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1.85: The Difference Between Fermions and Bosons

  • Page ID
    157653
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    \[
    \mathrm{n}_{1} :=1 \qquad \mathrm{n}_{2} :=2 \\ \Psi(x) :=\sqrt{2} \cdot \sin \left(n_{1} \cdot \pi \cdot x\right) \qquad \Phi(x) :=\sqrt{2} \cdot \sin \left(n_{2} \cdot \pi \cdot x\right)
    \nonumber \]

    Calculate the average separation, |x1 - x2|, for two fermions and two bosons in a 1D box of unit length.

    Fermions have antisymmetric wave functions:

    \[
    \Psi_{\mathrm{f}}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) :=\frac{\Psi\left(\mathrm{x}_{1}\right) \cdot \Phi\left(\mathrm{x}_{2}\right)-\Psi\left(\mathrm{x}_{2}\right) \cdot \Phi\left(\mathrm{x}_{1}\right)}{\sqrt{2}}
    \nonumber \]

    The average particle separation for indistinquishable fermions:

    \[
    \int_{0}^{1} \int_{0}^{1} \Psi_{\mathrm{f}}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) \cdot\left|\mathrm{x}_{1}-\mathrm{x}_{2}\right| \cdot \Psi_{\mathrm{f}}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) \mathrm{dx}_{1} \mathrm{dx}_{2}=0.383
    \nonumber \]

    The particles are correlated so as to keep them apart.

    \[
    \mathrm{N} :=40 \qquad \mathrm{i} :=0 . . \mathrm{N} \\ x_{1_{i}} :=\frac{1}{N} \quad j :=0 \ldots N \qquad x_{2} :=\frac{j}{N} \\ \Psi_{\mathrm{f}_{\mathrm{i}, \mathrm{j}}} :=\Psi_{\mathrm{f}\left(\mathrm{x}_{\mathrm{1_{i}}}, \mathrm{x}_{2_{j}}\right)^{2}}
    \nonumber \]

    clipboard_e33e094e182145d189200b2fd082dc67a.png

    Bosons have symmetric wave functions:

    \[
    \Psi_{\mathrm{b}}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=\frac{\Psi\left(\mathrm{x}_{1}\right) \cdot \Phi\left(\mathrm{x}_{2}\right)+\Psi\left(\mathrm{x}_{2}\right) \cdot \Phi\left(\mathrm{x}_{1}\right)}{\sqrt{2}}
    \nonumber \]

    The average particle separation for indistinquishable bosons:

    \[
    \int_{0}^{1} \int_{0}^{1} \Psi_{\mathrm{b}}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) \cdot\left|\mathrm{x}_{1}-\mathrm{x}_{2}\right| \cdot \Psi_{\mathrm{b}}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) \mathrm{dx}_{1} \mathrm{dx}_{2}=0.157
    \nonumber \]

    The particles are correlated so as to bring them closer together.

    \[
    \mathrm{N} :=40 \qquad \mathrm{i} :=0 . . \mathrm{N} \\ x_{1_{i}} :=\frac{1}{N} \quad j :=0 \ldots N \qquad x_{2} :=\frac{j}{N} \\ \Psi_{\mathrm{b}_{\mathrm{i}, \mathrm{j}}} :=\Psi_{\mathrm{b}\left(\mathrm{x}_{\mathrm{1_{i}}}, \mathrm{x}_{2_{j}}\right)^{2}}
    \nonumber \]

    clipboard_e787ff274e05dce8a4120840533d28279.png

    All fundamental particles (electrons, neutrons, protons, photons, etc.) are either bosons or fermions. Composite entities such as the elements also fall into these two categories. The fundamental distinction is spin: bosons have integer spin (0, 1, 2, ...) while fermions have half-integer spin (1/2, 3/2, ....).

    The dramatic difference in behavior between bosons and fermions has led to a sociology of fundamental particles. Bosons are social and gregarious, while fermions are antisocial and aloof.


    This page titled 1.85: The Difference Between Fermions and Bosons is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform.