Pure States

A pure state is defined as a state that can be described by a ket vector $\vert\psi>$. Such state evolves in time according to the time dependent Schrödinger equation,

$$i\hbar \frac{\partial \vert\psi>}{\partial t} = \hat{H} \vert\psi>,$$ (1)

where $H$ is the Hamiltonian operator. Note that Eq. (1) is a deterministic equation of motion that allows one to determine the state vector at any time, once the initial conditions are provided. The state vector $\vert\psi>$ provides the maximum possible information of the system. It can be expanded in the basis set of eigenstates $\vert\phi _k>$ of an arbitrary Hermitian operator $\hat{o}$ that represents an observable of the system,

$$\vert\psi> = \sum_k a_k \vert\phi_k>,$$ (2)

where $\vert\phi _k>$ are the eigenstates of $\hat{o}$, with eigenvalues $o_k$,

$$\hat{o}\vert\phi_k> = o_k \vert\phi_k>.$$ (3)

The expansion coefficients $a_k$, introduced by Eq. (2), are complex numbers that can be written in terms of real amplitudes $p_k$ and phases $\theta_k$ as follows,

$$a_k =\sqrt{p_k} e^{i\theta_k}.$$ (4)

The coefficients $p_k$ determine the probability of observing the eigenvalue $o_k$ when the system is in state $\vert\psi>$. The expectation value of $\hat{o}$ is

$$<\psi\vert\hat{o}\vert\psi> = \sum_k p_k o_k,$$ (5)

i.e., the average of expectation values associated with states $\vert\phi _k>$. The expectation value of any arbitrary operator $\hat{A}$, which does not share a common set of eigenstates with $\hat{o}$, can be computed in the basis set of eigenstates of $\hat{o}$ as follows,

$$<\psi\vert\hat{A}\vert\psi> = \sum_k p_k <\phi_k\vert\hat{A}\vert... ...neq k}\sqrt{p_kp_j} e^{i (\theta_j -\theta_k)} <\phi_k\vert\hat{A}\vert\phi_j>.$$ (6)

Note that such an expectation value is not only determined by the average of expectation values associated with states k (i.e., the first term in the r.h.s of Eq. (6)), but also by the second term in that equation. Such second term is responsible for interferences, or coherences, between states $\mid \phi_k>$ and $\mid \phi_j>$ as determined by the phases $\theta_k$ and $\theta_j$. Consider a large number of N replicas of the system, all of them described by the same state vector $\vert\psi>$. Note that such collection of N replica systems is also described by a pure state. Therefore, the ensemble averages associated with the observables $\hat{o}$ and $\hat{A}$ of such a pure state will coincide with the expectation values given by the equations Eq. (5) and Eq. (6), respectively.