# Quantum state

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.

## Bra-ket notation

Paul Dirac invented a powerful and intuitive mathematical notation to describe quantum states, known as bra-ket notation. For instance, one can refer to an |excited atom> or to $\displaystyle |\!\!\uparrow\rangle$ for a spin-up particle, hiding the underlying complexity of the mathematical description, which is revealed when the state is projected onto a coordinate basis. For instance, the simple notation |1s> describes the first hydrogen atom bound state, but becomes a complicated function in terms of Laguerre polynomials and spherical harmonics when projected onto the basis of position vectors |r>. The resulting expression Ψ(r)=<r|1s>, which is known as the wavefunction, is a special representation of the quantum state, namely, its projection into position space. Other representations, like the projection into momentum space, are possible. The various representations are simply different expressions of a single physical quantum state.

## Basis states

Any quantum state $\displaystyle |\psi\rangle$ can be expressed in terms of a sum of basis states (also called basis kets), $\displaystyle |k_i\rangle$

$\displaystyle | \psi \rangle = \sum_i c_i | k_i \rangle$

where $\displaystyle c_i$ are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, $\displaystyle \left | c_i \right | ^2$ is the probability of a measurement in terms of the basis states yielding the state $\displaystyle |k_i\rangle$ . The normalization condition mandates that the total sum of probabilities is equal to one,

$\displaystyle \sum_i \left | c_i \right | ^2 = 1$ .

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

## Superposition of states

If a quantum mechanical state $\displaystyle |\psi\rangle$ can be reached by more than one path, then $\displaystyle |\psi\rangle$ is said to be a linear superposition of states. In the case of two paths, if the states after passing through path $\displaystyle \alpha$ and path $\displaystyle \beta$ are

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

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

then $\displaystyle |\psi\rangle$ is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields

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

Note that in the states $\displaystyle |\alpha\rangle$ and $\displaystyle |\beta\rangle$ , the two states $\displaystyle |0\rangle$ and $\displaystyle |1\rangle$ each have a probability of $\displaystyle \begin{matrix}\frac{1}{2}\end{matrix}$ , as obtained by the absolute square of the probability amplitudes, which are $\displaystyle \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}$ and $\displaystyle \begin{matrix}\pm\frac{1}{\sqrt{2}}\end{matrix}$ . 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, $\displaystyle |0\rangle$ is said to constructively interfere, and $\displaystyle |1\rangle$ 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 $\displaystyle \langle a \rangle$ of a measurement $\displaystyle A$ on a pure quantum state is given by

$\displaystyle \langle a \rangle = \langle \psi | A | \psi \rangle = \sum_i a_i \langle \psi | \alpha_i \rangle \langle \alpha_i | \psi \rangle = \sum_i a_i | \langle \alpha_i | \psi \rangle |^2 = \sum_i a_i P(\alpha_i)$

where $\displaystyle |\alpha_i\rangle$ are basis kets for the operator $\displaystyle A$ , and $\displaystyle P(\alpha_i)$ is the probability of $\displaystyle | \psi \rangle$ being measured in state $\displaystyle |\alpha_i\rangle$ .

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

$\displaystyle \rho = \sum_s p_s | \psi_s \rangle \langle \psi_s |$

where $\displaystyle p_s$ is the fraction of each ensemble in pure state $\displaystyle |\psi_s\rangle$ . The ensemble average of a measurement $\displaystyle A$ on a mixed state is given by

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

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.