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Essential Quantum Mechanics$
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Gary Bowman

Print publication date: 2007

Print ISBN-13: 9780199228928

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780199228928.001.0001

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Time Evolution

Time Evolution

(p.145) 11 Time Evolution
Essential Quantum Mechanics

Gary E. Bowman

Oxford University Press

The time-dependent Schrödinger equation forms the chapter's starting point. For a time-independent Hamiltonian, a quantum state may be time evolved with the standard, unitary time evolution operator. After obtaining this operator, an algorithm is provided by which a state may be time evolved, and measurement probabilities determined. A spin 1/2 example illustrates time evolution concretely. All this makes close contact with the discussion of unitary operators in Chapter 10. The time dependence of expectation values forms the next topic, followed by the concept of a constant of the motion, which is carefully contrasted with that of a stationary state. The energy-time uncertainty relations, which rest on a quite different conceptual footing than the ‘standard’ uncertainty relations, occupy the rest of the chapter. An example from spin 1/2 illustrates the energy-time uncertainty relations.

Keywords:   time-dependent Schrödinger equation, Hamiltonian, time evolution operator, expectation value, constant of the motion, energy-time uncertainty relations

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