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3: Mostly 1-D Quantum Mechanics

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Mar 5, 2022
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- 2.4: Complex Variable, Stationary Path Integrals
- 3.1: 1-D Schrödinger Equation - Example Systems

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Topic hierarchy

3.1: 1-D Schrödinger Equation - Example Systems
3.2: General Uncertainty Principal
If two physical variables correspond to commuting Hermitian operators, they can be diagonalized simultaneously -- that is, they have a common set of eigenstates. In these eigenstates both variables have precise values at the same time, there is no “Uncertainty Principle” requiring that as we know one of them more accurately, we increasingly lose track of the other. For example, the energy and momentum of a free particle can both be specified exactly. More interesting examples will appear in the
3.3: Energy-Time Uncertainty Principle
The momentum-position uncertainty principle Δp⋅Δx≥ℏ has an energy-time analog, ΔE⋅Δt≥ℏ. Evidently, though, this must be a different kind of relationship to the momentum-position one, because t is not a dynamical variable, so this can’t have anything to do with non-commutation. To illustrate the meaning of the equation Δ E⋅Δ t≥ℏ, let us reconsider α-decay,
3.4: The Simple Harmonic Oscillator
The simple harmonic oscillator, a nonrelativistic particle in a quadratic potential , is an excellent model for a wide range of systems in nature. In fact, not long after Planck’s discovery that the black body radiation spectrum could be explained by assuming energy to be exchanged in quanta, Einstein applied the same principle to the simple harmonic oscillator, thereby solving a long-standing puzzle in solid state physics—the mysterious drop in specific heat of all solids at low temperatures.
3.5: Propagators and Representations
We’ve spent most of the course so far concentrating on the eigenstates of the Hamiltonian, states whose time-dependence is merely a changing phase. We did mention much earlier a superposition of two different energy states in an infinite well, resulting in a wave function sloshing backwards and forwards. It’s now time to cast the analysis of time dependent states into the language of bras, kets and operators.
3.6: Coherent States
For a given initial position and momentum, classical mechanics correctly predicts the future path, as confirmed by experiments with real (admittedly not perfect) systems. But from the Hamiltonian we could also write down Schrödinger’s equation, and from that predict the future behavior of the system. Since we already know the answer from classical mechanics and experiment, quantum mechanics must give us the same result in the limiting case of a large system.
3.7: Path Integrals
In quantum mechanics, such as the motion of an electron in an atom, we know that the particle does not follow a well-defined path, in contrast to classical mechanics. Where does the crossover to a well-defined path take place? Feynman (in Feynman and Hibbs) gives a nice picture to help think about summing over paths.
3.8: Path Integrals for the SHO
3.9: Appendix- Some Exponential Operator Algebra

Thumbnail: A particle in a 1D infinite potential well of dimension $L$ .

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