Skip to main content
\(\require{cancel}\)
Physics LibreTexts

6.5: Quantum Mechanics of Free Particles

  • Page ID
    6368
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    “Particle in a box.” Periodic boundary conditions. k-space. In the thermodynamic limit, the dots in k-space become densely packed, and it seems appropriate to replace sums over levels with integrals over k-space volumes. (In fact, there is at least one situation (see equation 6.83) in which this replacement is not correct.)

    Density of levels in k-space:

    \[ \frac{V}{8 \pi^{3}} \quad(\text { worth memorizing. })\]

    Energy density of levels:

    \[ \text { number of one-body levels with } \epsilon_{r} \text { from } \mathcal{E} \text { to } \mathcal{E}+d \mathcal{E} \equiv G(\mathcal{E}) d \mathcal{E}=V \frac{\sqrt{2} m^{3}}{2 \pi^{2} \hbar^{3}} \sqrt{\mathcal{E}} d \mathcal{E}\]

    How to use energy density of levels:

    \[ \sum_{r} f\left(\epsilon_{r}\right) \approx \int_{0}^{\infty} G(\mathcal{E}) f(\mathcal{E}) d \mathcal{E}\]

    and this approximation (usually) becomes exact in the thermodynamic limit.

    6.5.1 Problems

    6.14 Free particles in a box

    We argued that, for a big box, periodic boundary conditions would give the same results as “clamped boundary conditions”. Demonstrate this by finding the density of levels for three-dimensional particle in a box problem.

    6.15 Density of levels for bound particles

    What is the density of levels \(G (\mathcal{E})\) for a one-dimensional harmonic oscillator with spring constant K? For a three-dimensional isotropic harmonic oscillator?

    6.16 Density of levels in d dimensions

    What is the density of levels \(G (\mathcal{E})\) for free particles subject to periodic boundary conditions in a world of d dimensions?


    6.5: Quantum Mechanics of Free Particles is shared under a CC BY-SA license and was authored, remixed, and/or curated by Daniel F. Styer.

    • Was this article helpful?