# Density operator

The density operator (also statistical operator ) is a linear operator that describes the state of an ensemble of physical systems or an element of such an ensemble. This description is complete from a physical point of view. This means that with the help of the density operator, the expected value can be predicted for every possible measurement in the system or ensemble . If the system is in a mixed state , the density operator specifies in particular the probability with which a system selected from the ensemble is in a certain pure state. If the operator is represented as a matrix (with reference to a basis), one speaks of the density matrix (or the statistical matrix ); this is used a lot in quantum statistics .

The density operator was originally developed in the context of classical physics by George Gabriel Stokes for the polarization state of a light beam ( Stokes parameter ). He was introduced to quantum mechanics in 1927 by Lew Landau and John von Neumann and then presented in detail by Paul Dirac in Principles of Quantum Mechanics (1930) and by John von Neumann in Mathematical Fundamentals of Quantum Mechanics (1932).

## construction

### Density operator for a pure quantum mechanical state

For a pure state with a (normalized) state vector , the density operator is called (in Bra-Ket notation) ${\ displaystyle | \ psi \ rangle}$

${\ displaystyle {\ hat {\ rho}} = \ left | \ psi \ right \ rangle \ left \ langle \ psi \ right |}$.

This operator remains unchanged if the same state had been described by a state vector . Therefore, unlike the state vector, there is an unambiguous association in both directions between the physical state and its density operator. ${\ displaystyle \ mathrm {e} ^ {i \ varphi} | \ psi \ rangle}$

This operator is a projection operator , because applied to any state vector , it projects it onto the 1-dimensional subspace of the Hilbert space that is determined by: ${\ displaystyle {\ hat {\ rho}} = {\ hat {\ mathbb {P}}} _ {\ psi}}$${\ displaystyle | \ phi \ rangle}$${\ displaystyle | \ psi \ rangle}$

${\ displaystyle {\ hat {\ rho}} | \ phi \ rangle = \ left | \ psi \ right \ rangle \ left \ langle \ psi \ right | \ phi \ rangle}$,

where the number factor is the scalar product of both vectors. is Hermitian, unitary, and idempotent (i.e. ). Its eigenvalues ​​are 1 (for all vectors of the same pure state) and zero (all orthogonal vectors). ${\ displaystyle \ langle \ psi | \ phi \ rangle}$${\ displaystyle {\ hat {\ rho}}}$${\ displaystyle {\ hat {\ rho}} ^ {2} = {\ hat {\ rho}}}$

For a coherent, i.e. pure superposition

${\ displaystyle | \ Psi \ rangle = \ alpha | \ psi \ rangle + \ beta | \ phi \ rangle}$

the density operator can be expressed by the two superimposed states (with the complex conjugation and ): ${\ displaystyle {\ hat {\ rho}} = \ left | \ Psi \ right \ rangle \ left \ langle \ Psi \ right |}$${\ displaystyle x ^ {*} x = | x | ^ {2}}$

{\ displaystyle {\ begin {aligned} {\ hat {\ rho}} & = \ left | \ Psi \ right \ rangle \ left \ langle \ Psi \ right | \\ & = \ left (\ alpha | \ psi \ rangle + \ beta | \ phi \ rangle \ right) \ left (\ alpha ^ {*} \ langle \ psi | + \ beta ^ {*} \ langle \ phi | \ right) \\ & = | \ alpha | ^ {2} | \ psi \ rangle \ langle \ psi | + \ alpha \ beta ^ {*} | \ psi \ rangle \ langle \ phi | + \ alpha ^ {*} \ beta | \ phi \ rangle \ langle \ psi | + | \ beta | ^ {2} | \ phi \ rangle \ langle \ phi | \ end {aligned}}}.

If and are orthogonal and are taken as basis vectors then is through the matrix ${\ displaystyle | \ psi \ rangle}$${\ displaystyle | \ phi \ rangle}$${\ displaystyle {\ hat {\ rho}}}$

${\ displaystyle {\ hat {\ rho}} = {\ begin {pmatrix} | \ alpha | ^ {2} & \ alpha \ beta ^ {*} \\\ alpha ^ {*} \ beta & | \ beta | ^ {2} \ end {pmatrix}}}$

given. The coherent linear combination is expressed in the off-diagonal elements. All matrix elements are independent of whether a vector with a global phase has been chosen for the superposition state instead of the vector . The same off-diagonal matrix elements also appear in the formula for the expected value of an operator : ${\ displaystyle | \ Psi \ rangle}$${\ displaystyle \ mathrm {e} ^ {i \ varphi} | \ Psi \ rangle}$${\ displaystyle {\ hat {O}}}$

${\ displaystyle \ left \ langle \ Psi \ right | {\ hat {O}} \ left | \ Psi \ right \ rangle = | \ alpha | ^ {2} \ left \ langle \ psi \ right | {\ hat { O}} \ left | \ psi \ right \ rangle + | \ beta | ^ {2} \ left \ langle \ phi \ right | {\ hat {O}} \ left | \ phi \ right \ rangle + \ alpha ^ {*} \ beta \ left \ langle \ psi \ right | {\ hat {O}} \ left | \ phi \ right \ rangle + \ alpha \ beta ^ {*} \ left \ langle \ phi \ right | {\ has {O}} \ left | \ psi \ right \ rangle}$.

There they form the interference terms .

### Density operator for a mixture of states

An ensemble composed of sub-ensembles in which the systems are each in the same pure state is in a mixed state . Here the pure states are incoherently superimposed. If the states are orthogonal, the respective number of the ensembles concerned is the probability that a single system will be found in the state during a measurement . The weights are then normalized to 1 . Then the density operator is given by ${\ displaystyle | \ psi _ {i} \ rangle}$${\ displaystyle | \ psi _ {i} \ rangle}$${\ displaystyle p_ {i} \,}$${\ displaystyle | \ psi _ {i} \ rangle}$${\ displaystyle \ sum _ {i} p_ {i} = 1}$

${\ displaystyle {\ hat {\ rho}} = \ sum _ {i} p_ {i} \ left | \ psi _ {i} \ right \ rangle \ left \ langle \ psi _ {i} \ right | \ quad (1)}$.

With the help of the projection operators, the density operator can also be written as

${\ displaystyle {\ hat {\ rho}} = \ sum _ {i} p_ {i} \; {\ hat {\ mathbb {P}}} _ {\ psi _ {i}} \.}$

The expectation value of any operator is then ${\ displaystyle {\ hat {O}}}$

${\ displaystyle \ left \ langle {\ hat {O}} \ right \ rangle = \ sum _ {i} p_ {i} \; \ left \ langle \ psi _ {i} \ right | {\ hat {O} } \ left | \ psi _ {i} \ right \ rangle \,}$

i.e. the incoherent sum of the expected values ​​for the individual sub-ensembles, each weighted with the relative number of individual systems contained therein. There is no interference between the states of different individual systems.

For example, if the ensemble was composed of two sub-ensembles, each of which only has systems in one or the other of two orthogonal states and , then is the density operator ${\ displaystyle | \ psi \ rangle}$${\ displaystyle | \ phi \ rangle}$

${\ displaystyle {\ hat {\ rho}} = p _ {\ psi} \; {\ hat {\ mathbb {P}}} _ {\ psi} + p _ {\ phi} \; {\ hat {\ mathbb { P}}} _ {\ phi} \.}$

${\ displaystyle p _ {\ psi}}$and with are the relative frequencies. ${\ displaystyle p _ {\ phi}}$${\ displaystyle p _ {\ psi} + p _ {\ phi} = 1}$

With and as basis vectors, the density matrix of this mixture of states is through the diagonal matrix ${\ displaystyle | \ psi \ rangle}$${\ displaystyle | \ phi \ rangle}$

${\ displaystyle {\ hat {\ rho}} = {\ begin {pmatrix} p _ {\ psi} & 0 \\ 0 & p _ {\ phi} \ end {pmatrix}}}$

given. The incoherent superposition of systems is expressed in the disappearance of the off-diagonal elements when (as here) the systems each occupy one of the base states.

In another basis, the same density operator generally has an off-diagonal matrix, except for the case that all basis states are represented with equal frequency.

In the case of the same frequency of all incoherently superimposed base states, the density operator is -fold of the unit operator ? and has the matrix (here for :)} ${\ displaystyle N}$${\ displaystyle 1 / N}$${\ displaystyle {\ hat {\ rho}} = {\ tfrac {1} {N}}}$${\ displaystyle N = 2}$

${\ displaystyle {\ hat {\ rho}} = {\ begin {pmatrix} {\ tfrac {1} {N}} & 0 \\ 0 & {\ tfrac {1} {N}} \ end {pmatrix}} \. }$

This matrix is ​​independent of whether another basis was chosen within the subspace defined by the states involved. This expresses the fact that incoherent ensembles are physically identical if they are formed from orthogonal states with the same frequency in each case, but with a differently chosen base of the subspace formed by the superimposed states.

The density operator for the canonical ensemble is:

${\ displaystyle {\ hat {\ rho}} = {\ frac {e ^ {- \ beta {\ hat {H}}}} {\ rm {{track} \ {e ^ {- \ beta {\ hat { H}}}\}}}}}$

In the eigen basis of the Hamilton operator, the form (1) takes on. Analogously one obtains for the density operator of the grand canonical ensemble ${\ displaystyle {\ hat {\ rho}}}$

${\ displaystyle {\ hat {\ rho}} = {\ frac {e ^ {- \ beta ({\ hat {H}} - \ mu {\ hat {N}})}} {\ rm {{track} \ {e ^ {- \ beta ({\ hat {H}} - \ mu {\ hat {N}})} \}}}}}$.

### Mixed state in a single system

A mixed state is also present in only one single system if it was entangled with a second system before a measurement, so that certain pure states of the first system were fully correlated with certain pure states of the second system. If this measurement, which has no effect on the first system, reduces the state of the second system to a certain pure state , which as such did not belong to the correlated states, the first system must then be treated as a mixture of states.

This is often the case, for example when one atom collides with another, with a certain probability causing an excitation and then hits a detector at a certain deflection angle. The hit atom is then in a mixture of states in the form of an incoherent superposition of excited state and ground state. Conversely, if the direction of its recoil had been determined by measuring the collided atom, the colliding atom would now be in a mixture of states, formed from an incoherent superposition of the scattered waves of different energies. The reduced density operator is used for the description , which results from the density operator of the original overall system through partial tracking and no longer contains any information on the subsystem on which measurements were made. This change in the state of a system, mediated by entanglement, without it becoming the object of a physical influence, represents one of the most difficult aspects of quantum physics to look at (see e.g. quantum entanglement , EPR paradox , quantum eraser ).

For each individual component of the mixture of states, the mean value of the measurement results of a physical quantity is given by the expected value. This is the associated operator (see quantum mechanics , observable ). ${\ displaystyle | \ psi _ {i} \ rangle}$${\ displaystyle A}$${\ displaystyle \ langle A \ rangle _ {\ psi _ {i}} = \ langle \ psi _ {i} | {\ hat {A}} | \ psi _ {i} \ rangle \.}$${\ displaystyle {\ hat {A}}}$${\ displaystyle A}$

Since the ensemble is a mixture of systems in the various states involved , the mean value of all measurements on the individual systems is the weighted sum of the individual expected values: ${\ displaystyle | \ psi _ {i} \ rangle}$

${\ displaystyle \ langle A \ rangle _ {\ hat {\ rho}} = \ sum _ {i} \; p_ {i} \; \ langle \ psi _ {i} | {\ hat {A}} | \ psi _ {i} \ rangle \.}$

This is equal to the trace

${\ displaystyle \ langle A \ rangle _ {\ hat {\ rho}} = \ operatorname {Tr} ({\ hat {\ rho}} {\ hat {A}}) \,}$

as one can see with the help of a complete system of orthonormal basis vectors : paths (unit operator) is ${\ displaystyle | \ varphi _ {k} \ rangle}$${\ displaystyle {\ hat {1}} = \ sum _ {k} | \ varphi _ {k} \ rangle \ langle \ varphi _ {k} |}$

{\ displaystyle {\ begin {aligned} \ langle A \ rangle _ {\ hat {\ rho}} & = \ sum _ {i} \; p_ {i} \; \ langle \ psi _ {i} | {\ hat {A}} \ cdot {\ hat {1}} | \ psi _ {i} \ rangle = \ sum _ {i, k} \; p_ {i} \; \ langle \ psi _ {i} | { \ hat {A}} | \ varphi _ {k} \ rangle \ cdot \ langle \ varphi _ {k} | \ psi _ {i} \ rangle \\ & = \ sum _ {k} \ langle \ varphi _ { k} | \; \ left (\ sum _ {i} | \ psi _ {i} \ rangle p_ {i} \ langle \ psi _ {i} | {\ hat {A}} \ right) \; | \ varphi _ {k} \ rangle = \ sum _ {k} \ langle \ varphi _ {k} | \; {\ hat {\ rho}} {\ hat {A}} \; | \ varphi _ {k} \ rangle = \ operatorname {Tr} ({\ hat {\ rho}} {\ hat {A}}) \. \ end {aligned}}}

If they are the eigen-states of the observable (i.e. with the eigen-values ), then the following applies ${\ displaystyle | \ varphi _ {k} \ rangle}$${\ displaystyle A}$${\ displaystyle {\ hat {A}} | \ varphi _ {k} \ rangle = a_ {k} | \ varphi _ {k} \ rangle}$${\ displaystyle a_ {k}}$

{\ displaystyle {\ begin {aligned} \ langle A \ rangle _ {\ hat {\ rho}} & = \ sum _ {i, k} \; p_ {i} \; \ langle \ psi _ {i} | a_ {k} | \ varphi _ {k} \ rangle \ cdot \ langle \ varphi _ {k} | \ psi _ {i} \ rangle = \ sum _ {k} a_ {k} \ left (\ sum _ { i} p_ {i} \; \ langle \ varphi _ {k} | \ psi _ {i} \ rangle \; \ langle \ psi _ {i} | \ varphi _ {k} \ rangle \ right) \\ & = \ sum _ {k} a_ {k} \ left (\ sum _ {i} p_ {i} \; | \ langle \ varphi _ {k} | \ psi _ {i} \ rangle | ^ {2} \ right) \; = \ sum _ {k} a_ {k} P_ {k} \. \ end {aligned}}}

This is the mean weighted over the ensemble for the probability of encountering a singled out system in its own state . is also the probability of obtaining the eigenvalue as a result for a single measurement . It is characteristic that an incoherent sum is given by which is independent of the relative phases of the states involved in the ensemble . ${\ displaystyle P_ {k} = \ sum _ {i} p_ {i} \; | \ langle \ varphi _ {k} | \ psi _ {i} \ rangle | ^ {2} \}$${\ displaystyle | \ varphi _ {k} \ rangle}$${\ displaystyle P_ {k}}$${\ displaystyle a_ {k}}$${\ displaystyle P_ {k}}$${\ displaystyle | \ psi _ {i} \ rangle}$

Conversely, the operator can be represented by the sum formed from its eigenvalues and the density operators of the eigenstates : ${\ displaystyle {\ hat {A}}}$${\ displaystyle a_ {k}}$${\ displaystyle {\ hat {P}} _ {\ varphi _ {k}} = | \ varphi _ {k} \ rangle \ langle \ varphi _ {k} |}$

${\ displaystyle {\ hat {A}} = \ sum _ {k} a_ {k} \; {\ hat {P}} _ {\ varphi _ {k}} \.}$

## Example: density operator and density matrix for electron polarization

The density matrix is the matrix with which the operator can be represented with respect to an orthonormal basis : ${\ displaystyle {\ hat {\ rho}}}$${\ displaystyle | \ varphi _ {k} \ rangle}$

${\ displaystyle \ rho _ {mn} = \ langle \ varphi _ {m} | {\ hat {\ rho}} | \ varphi _ {n} \ rangle}$

### Base states

Hereinafter, the "numeral " means that a Bra, Ket or an operator of a base with respect illustrated is (see also Bra-Ket # representation ). The states “spin up” (with regard to the z-axis) and “spin down” are represented as ket vectors by columns. The associated bra vectors are then row vectors: or . The projection operators (through matrix multiplication ): ${\ displaystyle \ doteq}$${\ displaystyle \ left | {\ uparrow} \ right \ rangle {\ mathrel {\ doteq}} {\ bigl (} {\ begin {smallmatrix} 1 \\ 0 \ end {smallmatrix}} {\ bigr)}}$${\ displaystyle \ left | {\ downarrow} \ right \ rangle \ doteq {\ bigl (} {\ begin {smallmatrix} 0 \\ 1 \ end {smallmatrix}} {\ bigr)}}$${\ displaystyle \ left \ langle {\ uparrow} \ right | \ doteq (1 \ 0)}$${\ displaystyle \ left \ langle {\ downarrow} \ right | \ doteq (0 \ 1)}$

${\ displaystyle {\ hat {P}} _ {\ uparrow} \ doteq {\ bigl (} {\ begin {smallmatrix} 1 \\ 0 \ end {smallmatrix}} {\ bigr)} \ cdot (1 \ 0) \ = {\ bigl (} {\ begin {smallmatrix} 1 & 0 \\ 0 & 0 \ end {smallmatrix}} {\ bigr)} \ quad, \ {\ hat {P}} _ {\ downarrow} \ doteq {\ bigl ( } {\ begin {smallmatrix} 0 \\ 1 \ end {smallmatrix}} {\ bigr)} \ cdot (0 \ 1) \ = {\ bigl (} {\ begin {smallmatrix} 0 & 0 \\ 0 & 1 \ end {smallmatrix }} {\ bigr)}}$

These are also the density matrices for electrons that are completely polarized in - or - direction. ${\ displaystyle + z}$${\ displaystyle -z}$

### Polarization in z-direction

The -component of the spin has the diagonal matrix formed from the eigenvalues. The predicted measurement result is correct for the ensemble${\ displaystyle z}$${\ displaystyle {\ hat {s_ {z}}} \ doteq \ left ({\ begin {smallmatrix} 1/2 & 0 \\ 0 & -1 / 2 \ end {smallmatrix}} \ right) \.}$${\ displaystyle {\ hat {P _ {\ uparrow}}}}$

${\ displaystyle \ langle {\ hat {s_ {z}}} \ rangle = \ operatorname {Tr} ({\ hat {P}} _ {_ {\ uparrow}} \ cdot {\ hat {s}} _ { z}) \ doteq \ operatorname {Tr} \ left ({\ bigl (} {\ begin {smallmatrix} 1 & 0 \\ 0 & 0 \ end {smallmatrix}} {\ bigr)} \ cdot {\ bigl (} {\ begin { smallmatrix} 1/2 & 0 \\ 0 & -1 / 2 \ end {smallmatrix}} {\ bigr)} \ right) \ = \ operatorname {Tr} {\ bigl (} {\ begin {smallmatrix} 1/2 & 0 \\ 0 & 0 \ end {smallmatrix}} {\ bigr)} = {\ tfrac {1} {2}}.}$

For the ensemble it results ${\ displaystyle {\ hat {P}} _ {\ downarrow}}$${\ displaystyle \ langle {\ hat {s}} _ {z} \ rangle = \ operatorname {Tr} ({\ hat {P}} _ {\ downarrow} \ cdot {\ hat {s}} _ {z} ) \ doteq \ operatorname {Tr} {\ bigl (} {\ begin {smallmatrix} 0 & 0 \\ 0 & -1 / 2 \ end {smallmatrix}} {\ bigr)} = - {\ tfrac {1} {2}} .}$

### Different direction of polarization

The states of electrons polarized in - or - direction are The projection operators for this have (in the basis of the- eigen-states!) The matrices. It is characteristic that these are not diagonal matrices and that the different phases with which the- eigen-states as ket- Vectors that have been superimposed here can be found in the matrix elements outside the main diagonal . This is an expression of the coherent superposition, by those from -Eigenzuständen the -Eigenzustände be formed. ${\ displaystyle + x}$${\ displaystyle -x}$${\ displaystyle \ left | {\ rightarrow} \ right \ rangle \ doteq \ left ({\ begin {smallmatrix} {\ sqrt {1/2}} \\ {\ sqrt {1/2}} \ end {smallmatrix} } \ right) \;, \ \ left | {\ leftarrow} \ right \ rangle \ doteq \ left ({\ begin {smallmatrix} {\ sqrt {1/2}} \\ - {\ sqrt {1/2} } \ end {smallmatrix}} \ right).}$${\ displaystyle s_ {z}}$${\ displaystyle {\ hat {P}} _ {\ left | {\ rightarrow} \ right \ rangle} \ doteq {\ bigl (} {\ begin {smallmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \ end {smallmatrix}} {\ bigr)} \;, \ {\ hat {P}} _ {\ left | {\ leftarrow} \ right \ rangle} \ doteq {\ bigl (} {\ begin {smallmatrix} 1 / 2 & -1 / 2 \\ - 1/2 & 1/2 \ end {smallmatrix}} {\ bigr)} \.}$${\ displaystyle s_ {z}}$${\ displaystyle s_ {z}}$${\ displaystyle s_ {x}}$

### Unpolarized ensemble

If the electrons are each half polarized in -direction, the density matrix is ​​called: ${\ displaystyle \ pm z}$

${\ displaystyle {\ hat {\ rho}} \ doteq {\ tfrac {1} {2}} {\ bigl (} {\ begin {smallmatrix} 1 & 0 \\ 0 & 0 \ end {smallmatrix}} {\ bigr)} \ + {\ tfrac {1} {2}} {\ bigl (} {\ begin {smallmatrix} 0 & 0 \\ 0 & 1 \ end {smallmatrix}} {\ bigr)} = {\ bigl (} {\ begin {smallmatrix} 1 / 2 & 0 \\ 0 & 1/2 \ end {smallmatrix}} {\ bigr)} = {\ tfrac {1} {2}} \ cdot {\ hat {1}}}$

The same density matrix is ​​obtained for a mixture of electrons, each of which is polarized 50% in the direction (or in any other direction). This means that all possible measurement results are identical to those on the ensemble that was formed from polarized electrons. The original directions of polarization used to define the ensemble can no longer be physically (and therefore conceptually) differentiated: one and the same ensemble has always been created. ${\ displaystyle \ pm x}$${\ displaystyle \ pm z}$

### Mixture of different polarization directions

For example, for a mixture of electrons with spin in -direction and -direction with proportions or is called the density matrix ${\ displaystyle (+ z)}$${\ displaystyle (-x)}$${\ displaystyle p _ {\ uparrow}}$${\ displaystyle p _ {\ leftarrow}}$

${\ displaystyle {\ hat {\ rho}} _ {p _ {_ {\ uparrow}}, p _ {_ {\ leftarrow}}} = p _ {\ uparrow} \; {\ hat {P}} _ {\ left | {\ uparrow} \ right \ rangle} + p _ {\ leftarrow} \; {\ hat {P}} _ {\ left | {\ leftarrow} \ right \ rangle} \ doteq p _ {\ uparrow} \ cdot {\ bigl (} {\ begin {smallmatrix} 1 & 0 \\ 0 & 0 \ end {smallmatrix}} {\ bigr)} \ + p _ {\ leftarrow} \ cdot {\ bigl (} {\ begin {smallmatrix} 1/2 & -1 / 2 \\ - 1/2 & 1/2 \ end {smallmatrix}} {\ bigr)} = \ left ({\ begin {smallmatrix} p _ {_ {\ uparrow}} + {\ tfrac {p _ {_ {\ leftarrow} }} {2}} & - {\ tfrac {p _ {_ {\ leftarrow}}} {2}} \\ - {\ tfrac {p _ {_ {\ leftarrow}}} {2}} & {\ tfrac { p _ {_ {\ leftarrow}}} {2}} \ end {smallmatrix}} \ right)}$

The expected value of the spin in -direction is then ${\ displaystyle \ pm z}$

${\ displaystyle \ langle {\ hat {s}} _ {z} \ rangle = \ operatorname {Tr} ({\ hat {\ rho}} _ {p _ {_ {\ uparrow}}, p _ {_ {\ leftarrow }}} \ cdot {\ hat {s}} _ {z}) \ doteq \ operatorname {Tr} \ left (\ left ({\ begin {smallmatrix} p _ {_ {\ uparrow}} + {\ tfrac {p_ {_ {\ leftarrow}}} {2}} & - {\ tfrac {p _ {_ {\ leftarrow}}} {2}} \\ - {\ tfrac {p _ {_ {\ leftarrow}}} {2} } & {\ tfrac {p _ {_ {\ leftarrow}}} {2}} \ end {smallmatrix}} \ right) \ cdot {\ bigl (} {\ begin {smallmatrix} 1/2 & 0 \\ 0 & -1 / 2 \ end {smallmatrix}} {\ bigr)} \ right) = \ operatorname {Tr} \ left (\ left ({\ begin {smallmatrix} {\ tfrac {1} {2}} \ left (p _ {_ { \ uparrow}} + {\ tfrac {p _ {_ {\ leftarrow}}} {2}} \ right) & {\ tfrac {p _ {_ {\ uparrow}}} {4}} \\ - {\ tfrac { p _ {_ {\ leftarrow}}} {4}} & - {\ tfrac {p _ {_ {\ leftarrow}}} {4}} \ end {smallmatrix}} \ right) \ right) = {\ tfrac {1 } {2}} p _ {\ uparrow}.}$

As expected, the electrons polarized in the ( ) direction do not contribute anything to the expected value . ${\ displaystyle -x}$${\ displaystyle \ langle {\ hat {s}} _ {z} \ rangle}$

## Formal definition

A quantum mechanical system is given, which is modeled on a Hilbert space  . A bounded linear operator on is a density operator if: ${\ displaystyle \ mathbf {H}}$ ${\ displaystyle {\ hat {\ rho}} \, \;}$${\ displaystyle \ mathbf {H}}$

1. he is hermitian
2. it is positive semi-definite ,
3. it is track class with track equal to 1.

Although the terms density matrix and density operator are often used synonymously, there is a mathematical difference. Just as a matrix is the basic representation of a linear operator in linear algebra , in quantum mechanics a distinction can be made between an abstract density operator and a concrete density matrix in a specific representation. Is a density operator, so named ${\ displaystyle {\ hat {\ rho}}}$

${\ displaystyle \ rho (x, y) = \ langle x | {\ hat {\ rho}} | y \ rangle}$

the density matrix in position representation . However, it is not a real matrix, since the position representation is defined over a continuum of improper basis vectors , but a so-called integral kernel . ${\ displaystyle | x \ rangle}$

In finite-dimensional Hilbert spaces (e.g. in spin systems), on the other hand, a positive semidefinite matrix with track 1 results, i.e. a real density matrix, if an orthonormal basis is chosen: ${\ displaystyle \ mathbf {e} _ {i}}$

${\ displaystyle \ rho _ {ij} = \ langle \ mathbf {e} _ {i} | {\ hat {\ rho}} | \ mathbf {e} _ {j} \ rangle}$.

## properties

• The set of all density operators is a convex set whose edge is the set of pure (quantum mechanical) states. In contrast to classical theories, the set is not a simplex ; H. In general, a density operator cannot be represented uniquely as a convex combination of pure states.
• The probability of obtaining the measured value  when measuring an observable  on a system that is described by the density operator  is given by${\ displaystyle A \! \,}$${\ displaystyle {\ hat {\ rho}} \,}$${\ displaystyle a \,}$
${\ displaystyle p_ {a} = \ sum _ {i} \ left \ langle a_ {i} \ right | {\ hat {\ rho}} \ left | a_ {i} \ right \ rangle = \ operatorname {Tr} ({\ hat {\ mathbb {P}}} _ {a} {\ hat {\ rho}}),}$
where the orthonormal eigenvectors are to the eigenvalue and is the projection operator onto the corresponding eigenspace . The system is then in the state${\ displaystyle \ left | a_ {i} \ right \ rangle}$ ${\ displaystyle a \! \,}$${\ displaystyle {\ hat {\ mathbb {P}}} _ {a}}$${\ displaystyle {\ frac {{\ hat {\ mathbb {P}}} _ {a} {\ hat {\ rho}} {\ hat {\ mathbb {P}}} _ {a}} {\ operatorname { Tr} ({\ hat {\ mathbb {P}}} _ {a} {\ hat {\ rho}} {\ hat {\ mathbb {P}}} _ {a})}}}$
• The mean value of the measured values ​​( expected value ) when measuring an observable is${\ displaystyle A \! \,}$
${\ displaystyle \ left \ langle {\ hat {A}} \ right \ rangle = \ operatorname {Tr} ({\ hat {A}} {\ hat {\ rho}}).}$

### Density matrix for pure states

If the ensemble under consideration is a pure ensemble , i.e. if the system only consists of a pure state, then applies to the density matrix . ${\ displaystyle \ operatorname {Tr} \, ({\ hat {\ rho}} ^ {2}) = \ operatorname {Tr} \, ({\ hat {\ rho}}) = 1}$

The following always applies to mixed states . ${\ displaystyle \ operatorname {Tr} \, ({\ hat {\ rho}} ^ {2}) <1}$

### Density matrix for an evenly distributed ensemble

The density matrix has a level system in which all states are equally probable ${\ displaystyle N}$${\ displaystyle N \, \!}$

${\ displaystyle {\ hat {\ rho}} = {\ frac {1} {N}} \ \ mathbf {1} _ {N} \,}$

where denotes the -dimensional identity matrix . ${\ displaystyle \ mathbf {1} _ {N}}$${\ displaystyle N}$

## Reduced density operator

The reduced density operator was introduced in 1930 by Paul Dirac . It refers to a selected subsystem of a composite system and is used to predict the results of measurements on the subsystem if the other parts of the system are not being observed at all.

If and are two systems with (normalized) states in their respective Hilbert space , then the composite system has the tensor space for the Hilbert space. The overall system is in a separable state when it is certain that the two subsystems are in the states or . In general, the overall system is in one state ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle | \ psi _ {A} \ rangle \ ,, | \ varphi _ {B} \ rangle}$${\ displaystyle \ mathbb {H} _ {A}, \ \ mathbb {H} _ {B}}$${\ displaystyle A + B}$${\ displaystyle \ mathbb {H} _ {A} \ otimes \ mathbb {H} _ {B}}$${\ displaystyle | \ psi _ {A} \ rangle \, | \ varphi _ {B} \ rangle \ in \ mathbb {H} _ {A} \ otimes \ mathbb {H} _ {B}}$${\ displaystyle | \ psi _ {A} \ rangle}$${\ displaystyle | \ varphi _ {B} \ rangle}$

${\ displaystyle | \ Psi \ rangle = \ sum _ {ik} \, c_ {ik} \, | \ psi _ {Ai} \ rangle \, | \ varphi _ {Bk} \ rangle}$

(with orthonormal basis vectors and constants ), which is referred to as entangled if it can not be represented as a separable state. ${\ displaystyle | \ psi _ {Ai} \ rangle \ ,, \, | \ varphi _ {Bk} \ rangle}$${\ displaystyle c_ {ik}}$

For an observable of the subsystem , the operator is initially only defined in Hilbert space . For the measurement of these observables , which only affect the system, on the overall system, the operator must be expanded to an operator on , where the unit operator is in. ${\ displaystyle A}$${\ displaystyle {\ hat {O}} _ {\! A}}$${\ displaystyle \ mathbb {H} _ {A}}$${\ displaystyle A}$${\ displaystyle {\ hat {O}} _ {\! A} \ otimes {\ hat {\ mathbf {1}}} _ {B}}$${\ displaystyle \ mathbb {H} _ {A} \ otimes \ mathbb {H} _ {B}}$${\ displaystyle {\ hat {\ mathbf {1}}} _ {B}}$${\ displaystyle \ mathbb {H} _ {B}}$

If the state of the system is separable, then the expected value results

${\ displaystyle \ langle \ psi _ {A} | \, \ langle \ varphi _ {B} | \, \ left ({\ hat {O}} _ {\! A} \ otimes {\ hat {\ mathbf { 1}}} _ {B} \ right) \, | \ psi _ {A} \ rangle \, | \ varphi _ {B} \ rangle = \ langle \ psi _ {A} | \, {\ hat {O }} _ {\! A} | \ psi _ {A} \ rangle \ cdot \ langle \ varphi _ {B} | {\ hat {\ mathbf {1}}} _ {B} \, | \ varphi _ { B} \ rangle = \ langle \ psi _ {A} | \, {\ hat {O}} _ {\! A} | \ psi _ {A} \ rangle \.}$

This agrees with the result obtained by considering the subsystem from the outset as an isolated system. ${\ displaystyle A}$

In general, however, it follows for the expected value:

{\ displaystyle {\ begin {aligned} \ langle \ Psi | \, \ left ({\ hat {O}} _ {\! A} \ otimes {\ hat {\ mathbf {1}}} _ {B} \ right) \, | \ Psi \ rangle & = \ sum _ {ik \, i'k '} c_ {ik} c_ {i'k'} ^ {*} \ langle \ psi _ {Ai '} | \, {\ hat {O}} _ {\! A} | \ psi _ {Ai} \ rangle \ cdot \ langle \ varphi _ {Bk '} | {\ hat {\ mathbf {1}}} _ {B} \ , | \ varphi _ {Bk} \ rangle \\ & = \ sum _ {ii '} \ left (\ sum _ {k} c_ {ik} c_ {i'k} ^ {*} \ right) \ langle \ psi _ {Ai '} | \, {\ hat {O}} _ {\! A} | \ psi _ {Ai} \ rangle = \ operatorname {Tr} ({\ hat {\ rho}} _ {\! A} \, {\ hat {O}} _ {\! A}) \,. \ End {aligned}}}

That’s included

${\ displaystyle (\ rho _ {\! A}) _ {ii '} = \ sum _ {k} c_ {ik} c_ {i'k} ^ {*}}$

defines the reduced density operator for the subsystem when the overall system is in state . It is an operator in space and arises when in the matrix of the density operator for the overall system ${\ displaystyle A}$${\ displaystyle \ Psi}$${\ displaystyle \ mathbb {H} _ {A}}$

${\ displaystyle (\ rho _ {\! A + B}) _ {iki'k '} = c_ {ik} c_ {i'k'} ^ {*}}$

the partial trace is formed by summing up the index of the base states of the subsystem . ${\ displaystyle k = k '}$${\ displaystyle B}$

A simple interpretation arises for the case that the basis is the eigenvectors of the operator (with eigenvalues ). Then the expectation of is an incoherently weighted mean of its eigenvalues: ${\ displaystyle | \ psi _ {Ai} \ rangle}$${\ displaystyle {\ hat {O}} _ {\! A}}$${\ displaystyle X_ {i}}$${\ displaystyle {\ hat {O}} _ {\! A}}$

${\ displaystyle \ operatorname {Tr} ({\ hat {\ rho}} _ {\! A} \, {\ hat {O}} _ {\! A}) = \ sum _ {i} \ left (\ sum _ {k} | c_ {ik} | ^ {2} \ right) X_ {i}. \}$

In the event that the overall system is in a separable state, e.g. B. , this formula gives the expected result because all terms with index are zero and the sum is the norm of , i.e. equal to 1. ${\ displaystyle | \ psi _ {A {i_ {0}}} \ rangle | \ varphi _ {B} \ rangle}$${\ displaystyle \ operatorname {Tr} ({\ hat {\ rho}} _ {\! A} \, {\ hat {O}} _ {\! A}) = X_ {i_ {0}},}$${\ displaystyle i \ neq i_ {0}}$${\ displaystyle \ left (\ sum _ {k} | c_ {i_ {0} k} | ^ {2} \ right)}$${\ displaystyle | \ varphi _ {B} \ rangle}$

## One particle density operator

In a many-particle system, the single-particle density operator is the density operator reduced to the Hilbert space of a particle. For systems of identical particles, the knowledge of the single-particle density operator is sufficient to calculate expectation values ​​and transition matrix elements of each operator that is the sum of single-particle operators. This concerns z. B. the kinetic energy and the potential energy in an external field and is therefore an important tool when modeling the electron shell of atoms and molecules. The calculations are often carried out in spatial representation, i.e. based on the N-particle wave function . This contains the position and spin coordinates of the i-th particle. In the matrix representation they appear here as e.g. T. continuous indices and are therefore not written as a lower index, but like the argument of a function. The density matrix of the overall system is called ${\ displaystyle \ Psi ({\ vec {r}} _ {1}, m_ {s1}, \, {\ vec {r}} _ {2}, m_ {s2}, \ ldots, \, {\ vec {r}} _ {N}, m_ {sN},)}$${\ displaystyle {\ vec {r}} _ {i}, m_ {si},}$

${\ displaystyle \ rho ({\ vec {r}} _ {1} ', m_ {s1}', \, {\ vec {r}} _ {2} ', m_ {s2}', \ ldots, \ , {\ vec {r}} _ {N} ', m_ {sN}', \ {\ vec {r}} _ {1}, m_ {s1}, \, {\ vec {r}} _ {2 }, m_ {s2}, \ ldots, \, {\ vec {r}} _ {N}, m_ {sN})}$
${\ displaystyle = \ Psi ^ {*} ({\ vec {r}} _ {1} ', m_ {s1}', \, {\ vec {r}} _ {2} ', m_ {s2}' , \ ldots, \, {\ vec {r}} _ {N} ', m_ {sN}') \ cdot \ Psi ({\ vec {r}} _ {1}, m_ {s1}, \, { \ vec {r}} _ {2}, m_ {s2}, \ ldots, \, {\ vec {r}} _ {N}, m_ {sN})}$

The one-particle density matrix is ​​then

${\ displaystyle \ rho _ {1} ({\ vec {r}} ', s', \ {\ vec {r}}, s) = \ sum _ {m_ {s2}, \ ldots m_ {sN}} \ int _ {dV_ {2} \ ldots dV_ {N}} \ Psi ^ {*} ({\ vec {r}} ', s', \, {\ vec {r}} _ {2}, m_ { s2}, \ ldots, \, {\ vec {r}} _ {N}, m_ {sN}) \ cdot \ Psi ({\ vec {r}}, s, \, {\ vec {r}} _ {2}, m_ {s2}, \ ldots, \, {\ vec {r}} _ {N}, m_ {sN})}$

The choice of the (N-1) integration (or summation) variables with the numbers 2 to is arbitrary, since the wave function for identical particles changes at most the sign compared to renumbering and therefore always the same result comes out for the one-particle density matrix. ${\ displaystyle N}$

The diagonal element indicates the total density that the particles form at the location with the spin direction . ${\ displaystyle \ rho _ {1} ({\ vec {r}}, s, \, {\ vec {r}}, s)}$${\ displaystyle N}$${\ displaystyle {\ vec {r}}}$${\ displaystyle m_ {s}}$

Since the Einteilchendichteoperator is Hermitian, there is a basis of eigenstates: . For the eigenvalues and . The eigenstates with the greatest eigenvalues ​​are called natural orbitals . If every natural orbital is occupied with a particle, i.e. a state in the form of the Slater determinant , this represents the best approximation of the original N-particle wave function that can be achieved in the context of a single particle model with regard to the total particle density . ${\ displaystyle {\ hat {\ rho}} _ {1}}$${\ displaystyle \ {| \ chi _ {n} \ rangle \ ,, n = 1,2, \ ldots \}}$${\ displaystyle {\ hat {\ rho}} _ {1} | \ chi _ {n} \ rangle = \ lambda _ {n} | \ chi _ {n} \ rangle}$${\ displaystyle 0 \ leq \ lambda _ {n} \ leq 1}$${\ displaystyle \ sum _ {n} \ lambda _ {n} = N}$${\ displaystyle N}$${\ displaystyle \ Psi}$

## Time development

From the Schrödinger equation , which describes the time development (dynamics) of pure quantum states, the time development of mixed states can be derived directly. For this purpose, one uses any decomposition of the density matrix into pure states, the dynamics of which satisfy the Schrödinger equation, and from this one calculates the dynamics of the mixed state

${\ displaystyle {\ frac {\ partial {\ hat {\ rho}}} {\ partial t}} = {\ frac {i} {\ hbar}} \ left [{\ hat {\ rho}}, {\ has {H}} \ right],}$

where is the Hamilton operator of the system. This equation is known as the von Neumann equation of motion (not to be confused with Heisenberg's equation of motion ). ${\ displaystyle {\ hat {H}}}$

This differential equation can be solved for time-independent Hamilton operators and the equation is obtained with the unitary time expansion operator ${\ displaystyle {\ hat {U}} (t) = e ^ {- iHt / \ hbar}}$

${\ displaystyle {\ hat {\ rho}} (t) = {\ hat {U}} (t) \; {\ hat {\ rho}} (0) \; {\ hat {U}} ^ {\ dagger} (t)}$.

This solution can easily be checked by inserting it.

It is noteworthy here that the usual Heisenberg equation of motion does not apply to the operator, since the time evolution operator obeys the dynamics derived directly from the Schrödinger equation . The time development of the operator by the time development operator also does not follow the usual time development equation for operators ( for an ordinary observable A), which is understandable, however, since${\ displaystyle {\ hat {U}} (t)}$ ${\ displaystyle i \ hbar \ partial _ {t} U (t) = H (t) U (t)}$${\ displaystyle \ rho}$${\ displaystyle {\ hat {U}} (t)}$${\ displaystyle U (t) ^ {\ dagger} AU (t)}$${\ displaystyle {\ hat {\ rho}} (t) = {\ hat {U}} (t) \; {\ hat {\ rho}} (0) \; {\ hat {U}} ^ {\ dagger} (t) = \ sum _ {i} p_ {i} U (t) | \ psi (0) \ rangle \ langle \ psi (0) | U (t) ^ {\ dagger} = \ sum _ { i} p_ {i} | \ psi (t) \ rangle \ langle \ psi (t) |}$

## entropy

With the help of the density matrix , the Von Neumann entropy of a system can be defined as follows: ${\ displaystyle {\ hat {\ rho}} \, \!}$

${\ displaystyle S = -k _ {\ mathrm {B}} \ operatorname {Tr} \ left ({\ hat {\ rho}} \ ln {\ hat {\ rho}} \ right),}$

where is Boltzmann's constant , and the trace is taken over the space in which is operating. ${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle \ mathbf {H}}$${\ displaystyle {\ hat {\ rho}} \, \!}$

The entropy of any pure state is zero because the eigenvalues ​​of the density matrix are zero and one. This agrees with the heuristic argument that there is no uncertainty about the preparation of the condition.

One can show that unitary operators applied to a state (such as the time evolution operator obtained from the Schrödinger equation) do not change the entropy of the system. This combines the reversibility of a process with its change in entropy - a fundamental result that connects quantum mechanics with information theory and thermodynamics .