$ satisfying ${\partial }_{\mu }{J}^{\mu }=0$.
• In the Hamiltonian formalism of quantum mechanics, a symmetry is represented by a unitary operator $U$ that commutes with the Hamiltonian, $[U,H]=0$.
• The conserved quantity in quantum mechanics is the Hermitian generator $G$ of the symmetry transformation ($U=e^{-iϵG/\hslash }$), which satisfies $[G,H]=0$, ensuring its expectation value is constant in time.

This principle will be a recurring theme throughout our course. We will see how symmetries dictate the structure of theories, predict degeneracies in energy spectra, and determine selection rules for transitions.

Preview of the next lesson: We will now take one of the most important continuous symmetries—rotation—and study its structure in detail. We will analyze the rotation group, its relationship to the special unitary group SU(2), and the properties of its generator, the angular momentum operator.