quantum state

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.

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

c_i^2

is the probability of a measurement in terms of the basis states yielding the state

|k_i\rangle

. The normalization condition mandates that

\sum_i c_i^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 + {1\over 2}\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.

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

\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

|\alpha_i\rangle

are basis kets for the operator

A

, and

P(\alpha_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),

\rho

, 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

= \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.

Quantum State

Bra-ket notation

Basis States

Pure and Mixed States

See Also