San José State University

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Thayer Watkins
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The Nature and Derivation
of the Time-Energy
Uncertainty Relation

A particle in motion has probability density distributions for its location and momentum. Those distributions have variances and hence standard deviations, say σx and σp. According to the Uncertainty Principle of Werner Heisenberg the product of those standard deviations must equal or exceed Planck's constant divided by 4π.; i.e.,

σxσp ≥ h/(4π) = ½h

The dimensions of the product of the standard deviations are L(ML/T)=ML²/T. This suggests that the might be an uncertainty relation for pairs of variables whose product has the dimensions ML²/T). The dimensions of energy are ML²/T². The variable paired with energy in an uncertainty relation would have to have the dimension of time. Thus if ΔT and ΔE denote the uncertainty of time and energy respectively then the uncertainty relation would be

ΔEΔT ≥ ½h

But there is an essential difference between this relation and the one for location and momentum. Time does not have have a probability density distribution. Often in physical situations energy is constant so its uncertainty is zero. Thus an uncertainty relation for time and energy must have a quite different derivation and interpretation than Heisenberg's Uncertainty Principle for locattion and momentum.

Let Q(p, x) be any measurable quantity of a physical system and Q^ its operator. It is known by the Ehrenfest Theorem that the time rate of the expected value of Q is given by

d(E{Q: Ψ})/dt = (i/h)[H^, Q^]

where [H^, Q^] is the commutator of H^, the operator of the Hamiltonian of the system, with the operator Q^.

From the generalized uncertainty relation, this means that

σHσQ ≥ (h/2)|dE{Q: Ψ}/dt|

Then ΔT may be defined as the time it takes for Q to change by an amount σQ; i.e.,

ΔT = σQ/|dE{Q: Ψ}/dt|

ΔE is given by σH.

Then

ΔEΔT = σHQ/|dE{Q: Ψ}/dt|)
= (σHσQ)/|dE{Q: Ψ}/dt|)
≥ (h/2)|dE{Q: Ψ}/dt|/|dE{Q: Ψ}/dt|
= ½h

Thus

ΔEΔT ≥ ½h


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