# Understanding the product of partition functions by making sense of the maths and the physics

I have $N$ distinguishable particles in a 1D harmonic oscillator potential with 'proper' frequency $\omega$. The particles also have internal spin-$\frac12$ degrees of freedom in a magnetic field $B$ with magnetic dipole $\mu$. The spin induced energy level splitting is $$\varepsilon=2\mu B=0.1\hbar\omega$$

Question: Show that the partition function is a product of the oscillator ($Z_1$) and spin ($Z_2$) partition functions.

I have a response but I'm pretty dissatisfied with it.

I argued that $$Z=\sum_n\exp(\beta\varepsilon_n)$$ so for $\varepsilon_i$ (oscillation energy) and $\varepsilon_j$ (spin energy) we have:

$$Z=\sum_n\exp(\beta\varepsilon_n)=\sum_{i,j}\exp\bigl(\beta(\varepsilon_{i}+\varepsilon_{j})\bigr)=\sum_i\sum_j\exp(\beta\varepsilon_i)\exp(\beta\varepsilon_j)=Z_1Z_2$$

(Edit: I actually don't think $\sum_{i,j}\leftrightarrow \sum_i \sum_j$ is valid)

This is where I left things for the time being but I wonder if I should elaborate.

For example, I considered... $$\varepsilon=2\mu B$$ ...for the 3 level spin and argued that... \begin{align} Z_2 &=\sum \exp(\beta\varepsilon_2)=\exp(-0.5\beta\varepsilon_2)+1+\exp(0.5\beta\varepsilon_2) \\ &=\exp(-0.05\beta\hbar\omega)+1+\exp(0.05\beta\hbar\omega) \end{align} ...for 3 energy levels $-0.5\varepsilon$, $0$ and $0.5\varepsilon$

For the harmonic oscillator I argued that... $$\varepsilon_1=\biggl(i+\frac{1}{2}\biggr)\hbar\omega$$ ...such that... \begin{align} Z_1 &=\sum_i \exp(\beta\varepsilon_1)=\sum_i \exp\biggl[\beta \biggl(i+\frac{1}{2}\biggr)\hbar\omega\biggr] \\ &=\frac{1}{2}\hbar\omega \sum_i \exp(\beta i\hbar\omega)=\frac{1}{\exp(-0.5\beta\varepsilon)-\exp(0.5\beta\varepsilon)} \end{align}

If the above is correct then I should be able to arrive at $Z=Z_1Z_2$ easily. However, the outcome of the multiplication is unrecognisable to me. This is one of my challenges with this topic - trying to reconcile the maths with the physics

Clarification:

The above is the homework part. I'm also experiencing difficulty grasping some of the ideas in statistical physics, in particular, the relationship between the partition function, entropy and specific heat capacity

My understanding is that the partition function shows the number of microstates that are accessible within thermal energy range. Hence, the number of microstates that are accessible within thermal energy range is the product of microstates in the 1d oscillator and the 3 level system. Would this be 4 degrees of freedom in total (1 from the oscillator and 3 from the splitting)?

I have to say I'm very uncomfortable with my lack of physical understanding for the material. The relationship between the partition function, entropy and specific heat capacity is still very fuzzy in my mind.

• Hi FractualHallEffect, and welcome to Physics Stack Exchange! I edited your question title to improve it, but I'm not quite sure what you're asking so perhaps my edit to the title is wrong. Could you check on it, pick a better title if necessary, and perhaps also edit your question to make it really clear what conceptual issue you want to ask about? – David Z May 7 '14 at 18:01
• Hi David Z. Thanks for the edits. I've gone back and clarified my question. – FractualHallEffect May 8 '14 at 0:41
• For multiletter subscripts or superscripts you can use braces, like z_{oscillator}. Better yet, use z_{\text{oscillator}} so the subscript renders as text, not in italics. I'll make an edit that fixes all that shortly. – David Z May 8 '14 at 0:53