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Think of this not as an extremely rigorous way of solving the differential equation, but rather as using your intuition to guess a solution. Often when you are given a differential equation, the solution is not at all obvious, and perhaps the equation isn't even solvable analytically. Instead of giving up, though, sometimes you can identify a parameter ...


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When calculating expectation values, you need to know a few things: What is my random variable? What is my distribution function? What is my desired quantity in terms of the random variable? The general form in one dimension would look like this $$ \langle G(x) \rangle = \frac{\int_{x_{min}}^{x_{max}} f(x) G(x) dx}{\int_{x_{min}}^{x_{max}} f(x) dx} \, ...


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The first thing we can do is to split up $\Gamma$ according to the number of particles in the given states. Let $\gamma_N$ be a state with $N$ particles. The grand canonical partition function is then \begin{align} \mathcal{Z} = & \sum_\Gamma \exp\left(-\beta(\mathcal{H} - \mu N)\right)\\ =& \sum_{N=0}^\infty\exp\left(\beta \mu N ...



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