Relation between 1D and 3D partition function In a solid state physics text author said $$Z_{3D} = [ Z_{1D} ]^{3} $$ $$<E_{3D}>=3<E_{1D}>$$
Where $Z$ is partition function. Can anyone convince me why it is so? 
 A: You probably (you should really give more details when posting questions, so that those without your book can still help you) came across this in the Einstein model of the harmonic oscillators for the heat capacity estimation, in which case the energy eigenvalues in 1D (say for x-component) are given by:
$$
E_{nx} =\hbar \omega (1/2+n_x)
$$
With $n_x\ge 0$ we know all the possible energy states of the system, so the partition function is simply given by (sum of the weight of each state):
$$
Z_{1D} = \sum e^{-\beta E_{nx}} = \sum \exp{-\beta \hbar \omega (1/2+n)}
$$
In 3D, $E_n$ is given by:
$$
E_{nx,ny,nz} = \hbar \omega \left((1/2+n_x)+(1/2+n_y)+(1/2+n_z)\right)
$$
Substitute the above in the partition function expression again, with $n_x=n_y=n_z=n$:
$$
Z_{3D} = \sum \exp{-\beta \hbar \omega (3/2+3n)}=Z_{1D}^3
$$
As for the average of the energy $E,$ it is really straighforward to show for 3D once you've solved the 1D case (replace $Z$ by $Z_{1D}$):
$$
\langle E \rangle  = -\frac{1}{Z}\frac{\partial Z}{\partial \beta}=...
$$
It is rather trivial from here, so I leave the rest up to you, or simply read your book more carefully, as I know Steve H. Simon's book is really well contained.

An edit to include a discussion from comments:

Self-Made Man: Does it also hold for the classical Einstein solid?

Yes, because to solve $Z=\int d\mathbf{p}\int d\mathbf{r}\exp(-\beta\mathcal{H}(\mathbf{p},\mathbf{r}))$ for the Hamiltonian of the harmonic oscillator, you have $6$ independent Gaussian integrals to solve, specifically because $\mathcal{H}\propto \mathbf{p}^2+\mathbf{r}^2$, so just solve the 1D case, which should give you $2\pi/\beta \left(\frac{m}{k}\right)^{1/2}$, and you will straightforwardly find for the 3D case that $Z_{3D}\propto \left(\frac{m}{k}\right)^{3/2}$ hence the same relation $Z_{3D}=Z_{1D}^3$ is obtained, I have not done the calculations exactly, so I have not included all constants and coefficients.
