States

A quantum state is any possible state in which a quantum mechanical system can be. A fully specified quantum state can be described by a state vector, a wavefunction, or a complete set of quantum numbers for a specific system. A partially known quantum state, such as an ensemble with some quantum numbers fixed, can be described by a density operator.

Paul A. M. Dirac invented a powerful and intuitive mathematical notation to describe quantum states, known as bra-ket notation.

Basis states

Any quantum state ∣ψ⟩ can be expressed in terms of a sum of basis states (also called basis kets), ∣ki

ψ⟩ = ∑iciki

where ci are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, ∣ci∣2 is the probability of a measurement in terms of the basis states yielding the state ∣ki⟩. The normalization condition mandates that the total sum of probabilities is equal to one,

ici∣2 = 1.

The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state ∣n⟩ has an energy $E_n = \hbar \omega \left(n + {\begin{matrix}\frac{1}{2}\end{matrix}}\right)$. The set of basis states can be extracted using a construction operator a †  and a destruction operator a in what is called the ladder operator method.

Superposition of states

If a quantum mechanical state ∣ψ⟩ can be reached by more than one path, then ∣ψ⟩ is said to be a linear superposition of states. In the case of two paths, if the states after passing through path α and path β are

$|\alpha\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle$, and

$|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle$,

then ∣ψ⟩ is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields

$|\psi\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\alpha\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) = |0\rangle$.

Note that in the states ∣α⟩ and ∣β⟩, the two states ∣0⟩ and ∣1⟩ each have a probability of $\begin{matrix}\frac{1}{2}\end{matrix}$, as obtained by the absolute square of the probability amplitudes, which are $\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}$ and $\begin{matrix}\pm\frac{1}{\sqrt{2}}\end{matrix}$. In a superposition, it is the probability amplitudes which add, and not the probabilities themselves. The pattern which results from a superposition is often called an interference pattern. In the above case, ∣0⟩ is said to constructively interfere, and ∣1⟩ is said to destructively interfere.

For more about superposition of states, see the double-slit experiment.

Pure and mixed states

A pure quantum state is a state which can be described by a single ket vector, or as a sum of basis states. A mixed quantum state is a statistical distribution of pure states.

The expectation value ⟨a⟩ of a measurement A on a pure quantum state is given by

a⟩ = ⟨ψAψ⟩ = ∑iaiψαi⟩⟨αiψ⟩ = ∑iai∣⟨αiψ⟩∣2 = ∑iaiP(αi)

where ∣αi⟩ are basis kets for the operator A, and P(αi) is the probability of ∣ψ⟩ being measured in state ∣αi⟩.

In order to describe a statistical distribution of pure states, or mixed state, the density operator (or density matrix), ρ, is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as

ρ = ∑spsψs⟩⟨ψs

where ps is the fraction of each ensemble in pure state ∣ψs⟩. The ensemble average of a measurement A on a mixed state is given by

$\left [ A \right ] = \langle \overline{A} \rangle = \sum_s p_s \langle \psi_s | A | \psi_s \rangle = \sum_s \sum_i p_s a_i | \langle \alpha_i | \psi_s \rangle |^2 = tr(\rho A)$

where it is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average over the ensemble of pure states.