12.2 Noether's theorem and conserved quantities

Noether's theorem connects continuous symmetries to conserved quantities in physics. It unifies conservation laws across classical mechanics, field theory, and quantum mechanics, providing deep insights into the fundamental structure of physical laws.

The theorem's applications are vast, from simple harmonic oscillators to complex quantum systems. By identifying symmetries and applying Noether's theorem, we can derive conserved quantities like energy, momentum, and charge, simplifying analysis of physical phenomena.

Noether's Theorem and Symmetries

Noether's theorem significance

Top images from around the web for Noether's theorem significance

• 

  Static Electricity and Charge: Conservation of Charge · Physics View original

  Is this image relevant?

• 

  Noether's Theorems [The Physics Travel Guide] View original

  Is this image relevant?

• 

  homework and exercises - Finding conserved quantity using Noether's theorem - Physics Stack Exchange View original

  Is this image relevant?

• 

  Static Electricity and Charge: Conservation of Charge · Physics View original

  Is this image relevant?

• 

  Noether's Theorems [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

Top images from around the web for Noether's theorem significance

• 

  Static Electricity and Charge: Conservation of Charge · Physics View original

  Is this image relevant?

• 

  Noether's Theorems [The Physics Travel Guide] View original

  Is this image relevant?

• 

  homework and exercises - Finding conserved quantity using Noether's theorem - Physics Stack Exchange View original

  Is this image relevant?

• 

  Static Electricity and Charge: Conservation of Charge · Physics View original

  Is this image relevant?

• 

  Noether's Theorems [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

• Noether's theorem links continuous symmetries to conserved quantities in physical systems described by Lagrangian or Hamiltonian
• Unifies conservation laws under single principle applicable in classical mechanics, field theory, and quantum mechanics (energy, momentum, charge)
• Identifies conserved quantities in complex systems facilitates analysis of physical phenomena
• Provides deep insight into fundamental structure of physical laws and symmetries

Conserved quantities from symmetries

• General procedure identifies symmetry transformation, expresses it in infinitesimal parameters, calculates action variation, derives conserved current
• Key steps utilize principle of least action $\delta S = 0$, express Lagrangian variation $\delta L = \frac{\partial L}{\partial q_i} \delta q_i + \frac{\partial L}{\partial \dot{q_i}} \delta \dot{q_i}$
• Apply Euler-Lagrange equations to simplify expressions
• Identify conserved quantity from resulting mathematical form

Symmetries and conservation laws

• Time translation symmetry invariance of Lagrangian under time shifts leads to energy conservation (planetary orbits)
• Space translation symmetry invariance under spatial translations results in linear momentum conservation (collisions)
• Rotational symmetry invariance under rotations implies angular momentum conservation (spinning ice skater)
• Gauge symmetry invariance under phase transformations in quantum mechanics leads to charge conservation (electron interactions)

Applications of Noether's theorem

• Classical mechanics: Simple harmonic oscillator energy conservation from time translation symmetry, central force problem angular momentum conservation from rotational symmetry (planetary motion)
• Quantum mechanics: Schrödinger equation probability conservation from global phase invariance, Klein-Gordon field charge conservation from U(1) gauge symmetry
• Field theory: Electromagnetic field energy and momentum conservation from spacetime translation symmetry, Yang-Mills theory color charge conservation from SU(3) gauge symmetry (strong nuclear force)
• Practical considerations:
  • Identify relevant symmetries in system (translational, rotational, gauge)
  • Construct appropriate Lagrangian or Hamiltonian
  • Apply Noether's theorem to derive conserved quantities
  • Interpret physical meaning of conserved quantities in context of system (energy, momentum, charge)