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4: One-Dimensional Potentials

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    In this chapter, we shall investigate the interaction of a non-relativistic particle of mass \(m\) and energy \(E\) with various one-dimensional potentials, \(V(x)\). Because we are searching for stationary solutions with unique energies, we can write the wavefunction in the form (see Section [sstat])

    \[\psi(x,t) = \psi(x)\,\rm e^{-{\rm i}\,E\,t/\hbar}, \nonumber \]

    where \(\psi(x)\) satisfies the time-independent Schrödinger equation:

    \[ \frac{d^2 \psi}{d x^2} = \frac{2\,m}{\hbar^2} \left[V(x)-E\right]\psi. \label{e5.2} \]

    In general, the solution, \(\psi(x)\), to the previous equation must be finite, otherwise the probability density \(|\psi|^2\) would become infinite (which is unphysical). Likewise, the solution must be continuous, otherwise the probability current \ref{eprobc} would become infinite (which is also unphysical).

    • 4.1: Infinite Potential Well
      This page explains a particle of mass \(m\) in a one-dimensional potential box, where the wave function is zero at the boundaries due to infinite potential outside. This leads to discrete energy eigenvalues and solutions of the form \(\psi_n(x) = \sqrt{\frac{2}{a}}\,\sin\left(n\,\pi\,\frac{x}{a}\right)\). These solutions exhibit orthonormal properties, and the overall wave function can be expressed as a superposition of stationary states.
    • 4.2: Square Potential Barrier
      This page examines the interaction of a particle with mass \(m\) and energy \(E\) with a square potential barrier, emphasizing wavefunction solutions and the concepts of reflection and transmission probabilities. It covers the continuity conditions at barrier edges, leading to expressions for \(|R|^2\) and \(|T|^2\) while affirming the relation \(|R|^2 + |T|^2 = 1\).
    • 4.3: WKB Approximation
      This page explains the behavior of a quantum particle in a slowly varying potential using the WKB approximation. It highlights that the wavefunction remains constant in allowed regions and decays in potential barriers. Transmission probability for tunneling is derived, indicating that the WKB approximation is most accurate when the particle's de Broglie wavelength is much smaller than the barrier's width.
    • 4.4: Cold Emission
      This page explains how an unheated metal surface in a uniform electric field influences electron emission through the Fowler-Nordheim formula. It describes the triangular potential barrier confining electrons, where increased electric field strength enhances tunneling probabilities. This principle is crucial for the operation of scanning tunneling microscopes (STMs), which leverage electron tunneling currents to achieve atomic-level surface mapping.
    • 4.5: Alpha Decay
      This page covers alpha decay in heavy atomic nuclei, explaining how an alpha particle emission transforms the nucleus into a daughter nucleus. It outlines George Gamow's 1928 theory of tunneling through a potential barrier using the WKB approximation.
    • 4.6: Square Potential Well
      This page examines particles in a square potential well with a focus on energy conditions, distinguishing between unbounded (E>0) and bounded (E<0) states. It classifies Schrödinger's equation solutions as symmetric, resulting in two bound states, or anti-symmetric, leading to one bound state.
    • 4.7: Simple Harmonic Oscillator
      This page covers the quantum mechanics of a simple harmonic oscillator, detailing the Hamiltonian and Schrödinger equation application. It derives quantized energy levels spaced by \(\hbar \omega\), identifies the ground state energy as \((1/2) \hbar \omega\), and introduces raising and lowering operators (\(a_+\) and \(a_-\)) for energy state transitions.
    • 4.E: One-Dimensional Potentials (Exercises)
      This page covers quantum mechanics scenarios involving wavefunctions, starting with a particle in an infinite one-dimensional square well, its revival time, and time evolution. It addresses expanding the well and energy measurement, along with particle behavior at potential steps and delta-function potentials, including reflection probabilities and bound states.

    This page titled 4: One-Dimensional Potentials is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.