6.5: Quantum Mechanics of Free Particles

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“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?