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Why is energy expressed as the second-order partial differential of a wave function in quantum mechanics?
Wiki User
∙ 14y ago
You are referring to the Schrodinger Equation. This is because it comes from the classical view that the total energy is equal to the hamiltonian of a system:
Kinetic Energy + Potential Energy = Total energy.
Classically the kinetic energy is (1/2)mv2 = p2/(2m) ; where m is mass, v is velocity, p is momentum (p=mv).
Now the momentum operator in QM is p=iħ∇ ;where ∇ is the gradient operator.
This therefore yields the QM hamiltonian [-ħ2∇2/(2m) + V(x,y,z)]Ψ = EΨ
Now a more fun question to ask would be "Why is the Hamiltonian a function of the second-order partial differential with respect to position but the time dependent is only a function of a first-order differential with respect to time?"
meaning
HΨ = -iħ(dΨ/dt) or
[-ħ2∇2/(2m) + V(x,y,z)]Ψ = -iħ(dΨ/dt)
hint: Think Maxwell's Equations!
Wiki User
∙ 14y ago
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