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UEA05_2

# UEA05_2 - 10/PH04/QM Markus Bobrowski Abstracts on Quantum...

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10/PH04/QM.- Markus Bobrowski Abstracts on Quantum Mechanics Ehrenfest’s Theorem and Quantum Virial Theorem We first state a formula representing a general form of Ehrenfest’s theorem: Theorem 1 (Ehrenfest) For any quantum mechanical operator, the equation d dt D ˆ O E = i ~ Dh ˆ H, ˆ O iE + * ˆ O ∂t + (1) holds, where ˆ H ist the Hamiltonian Operator of the System. proof. This can most easily derived in the Heisenberg picture. Per definition ˆ O H = ˆ U ˆ O S ˆ U , with ˆ U being the (unitary) time evolution operator. Hence, d ˆ O H dt = ˆ U ∂t ˆ O S ˆ U + ˆ U ˆ O S ˆ U ∂t + ˆ U ˆ O S ∂t ˆ U = - 1 i ~ ˆ U ˆ H ˆ U ˆ U | {z } 1 ˆ O S ˆ U + ˆ U ˆ O S ˆ U ˆ U | {z } 1 1 i ~ ˆ H ˆ U + ˆ O S ∂t ! H = 1 i ~ h ˆ O H , ˆ U ˆ H ˆ U i + ˆ O ∂t ! H = 1 i ~ h ˆ O H , ˆ H i + ˆ O S ∂t ! H . Since the state kets are time-independent in the Heisenberg picture, the time derivations

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UEA05_2 - 10/PH04/QM Markus Bobrowski Abstracts on Quantum...

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