Why can't a particle at rest decay when its energy is real-valued?

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SUMMARY

A particle at rest cannot decay when its energy is real-valued due to the implications of the relativistic Breit-Wigner distribution and the lifetime of the particle. The width of resonance, represented by Γ, is defined as Γ = ħ/τ, where τ is the particle's lifetime. If τ is infinite, the decay width approaches zero, indicating no decay occurs. Introducing an imaginary component to the energy E allows for a non-zero decay width, facilitating the decay process.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly wave functions
  • Familiarity with the relativistic Breit-Wigner distribution
  • Knowledge of resonance width and its relation to particle lifetime
  • Basic grasp of complex numbers in the context of energy values
NEXT STEPS
  • Study the implications of complex energy values in quantum mechanics
  • Explore the derivation and applications of the relativistic Breit-Wigner distribution
  • Investigate the relationship between particle lifetime and decay width in detail
  • Learn about the role of imaginary components in quantum state evolution
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Students and professionals in physics, particularly those focusing on quantum mechanics and particle physics, will benefit from this discussion.

Tiberius47
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Homework Statement


Show that for real valued E a particle at rest can not decay.


Homework Equations


Width of resonance.
\Gamma=\hbar/\tau
Relativistic Breit-Wigner distribution
P_rel_=\Gamma/((E^2-M^2)^2+M^2\Gamma^2)
Wave function of particle at rest.
ψ(t)=ψ(0)e^(-iEt/\hbar)

The Attempt at a Solution


I guess Lambda would be 0 if tau (the lifetime of the particle) were infinite but I have no idea what that has to do with real valued E.
 
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Put a imaginary part on E and see the effect on |ψ|² and the total ∫|ψ|²dx .
That's just a simple model to represent the decay of a particle.
The total probability of presence would decay.
 
Oh wow, duh. It's been over a year since I've taken quantum and I assumed the distribution would be relevant somehow.
 

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