BIGpedia - Quantum state - Encyclopedia and Dictionary Online

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A quantum state is any possible state in which a quantum mechanical system can be. A quantum state can be described by a state vector or a wavefunction, or in the case of an ensemble, quantum states can be described by a density operator.

Contents

1 Bra-ket notation

2 Basis States

3 Superposition of States

4 Pure and Mixed States

5 See Also

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 $|\!\!\uparrow\rangle$ 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 $|\psi\rangle$ can be expressed in terms of a sum of basis states (also called basis kets), $|k_i\rangle$

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

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

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

The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state $|n\rangle$ 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 $a^{\dagger}$ and a destruction operator $a$ in what is called the ladder operator method.

Superposition of States

If a quantum mechanical state $|\psi\rangle$ can be reached by more than one path, then $|\psi\rangle$ 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 $|\psi\rangle$ 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 $|\alpha\rangle$ and $|\beta\rangle$ , the two states $|0\rangle$ and $|1\rangle$ both 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, $|0\rangle$ is said to constructively interfere, and $|1\rangle$ 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 $\langle a \rangle$ of a measurement $A$ on a pure quantum state is given by

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

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

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

where $p s$ is the fraction of each ensemble in pure state $|\psi_s\rangle$ . 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.