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+dE≡G(E)dE=V√2m32π2ℏ3√EdE
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?