Specification of the state of a many particle system

How do we determine the state of a many particle system? Well, let us, first of all, consider the simplest possible many particle system, which consists of a single spinless particle moving classically in one dimension. Assuming that we know the particle's equation of motion, the state of the system is fully specified once we simultaneously measure the particle's position $q$ and momentum $p$. In principle, if we know $q$ and $p$ then we can calculate the state of the system at all subsequent times using the equation of motion. In practice, it is impossible to specify $q$ and $p$ exactly, since there is always an intrinsic error in any experimental measurement.

Consider the time evolution of $q$ and $p$. This can be visualized by plotting the point ($q$, $p$) in the $q$-$p$ plane. This plane is generally known as phase-space. In general, the point ($q$, $p$) will trace out some very complicated pattern in phase-space. Suppose that we divide phase-space into rectangular cells of uniform dimensions $\delta q$ and $\delta p$. Here, $\delta q$ is the intrinsic error in the position measurement, and $\delta p$ the intrinsic error in the momentum measurement. The ``area'' of each cell is

$$\delta q\, \delta p = h_0,$$ (85)

where $h_0$ is a small constant having the dimensions of angular momentum. The coordinates $q$ and $p$ can now be conveniently specified by indicating the cell in phase-space into which they plot at any given time. This procedure automatically ensures that we do not attempt to specify $q$ and $p$ to an accuracy greater than our experimental error, which would clearly be pointless.

Let us now consider a single spinless particle moving in three dimensions. In order to specify the state of the system we now need to know three $q$-$p$ pairs: i.e., $q_x$-$p_x$, $q_y$-$p_y$, and $q_z$-$p_z$. Incidentally, the number of $q$-$p$ pairs needed to specify the state of the system is usually called the number of degrees of freedom of the system. Thus, a single particle moving in one dimension constitutes a one degree of freedom system, whereas a single particle moving in three dimensions constitutes a three degree of freedom system.

Consider the time evolution of ${\bf q}$ and ${\bf p}$, where ${\bf q}=(q_x,q_y,q_z)$, etc. This can be visualized by plotting the point (${\bf q}$, ${\bf p}$) in the six dimensional ${\bf q}$-${\bf p}$ phase-space. Suppose that we divide the $q_x$-$p_x$ plane into rectangular cells of uniform dimensions $\delta q$ and $\delta p$, and do likewise for the $q_y$-$p_y$ and $q_z$-$p_z$ planes. Here, $\delta q$ and $\delta p$ are again the intrinsic errors in our measurements of position and momentum, respectively. This is equivalent to dividing phase-space up into regular six dimensional cells of volume $h_0^{~3}$. The coordinates ${\bf q}$ and ${\bf p}$ can now be conveniently specified by indicating the cell in phase-space into which they plot at any given time. Again, this procedure automatically ensures that we do not attempt to specify ${\bf q}$ and ${\bf p}$ to an accuracy greater than our experimental error.

Finally, let us consider a system consisting of $N$ spinless particles moving classically in three dimensions. In order to specify the state of the system, we need to specify a large number of $q$-$p$ pairs. The requisite number is simply the number of degrees of freedom, $f$. For the present case, $f=3\,N$. Thus, phase-space (i.e., the space of all the $q$-$p$ pairs) now possesses $2\,f=6\,N$ dimensions. Consider a particular pair of conjugate coordinates, $q_i$ and $p_i$. As before, we divide the $q_i$-$p_i$ plane into rectangular cells of uniform dimensions $\delta q$ and $\delta p$. This is equivalent to dividing phase-space into regular $2\,f$ dimensional cells of volume $h_0^{~f}$. The state of the system is specified by indicating which cell it occupies in phase-space at any given time.

In principle, we can specify the state of the system to arbitrary accuracy by taking the limit $h_0\rightarrow 0$. In reality, we know from quantum mechanics that it is impossible to simultaneously measure a coordinate $q_i$ and its conjugate momentum $p_i$ to greater accuracy than $\delta q_i\, \delta p_i=\hbar$. This implies that

$$h_0\geq \hbar.$$ (86)

In other words, the uncertainty principle sets a lower limit on how finely we can chop up classical phase-space.

In quantum mechanics we can specify the state of the system by giving its wave-function at time $t$,

$$\psi(q_1,\cdots, q_f, s_1,\cdots, s_g, t),$$ (87)

where $f$ is the number of translational degrees of freedom, and $g$ the number of internal (e.g., spin) degrees of freedom. For instance, if the system consists of $N$ spin-one-half particles then there will be $3\,N$ translational degrees of freedom, and $N$ spin degrees of freedom (i.e., the spin of each particle can either point up or down along the $z$-axis). Alternatively, if the system is in a stationary state (i.e., an eigenstate of the Hamiltonian) then we can just specify $f+g$ quantum numbers. Either way, the future time evolution of the wave-function is fully determined by Schrödinger's equation.

In reality, this approach does not work because the Hamiltonian of the system is only known approximately. Typically, we are dealing with a system consisting of many weakly interacting particles. We usually know the Hamiltonian for completely non-interacting particles, but the component of the Hamiltonian associated with particle interactions is either impossibly complicated, or not very well known (often, it is both!). We can define approximate stationary eigenstates using the Hamiltonian for non-interacting particles. The state of the system is then specified by the quantum numbers identifying these eigenstates. In the absence of particle interactions, if the system starts off in a stationary state then it stays in that state for ever, so its quantum numbers never change. The interactions allow the system to make transitions between different ``stationary'' states, causing its quantum numbers to change in time.