12 Position and the Free Particle
Our discussion of the free particle has so far been largely in terms of one observable, the momentum operator. The free particle Hamiltonian is given in terms of this operator and we have seen in section 11.5 that solutions of the Schr¨odinger equation behave very simply in momentum space. Since , momentum is a conserved quantity, and momentum eigenstates will remain momentum eigenstates under time evolution.
The Fourier transform interchanges momentum and position space, and a position operator can be defined that will play the role of the Fourier transform of the momentum operator. Position eigenstates will be position space -functions, but and the position will not be a conserved quantity. The time evolution of a state initially in a position eigenstate can be calculated in terms of a quantity called the propagator, which we will compute and study.
Chapter contents
- 12.1 The position operator
- 12.2 Momentum space representation
- 12.3 Dirac notation
- 12.4 Heisenberg uncertainty
- 12.5 The propagator in position space
- 12.6 Propagators in frequency-momentum space
- 12.7 Green’s functions and solutions to the Schr¨odinger equations
- 12.8 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 143、144、145、146、147、148、149、150、151、152、153、154、155
来源版本：2025-10-20
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