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6.5: Quantum Mechanics of Free Particles

( \newcommand{\kernel}{\mathrm{null}\,}\)

“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:

V8π3( worth memorizing. )

Energy density of levels:

 number of one-body levels with ϵr from E to E+dEG(E)dE=V2m32π23EdE

How to use energy density of levels:

rf(ϵr)0G(E)f(E)dE

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(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(E) for free particles subject to periodic boundary conditions in a world of d dimensions?


This page titled 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.

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