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The Klein-Gordon and Dirac Equations

Science · Quantum & Modern Physics · ages 23-24

Name ______________________ Date ____________

What two obstacles block the unrestricted one-particle reading of the Klein-Gordon equation?
- A negative energy branch with no bottom, and an indefinite density
- Too many solutions and too few variables
- Faster-than-light speeds and imaginary masses, always, in every single case
What relation does the Klein-Gordon equation quantize?
- Half the mass times velocity squared
- The relativistic energy momentum relation
- The force equals mass times acceleration rule
For a local massive relativistic field in 3+1 dimensions, demanding first order in both time and space forces the Dirac coefficients to be matrices rather than plain numbers.
Circle one: True False
A beam of neutral silver atoms in the orbital ground state splits into two bands in Stern-Gerlach. What does that support?
- Electrons carrying no spin at all
- Four components of every atom
- Two spin states of the electron
In the standard massive Dirac representation, what are the four spinor components?
- Spinor entries, not universal particle labels
- Four different particles with unrelated masses
- Two particles and their two decay modes, always
For a complex scalar field of mass m > 0, why can the usual conserved Klein-Gordon density not serve as a probability for all solutions?
- Its conserved density is not positive definite
- It uses too many components
- It forbids all negative energies, every single one
The standard massive Dirac equation in 3+1 dimensions uses 4x4 coefficient matrices. Type the number of components of its spinor wavefunction.
Answer: ______________
For a massive charged Dirac field, what do the negative-frequency solutions become after quantization?
- They are removed from the theory as unphysical states, always
- Positive-energy antiparticles, same mass, opposite charge
- They become photons

Answer key

For grown-ups. Fold this page away before handing over the rest.

The Klein-Gordon and Dirac Equations W1-mt_njCW9OEevW-s1

A negative energy branch with no bottom, and an indefinite density · The formal one-particle energies E = ±√(p²c² + m²c⁴) have a negative branch with no bottom, and for a complex scalar of mass m > 0 the conserved density is proportional to the U(1) charge density, not a positive probability for arbitrary solutions.
The relativistic energy momentum relation · It quantizes E² = p²c² + m²c⁴ with E → iħ∂/∂t and p → −iħ∇, giving (□ + (mc/ħ)²)ψ = 0.
True · The Schrödinger equation is first order in time with scalar coefficients; the Dirac demand is first order in both time and space for a local massive field in 3+1 dimensions, whose Clifford relations nonzero scalars cannot meet.
Two spin states of the electron · The silver atoms are neutral with a single unpaired electron in the orbital ground state, so the splitting reflects the electron's two spin states.
Spinor entries, not universal particle labels · The four components are representation-dependent entries of the spinor; at fixed momentum there are two positive-frequency and two negative-frequency independent plane-wave solutions, and the particle or antiparticle reading applies only after quantization of a charged field.
Its conserved density is not positive definite · The density (iħ/2mc²)(ψ*∂ψ/∂t − ψ∂ψ*/∂t) is proportional to the U(1) charge density up to charge and normalisation, not a positive probability for arbitrary solutions.
4 · The smallest matrices satisfying the Clifford relations are 4x4, so the spinor carries four components in the standard representation.
Positive-energy antiparticles, same mass, opposite charge · For a massive charged Dirac field, the negative-frequency classical solutions are reinterpreted after quantization as positive-energy antiparticles of the same mass and opposite charge.