15.3 Relativistic quantum mechanics and Klein-Gordon equation

Relativistic quantum mechanics bridges the gap between quantum theory and special relativity. It's crucial for understanding high-energy particles and phenomena like antimatter, addressing the limitations of non-relativistic quantum mechanics at extreme speeds and energies.

The Klein-Gordon equation, derived from the relativistic energy-momentum relation, is a key development. While it introduces important concepts like antiparticles, it has limitations in describing spin and probability density, paving the way for the more comprehensive Dirac equation.

Foundations of Relativistic Quantum Mechanics

Need for relativistic quantum mechanics

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• Non-relativistic quantum mechanics breaks down at high energies and fails to describe particles approaching light speed
• Special relativity principles (mass-energy equivalence $E = mc^2$, Lorentz transformations) must be incorporated
• Unified theory needed to describe high-energy or high-velocity particles (particle accelerators, cosmic rays)
• Historical context: Dirac sought to reconcile quantum mechanics with special relativity
• Relativistic treatment required for spin-orbit coupling, fine structure in atomic spectra, antimatter prediction

Derivation of Klein-Gordon equation

• Starts with relativistic energy-momentum relation $E^2 = p^2c^2 + m^2c^4$
• Quantum operators replace classical variables: $E \rightarrow i\hbar\frac{\partial}{\partial t}$, $p \rightarrow -i\hbar\nabla$
• Substituting operators into energy-momentum relation yields Klein-Gordon equation
• Resulting equation: $(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 + \frac{m^2c^2}{\hbar^2})\phi = 0$
• Equation properties: second-order in time, Lorentz invariant
• Plane wave solutions take form $\phi(x,t) = Ae^{i(k\cdot x - \omega t)}$

Solutions of Klein-Gordon equation

• Positive and negative energy solutions: $E = \pm\sqrt{p^2c^2 + m^2c^4}$
• Probability density not positive-definite, challenging interpretation
• Wave function describes scalar particles (pions, Higgs boson)
• Predicts existence of antiparticles (positron)
• Klein paradox emerges for strong potentials, transmission coefficient exceeds unity
• Zitterbewegung observed in wave packet solutions, rapid oscillatory behavior

Limitations of Klein-Gordon equation

• Negative probability density violates Born interpretation of quantum mechanics
• Fails to describe particles with non-zero spin (electrons, protons)
• Negative energy states lack clear physical interpretation
• Second-order time derivative complicates single-particle interpretation
• Cannot predict magnetic moment or explain fine structure in atomic spectra
• Dirac equation addresses these issues: first-order in time and space, incorporates spin naturally
• Dirac equation successfully predicts electron's magnetic moment, explains spin-orbit coupling