- #1
Diracobama2181
- 75
- 2
- Homework Statement
- Suppos that$$ D(\epsilon) ∝ \epsilon^
α $$ as $$\epsilon →0$$.
a) Show that for a free massive particle in D dimensions, α = (D − 2)/2 holds true.
b) The density of states is related to the number of energy eigenstates with energy less than or
equal to $$\epsilon$$, $$N(\epsilon)$$, by $$D(\epsilon) = \frac{dN(\epsilon)}
{d\epsilon} $$. On the basis of this observation, argue that for massive
particles in a harmonic-oscillator potential in D dimensions the exponent is α = D − 1.
- Relevant Equations
- $$\int D(\epsilon)d\epsilon \propto V$$ where $$D(\epsilon)$$ is the density of states.
The dubious assumption I am making is that the integral over the density of states is proportional to the volume in k space.
Since $$\epsilon=\frac{(\hbar)^2k^2}{2m}$$ for part a, and $$\epsilon=(\hbar)\omega k$$ for part b, and $$V\propto k^d$$ for d dimensions in k space.
So, $$\int D(\epsilon)d\epsilon \propto \epsilon^{\frac{D}{2}}$$ for part a and $$\int D(\epsilon)d\epsilon \propto \epsilon^{D}$$ for part b.
If I take the derivative with respect to $$\epsilon$$, I get $$ D(\epsilon) \propto \epsilon^{\frac{D-2}{2}}$$ and $$ D(\epsilon) \propto \epsilon^{D-1}$$. Is my reasoning correct?
Since $$\epsilon=\frac{(\hbar)^2k^2}{2m}$$ for part a, and $$\epsilon=(\hbar)\omega k$$ for part b, and $$V\propto k^d$$ for d dimensions in k space.
So, $$\int D(\epsilon)d\epsilon \propto \epsilon^{\frac{D}{2}}$$ for part a and $$\int D(\epsilon)d\epsilon \propto \epsilon^{D}$$ for part b.
If I take the derivative with respect to $$\epsilon$$, I get $$ D(\epsilon) \propto \epsilon^{\frac{D-2}{2}}$$ and $$ D(\epsilon) \propto \epsilon^{D-1}$$. Is my reasoning correct?