In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (the "external world") that acts as a generator of random phase. The introduction, however, is not quite clear to me:

The wave function $\Psi$ of a truly isolated system may be expressed as a linear superposition of a complete orthonormal set of stationary wave functions $\{\phi_n\}$

$$\Psi=\sum_n c_n \phi_n$$

Here we can regard the system plus the external world as a truly isolated system. The wave function $\Psi$ for this whole system will depend on both the coordinates of the system under consideration and the coordinates of the external world. If $\{\phi_n\}$ denotes a complete set of orthonormal wave functions of the system, then $\Psi$ is still formally given by the above equation, but the $c_n$ are to be interpreted as wave functions of the external world.

Why is that? The definition of $c_n$is

$$c_n= \langle \phi_n|\Psi \rangle$$

which is a complex number. How do I get from there to the $c_n$ being wave functions of the outside world?


Huang seems to conceal the fact that he splits the Hilbert space in two $\mathcal{H}=\mathcal{H}_S\otimes\mathcal{H}_E$, where $S$ is the system and $E$ is the envoirement. A general state in the total Hilber space can be written as $$\rvert\Psi\rangle=\sum_{n,m} \gamma_{nm}\ \rvert\phi_n\rangle\otimes\rvert\psi_m\rangle$$ where $\{\phi\}$ are a basis of $\mathcal{H}_S$ and $\{\psi\}$ are a basis of $\mathcal{H}_E$ and where $\gamma_{nm}$ are complex numbers defined as $$\gamma_{nm}=\langle \phi_n\otimes\psi_m\rvert\Psi\rangle$$ in general, $\Psi$ is an entangled state.

If you define $\rvert c_n\rangle= \sum_{m} \gamma_{nm}\ \rvert\psi_m\rangle$ then you can write $$\rvert\Psi\rangle=\sum_{n} \rvert c_n\rangle\otimes\rvert\phi_n\rangle$$ where the "coefficients" $c_n$ now are linear combinations of states describing the envoirement.

If you project onto the coordinate basis to obtain the wave functions you recover the Huang expression $$\Psi=\sum_{n}c_n\ \phi_n$$ where now $c_n=\langle x_E\rvert c_n\rangle$ and $\phi_n=\langle x_S\rvert \phi_n\rangle$


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