Franck–Condon principle

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Figure 1. Franck–Condon principle energy diagram. Since electronic transitions are very fast compared with nuclear motions, the vibrational states to and from which absorption and emission occur are those that correspond to a minimal change in the nuclear coordinates. As a result, both absorption and emission produce molecules in vibrationally excited states. The potential wells are shown favoring transitions with changes in ν.

The Franck–Condon principle describes the intensities of vibronic transitions, or the absorption or emission of a photon. It states that when a molecule is undergoing an electronic transition, such as ionization, the nuclear configuration of the molecule experiences no significant change.

Overview

Figure 2. Schematic representation of the absorption and fluorescence spectra corresponding to the energy diagram in Figure 1. The symmetry is due to the equal shape of the ground and excited state potential wells. The narrow lines can usually only be observed in the spectra of dilute gases. The darker curves represent the inhomogeneous broadening of the same transitions as occurs in liquids and solids. Electronic transitions between the lowest vibrational levels of the electronic states (the 0–0 transition) have the same energy in both absorption and fluorescence.

Figure 3. Semiclassical pendulum analogy of the Franck–Condon principle. Vibronic transitions are allowed at the classical turning points because both the momentum and the nuclear coordinates correspond in the two represented energy levels. In this illustration, the 0–2 vibrational transitions are favored.

The Franck–Condon principle has a well-established semiclassical interpretation based on the original contributions of James Franck. Electronic transitions are relatively instantaneous compared with the time scale of nuclear motions, therefore if the molecule is to move to a new vibrational level during the electronic transition, this new vibrational level must be instantaneously compatible with the nuclear positions and momenta of the vibrational level of the molecule in the originating electronic state. In the semiclassical picture of vibrations (oscillations) of a simple harmonic oscillator, the necessary conditions can occur at the turning points, where the momentum is zero.

Classically, the Franck–Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in the molecular entity and its environment. The resulting state is called a Franck–Condon state, and the transition involved, a vertical transition. The quantum mechanical formulation of this principle is that the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational wavefunctions of the two states that are involved in the transition.

— IUPAC Compendium of Chemical Terminology, 2nd Edition (1997)

In the quantum mechanical picture, the vibrational levels and vibrational wavefunctions are those of quantum harmonic oscillators, or of more complex approximations to the potential energy of molecules, such as the Morse potential. Figure 1 illustrates the Franck–Condon principle for vibronic transitions in a molecule with Morse-like potential energy functions in both the ground and excited electronic states. In the low temperature approximation, the molecule starts out in the v = 0 vibrational level of the ground electronic state and upon absorbing a photon of the necessary energy, makes a transition to the excited electronic state. The electron configuration of the new state may result in a shift of the equilibrium position of the nuclei constituting the molecule. In Figure 3 this shift in nuclear coordinates between the ground and the first excited state is labeled as q₀₁. In the simplest case of a diatomic molecule the nuclear coordinates axis refers to the internuclear separation. The vibronic transition is indicated by a vertical arrow due to the assumption of constant nuclear coordinates during the transition. The probability that the molecule can end up in any particular vibrational level is proportional to the square of the (vertical) overlap of the vibrational wavefunctions of the original and final state (see Quantum mechanical formulation section below). In the electronic excited state molecules quickly relax to the lowest vibrational level of the lowest electronic excitation state (Kasha's rule), and from there can decay to the electronic ground state via photon emission. The Franck–Condon principle is applied equally to absorption and to fluorescence.

The applicability of the Franck–Condon principle in both absorption and fluorescence, along with Kasha's rule leads to an approximate mirror symmetry shown in Figure 2. The vibrational structure of molecules in a cold, sparse gas is most clearly visible due to the absence of inhomogeneous broadening of the individual transitions. Vibronic transitions are drawn in Figure 2 as narrow, equally spaced Lorentzian line shapes. Equal spacing between vibrational levels is only the case for the parabolic potential of simple harmonic oscillators, in more realistic potentials, such as those shown in Figure 1, energy spacing decreases with increasing vibrational energy. Electronic transitions to and from the lowest vibrational states are often referred to as 0–0 (zero zero) transitions and have the same energy in both absorption and fluorescence.

Development of the principle

In a report published in 1926 in Transactions of the Faraday Society, James Franck was concerned with the mechanisms of photon-induced chemical reactions. The presumed mechanism was the excitation of a molecule by a photon, followed by a collision with another molecule during the short period of excitation. The question was whether it was possible for a molecule to break into photoproducts in a single step, the absorption of a photon, and without a collision. In order for a molecule to break apart, it must acquire from the photon a vibrational energy exceeding the dissociation energy, that is, the energy to break a chemical bond. However, as was known at the time, molecules will only absorb energy corresponding to allowed quantum transitions, and there are no vibrational levels above the dissociation energy level of the potential well. High-energy photon absorption leads to a transition to a higher electronic state instead of dissociation. In examining how much vibrational energy a molecule could acquire when it is excited to a higher electronic level, and whether this vibrational energy could be enough to immediately break apart the molecule, he drew three diagrams representing the possible changes in binding energy between the lowest electronic state and higher electronic states.

Diagram I. shows a great weakening of the binding on a transition from the normal state n to the excited states a and a'. Here we have D > D' and D' > D". At the same time the equilibrium position of the nuclei moves with the excitation to greater values of r. If we go from the equilibrium position (the minimum of potential energy) of the n curve vertically [emphasis added] upwards to the a curves in Diagram I. the particles will have a potential energy greater than D' and will fly apart. In this case we have a very great change in the oscillation energy on excitation by light...

— James Franck, 1926

James Franck recognized that changes in vibrational levels could be a consequence of the instantaneous nature of excitation to higher electronic energy levels and a new equilibrium position for the nuclear interaction potential. Edward Condon extended this insight beyond photoreactions in a 1926 Physical Review article titled "A Theory of Intensity Distribution in Band Systems".^[3] Here he formulates the semiclassical formulation in a manner quite similar to its modern form. The first joint reference to both Franck and Condon in regard to the new principle appears in the same 1926 issue of Physical Review in an article on the band structure of carbon monoxide by Raymond Birge.

Figure 5. Figure 1 in Edward Condon's first publication on what is now the Franck–Condon principle [Condon 1926]. Condon chose to superimpose the potential curves to illustrate the method of estimating vibrational transitions.

Quantum mechanical formulation

Consider an electrical dipole transition from the initial vibrational state (υ) of the ground electronic level (ε), ${\displaystyle |ϵv⟩}$, to some vibrational state (υ′) of an excited electronic state (ε′), ${\displaystyle |{ϵ}^{\prime }{v}^{\prime }⟩}$ (see bra–ket notation). The molecular dipole operator μ is determined by the charge (−e) and locations (r_i) of the electrons as well as the charges (+Z_je) and locations (R_j) of the nuclei:

${\displaystyle \mathit{\mu }={\mathit{\mu }}_e+{\mathit{\mu }}_N=-e\sum _i{\mathit{r}}_i+e\sum _j{Z}_j{\mathit{R}}_j.}$

The probability amplitude P for the transition between these two states is given by

${\displaystyle P=⟨{\psi }^{\prime }|\mathit{\mu }|\psi ⟩=\int {\psi }^{\prime \ast }\mathit{\mu }\psi d\tau ,}$

where ${\displaystyle \psi }$ and ${\displaystyle {\psi }^{\prime }}$ are, respectively, the overall wavefunctions of the initial and final state. The overall wavefunctions are the product of the individual vibrational (depending on spatial coordinates of the nuclei) and electronic space and spin wavefunctions:

${\displaystyle \psi ={\psi }_e{\psi }_v{\psi }_s.}$

This separation of the electronic and vibrational wavefunctions is an expression of the Born–Oppenheimer approximation and is the fundamental assumption of the Franck–Condon principle. Combining these equations leads to an expression for the probability amplitude in terms of separate electronic space, spin and vibrational contributions:

${\displaystyle P=⟨{\psi }_e^{\prime }{\psi }_v^{\prime }{\psi }_s^{\prime }|\mathit{\mu }|{\psi }_e{\psi }_v{\psi }_s⟩=\int {\psi }_e^{\prime \ast }{\psi }_v^{\prime \ast }{\psi }_s^{\prime \ast }({\mathit{\mu }}_e+{\mathit{\mu }}_N){\psi }_e{\psi }_v{\psi }_{s}d\tau }$

${\displaystyle =\int {\psi }_e^{\prime \ast }{\psi }_v^{\prime \ast }{\psi }_s^{\prime \ast }{\mathit{\mu }}_e{\psi }_e{\psi }_v{\psi }_{s}d\tau +\int {\psi }_e^{\prime \ast }{\psi }_v^{\prime \ast }{\psi }_s^{\prime \ast }{\mathit{\mu }}_N{\psi }_e{\psi }_v{\psi }_{s}d\tau }$

${\displaystyle =\underset{{\displaystyle \genfrac{}{}{0ex}{}{\text{Franck–Condon}}{\text{factor}}}}{\underbrace{\int {\psi }_v^{\prime \ast }{\psi }_{v}d{\tau }_n}}\underset{{\displaystyle \genfrac{}{}{0ex}{}{\text{orbital}}{\text{selection rule}}}}{\underbrace{\int {\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_{e}d{\tau }_e}}\underset{{\displaystyle \genfrac{}{}{0ex}{}{\text{spin}}{\text{selection rule}}}}{\underbrace{\int {\psi }_s^{\prime \ast }{\psi }_{s}d{\tau }_s}}+\underset{{\displaystyle 0}}{\underbrace{\int {\psi }_e^{\prime \ast }{\psi }_{e}d{\tau }_e}}\int {\psi }_v^{\prime \ast }{\mathit{\mu }}_N{\psi }_{v}d{\tau }_v\int {\psi }_s^{\prime \ast }{\psi }_{s}d{\tau }_s.}$

The spin-independent part of the initial integral is here approximated as a product of two integrals:

${\displaystyle \iint {\psi }_v^{\prime \ast }{\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_e{\psi }_{v}d{\tau }_{e}d{\tau }_n\approx \int {\psi }_v^{\prime \ast }{\psi }_{v}d{\tau }_n\int {\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_{e}d{\tau }_e.}$

This factorization would be exact if the integral ${\displaystyle \int {\psi }_e^{\prime \ast }{\mathit{\mu }}_e{\psi }_{e}d{\tau }_e}$ over the spatial coordinates of the electrons would not depend on the nuclear coordinates. However, in the Born–Oppenheimer approximation ${\displaystyle {\psi }_e}$ and ${\displaystyle {\psi }_e^{\prime }}$ do depend (parametrically) on the nuclear coordinates, so that the integral (a so-called transition dipole surface) is a function of nuclear coordinates. Since the dependence is usually rather smooth it is neglected (i.e., the assumption that the transition dipole surface is independent of nuclear coordinates, called the Condon approximation is often allowed).

The first integral after the plus sign is equal to zero because electronic wavefunctions of different states are orthogonal. Remaining is the product of three integrals. The first integral is the vibrational overlap integral, also called the Franck–Condon factor. The remaining two integrals contributing to the probability amplitude determine the electronic spatial and spin selection rules.

The Franck–Condon principle is a statement on allowed vibrational transitions between two different electronic states; other quantum mechanical selection rules may lower the probability of a transition or prohibit it altogether. Rotational selection rules have been neglected in the above derivation. Rotational contributions can be observed in the spectra of gases but are strongly suppressed in liquids and solids.

It should be clear that the quantum mechanical formulation of the Franck–Condon principle is the result of a series of approximations, principally the electrical dipole transition assumption and the Born–Oppenheimer approximation. Weaker magnetic dipole and electric quadrupole electronic transitions along with the incomplete validity of the factorization of the total wavefunction into nuclear, electronic spatial and spin wavefunctions means that the selection rules, including the Franck–Condon factor, are not strictly observed. For any given transition, the value of P is determined by all of the selection rules, however spin selection is the largest contributor, followed by electronic selection rules. The Franck–Condon factor only weakly modulates the intensity of transitions, i.e., it contributes with a factor on the order of 1 to the intensity of bands whose order of magnitude is determined by the other selection rules. The table below gives the range of extinction coefficients for the possible combinations of allowed and forbidden spin and orbital selection rules.

Intensities of electronic transitions

	Range of extinction coefficient (ε) values (mol−1 cm−1)
Spin and orbitally allowed	103 to 105
Spin allowed but orbitally forbidden	100 to 103
Spin forbidden but orbitally allowed	10−5 to 100

Franck–Condon metaphors in spectroscopy

The Franck–Condon principle, in its canonical form, applies only to changes in the vibrational levels of a molecule in the course of a change in electronic levels by either absorption or emission of a photon. The physical intuition of this principle is anchored by the idea that the nuclear coordinates of the atoms constituting the molecule do not have time to change during the very brief amount of time involved in an electronic transition. However, this physical intuition can be, and is indeed, routinely extended to interactions between light-absorbing or emitting molecules (chromophores) and their environment. Franck–Condon metaphors are appropriate because molecules often interact strongly with surrounding molecules, particularly in liquids and solids, and these interactions modify the nuclear coordinates of the chromophore in ways closely analogous to the molecular vibrations considered by the Franck–Condon principle.

Figure 6. Energy diagram of an electronic transition with phonon coupling along the configurational coordinate q _i, a normal mode of the lattice. The upwards arrows represent absorption without phonons and with three phonons. The downwards arrows represent the symmetric process in emission.

Franck–Condon principle for phonons

The closest Franck–Condon analogy is due to the interaction of phonons (quanta of lattice vibrations) with the electronic transitions of chromophores embedded as impurities in the lattice. In this situation, transitions to higher electronic levels can take place when the energy of the photon corresponds to the purely electronic transition energy or to the purely electronic transition energy plus the energy of one or more lattice phonons. In the low-temperature approximation, emission is from the zero-phonon level of the excited state to the zero-phonon level of the ground state or to higher phonon levels of the ground state. Just like in the Franck–Condon principle, the probability of transitions involving phonons is determined by the overlap of the phonon wavefunctions at the initial and final energy levels. For the Franck–Condon principle applied to phonon transitions, the label of the horizontal axis of Figure 1 is replaced in Figure 6 with the configurational coordinate for a normal mode. The lattice mode ${\displaystyle q_i}$ potential energy in Figure 6 is represented as that of a harmonic oscillator, and the spacing between phonon levels (${\displaystyle \hslash {\mathrm{\Omega }}_i}$) is determined by lattice parameters. Because the energy of single phonons is generally quite small, zero- or few-phonon transitions can only be observed at temperatures below about 40 kelvins.

See Zero-phonon line and phonon sideband for further details and references.

Franck–Condon principle in solvation

Figure 7. Energy diagram illustrating the Franck–Condon principle applied to the solvation of chromophores. The parabolic potential curves symbolize the interaction energy between the chromophores and the solvent. The Gaussian curves represent the distribution of this interaction energy.

Franck–Condon considerations can also be applied to the electronic transitions of chromophores dissolved in liquids. In this use of the Franck–Condon metaphor, the vibrational levels of the chromophores, as well as interactions of the chromophores with phonons in the liquid, continue to contribute to the structure of the absorption and emission spectra, but these effects are considered separately and independently.

Consider chromophores surrounded by solvent molecules. These surrounding molecules may interact with the chromophores, particularly if the solvent molecules are polar. This association between solvent and solute is referred to as solvation and is a stabilizing interaction, that is, the solvent molecules can move and rotate until the energy of the interaction is minimized. The interaction itself involves electrostatic and van der Waals forces and can also include hydrogen bonds. Franck–Condon principles can be applied when the interactions between the chromophore and the surrounding solvent molecules are different in the ground and in the excited electronic state. This change in interaction can originate, for example, due to different dipole moments in these two states. If the chromophore starts in its ground state and is close to equilibrium with the surrounding solvent molecules and then absorbs a photon that takes it to the excited state, its interaction with the solvent will be far from equilibrium in the excited state. This effect is analogous to the original Franck–Condon principle: the electronic transition is very fast compared with the motion of nuclei—the rearrangement of solvent molecules in the case of solvation. We now speak of a vertical transition, but now the horizontal coordinate is solvent-solute interaction space. This coordinate axis is often labeled as "Solvation Coordinate" and represents, somewhat abstractly, all of the relevant dimensions of motion of all of the interacting solvent molecules.

In the original Franck–Condon principle, after the electronic transition, the molecules which end up in higher vibrational states immediately begin to relax to the lowest vibrational state. In the case of solvation, the solvent molecules will immediately try to rearrange themselves in order to minimize the interaction energy. The rate of solvent relaxation depends on the viscosity of the solvent. Assuming the solvent relaxation time is short compared with the lifetime of the electronic excited state, emission will be from the lowest solvent energy state of the excited electronic state. For small-molecule solvents such as water or methanol at ambient temperature, solvent relaxation time is on the order of some tens of picoseconds whereas chromophore excited state lifetimes range from a few picoseconds to a few nanoseconds. Immediately after the transition to the ground electronic state, the solvent molecules must also rearrange themselves to accommodate the new electronic configuration of the chromophore. Figure 7 illustrates the Franck–Condon principle applied to solvation. When the solution is illuminated by light corresponding to the electronic transition energy, some of the chromophores will move to the excited state. Within this group of chromophores there will be a statistical distribution of solvent-chromophore interaction energies, represented in the figure by a Gaussian distribution function. The solvent-chromophore interaction is drawn as a parabolic potential in both electronic states. Since the electronic transition is essentially instantaneous on the time scale of solvent motion (vertical arrow), the collection of excited state chromophores is immediately far from equilibrium. The rearrangement of the solvent molecules according to the new potential energy curve is represented by the curved arrows in Figure 7. Note that while the electronic transitions are quantized, the chromophore-solvent interaction energy is treated as a classical continuum due to the large number of molecules involved. Although emission is depicted as taking place from the minimum of the excited state chromophore-solvent interaction potential, significant emission can take place before equilibrium is reached when the viscosity of the solvent is high, or the lifetime of the excited state is short. The energy difference between absorbed and emitted photons depicted in Figure 7 is the solvation contribution to the Stokes shift.

Density matrix

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https://en.wikipedia.org/wiki/Beam-powered_propulsion

In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. These arise in quantum mechanics in two different situations:

1. when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and
2. when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.

Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems and quantum information.

Definition and motivation

The density matrix is a representation of a linear operator called the density operator. The density matrix is obtained from the density operator by a choice of an orthonormal basis in the underlying space. In practice, the terms density matrix and density operator are often used interchangeably.

Pick a basis with states ${\displaystyle |0⟩}$, ${\displaystyle |1⟩}$ in a two-dimensional Hilbert space, then the density operator is represented by the matrix $${\displaystyle ({\rho }_{ij})=\left(\begin{array}{cc}{\rho }_{00}& {\rho }_{01}\\ {\rho }_{10}& {\rho }_{11}\end{array}\right)=\left(\begin{array}{cc}{p}_0& {\rho }_{01}\\ {\rho }_{01}^{\ast }& p_1\end{array}\right)}$$ where the diagonal elements are real numbers that sum to one (also called populations of the two states ${\displaystyle |0⟩}$, ${\displaystyle |1⟩}$). The off-diagonal elements are complex conjugates of each other (also called coherences); they are restricted in magnitude by the requirement that ${\displaystyle ({\rho }_{ij})}$ be a positive semi-definite operator, see below.

A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system. This definition can be motivated by considering a situation where some pure states ${\displaystyle |{\psi }_j⟩}$ (which are not necessarily orthogonal) are prepared with probability ${\displaystyle p_j}$ each. This is known as an ensemble of pure states. The probability of obtaining projective measurement result ${\displaystyle m}$ when using projectors ${\displaystyle {\mathrm{\Pi }}_m}$ is given by $${\displaystyle p(m)=\sum _j{p}_j⟨{\psi }_j|{\mathrm{\Pi }}_m|{\psi }_j⟩=\mathrm{tr}\left[{\mathrm{\Pi }}_m\left(\sum _j{p}_j|{\psi }_j⟩⟨{\psi }_j|\right)\right],}$$ which makes the density operator, defined as $${\displaystyle \rho =\sum _j{p}_j|{\psi }_j⟩⟨{\psi }_j|,}$$ a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, self-adjoint, and has trace one. Conversely, it follows from the spectral theorem that every operator with these properties can be written as $\sum _j{p}_j|{\psi }_j⟩⟨{\psi }_j|$ for some states ${\displaystyle |{\psi }_j⟩}$ and coefficients ${\displaystyle p_j}$ that are non-negative and add up to one. However, this representation will not be unique, as shown by the Schrödinger–HJW theorem.

Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let ${\displaystyle |\mathrm{\Psi }⟩}$ be a pure entangled state in the composite Hilbert space ${\displaystyle {\mathcal{H}}_1\otimes {\mathcal{H}}_2}$. The probability of obtaining measurement result ${\displaystyle m}$ when measuring projectors ${\displaystyle {\mathrm{\Pi }}_m}$ on the Hilbert space ${\displaystyle {\mathcal{H}}_1}$ alone is given by $${\displaystyle p(m)=⟨\mathrm{\Psi }|\left({\mathrm{\Pi }}_m\otimes I\right)|\mathrm{\Psi }⟩=\mathrm{tr}\left[{\mathrm{\Pi }}_m\left({\mathrm{tr}}_2|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|\right)\right],}$$ where ${\displaystyle {\mathrm{tr}}_2}$ denotes the partial trace over the Hilbert space ${\displaystyle {\mathcal{H}}_2}$. This makes the operator $${\displaystyle \rho ={\mathrm{tr}}_2|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|}$$ a convenient tool to calculate the probabilities of these local measurements. It is known as the reduced density matrix of ${\displaystyle |\mathrm{\Psi }⟩}$ on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, the Schrödinger–HJW theorem implies that all density operators can be written as ${\displaystyle {\mathrm{tr}}_2|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|}$ for some state ${\displaystyle |\mathrm{\Psi }⟩}$.

Pure and mixed states

A pure quantum state is a state that can not be written as a probabilistic mixture, or convex combination, of other quantum states. There are several equivalent characterizations of pure states in the language of density operators. A density operator represents a pure state if and only if:

• it can be written as an outer product of a state vector ${\displaystyle |\psi ⟩}$ with itself, that is, $${\displaystyle \rho =|\psi ⟩⟨\psi |.}$$
• it is a projection, in particular of rank one.
• it is idempotent, that is $${\displaystyle \rho ={\rho }^2.}$$
• it has purity one, that is, $${\displaystyle \mathrm{tr}({\rho }^2)=1.}$$

It is important to emphasize the difference between a probabilistic mixture (i.e. an ensemble) of quantum states and the superposition of two states. If an ensemble is prepared to have half of its systems in state ${\displaystyle |{\psi }_1⟩}$ and the other half in ${\displaystyle |{\psi }_2⟩}$, it can be described by the density matrix:

${\displaystyle \rho =\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),}$

where ${\displaystyle |{\psi }_1⟩}$ and ${\displaystyle |{\psi }_2⟩}$ are assumed orthogonal and of dimension 2, for simplicity. On the other hand, a quantum superposition of these two states with equal probability amplitudes results in the pure state ${\displaystyle |\psi ⟩=(|{\psi }_1⟩+|{\psi }_2⟩)/\sqrt{2},}$ with density matrix

${\displaystyle |\psi ⟩⟨\psi |=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right).}$

Unlike the probabilistic mixture, this superposition can display quantum interference.

In the Bloch sphere representation of a qubit, each point on the unit sphere stands for a pure state. All other density matrices correspond to points in the interior.

Geometrically, the set of density operators is a convex set, and the pure states are the extremal points of that set. The simplest case is that of a two-dimensional Hilbert space, known as a qubit. An arbitrary mixed state for a qubit can be written as a linear combination of the Pauli matrices, which together with the identity matrix provide a basis for ${\displaystyle 2\times 2}$ self-adjoint matrices:

${\displaystyle \rho =\frac{1}{2}\left(I+r_x{\sigma }_x+r_y{\sigma }_y+r_z{\sigma }_z\right),}$

where the real numbers ${\displaystyle (r_x,r_y,r_z)}$ are the coordinates of a point within the unit ball and

${\displaystyle {\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).}$

Points with ${\displaystyle r_x^2+r_y^2+r_z^2=1}$ represent pure states, while mixed states are represented by points in the interior. This is known as the Bloch sphere picture of qubit state space.

Example: light polarization

The incandescent light bulb (1) emits completely random polarized photons (2) with mixed state density matrix:

${\displaystyle \left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]}$.

After passing through vertical plane polarizer (3), the remaining photons are all vertically polarized (4) and have pure state density matrix:

${\displaystyle \left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]}$.

An example of pure and mixed states is light polarization. An individual photon can be described as having right or left circular polarization, described by the orthogonal quantum states ${\displaystyle |\mathrm{R}⟩}$ and ${\displaystyle |\mathrm{L}⟩}$ or a superposition of the two: it can be in any state ${\displaystyle \alpha |\mathrm{R}⟩+\beta |\mathrm{L}⟩}$ (with ${\displaystyle |\alpha {|}^2+|\beta {|}^2=1}$), corresponding to linear, circular, or elliptical polarization. Consider now a vertically polarized photon, described by the state ${\displaystyle |\mathrm{V}⟩=(|\mathrm{R}⟩+|\mathrm{L}⟩)/\sqrt{2}}$. If we pass it through a circular polarizer that allows either only ${\displaystyle |\mathrm{R}⟩}$ polarized light, or only ${\displaystyle |\mathrm{L}⟩}$ polarized light, half of the photons are absorbed in both cases. This may make it seem like half of the photons are in state ${\displaystyle |\mathrm{R}⟩}$ and the other half in state ${\displaystyle |\mathrm{L}⟩}$, but this is not correct: if we pass ${\displaystyle (|\mathrm{R}⟩+|\mathrm{L}⟩)/\sqrt{2}}$ through a linear polarizer there is no absorption whatsoever, but if we pass either state ${\displaystyle |\mathrm{R}⟩}$ or ${\displaystyle |\mathrm{L}⟩}$ half of the photons are absorbed.

Unpolarized light (such as the light from an incandescent light bulb) cannot be described as any state of the form ${\displaystyle \alpha |\mathrm{R}⟩+\beta |\mathrm{L}⟩}$ (linear, circular, or elliptical polarization). Unlike polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and it cannot be made polarized by passing it through any wave plate. However, unpolarized light can be described as a statistical ensemble, e. g. as each photon having either ${\displaystyle |\mathrm{R}⟩}$ polarization or ${\displaystyle |\mathrm{L}⟩}$ polarization with probability 1/2. The same behavior would occur if each photon had either vertical polarization ${\displaystyle |\mathrm{V}⟩}$ or horizontal polarization ${\displaystyle |\mathrm{H}⟩}$ with probability 1/2. These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. For this example of unpolarized light, the density operator equals.

${\displaystyle \rho =\frac{1}{2}|\mathrm{R}⟩⟨\mathrm{R}|+\frac{1}{2}|\mathrm{L}⟩⟨\mathrm{L}|=\frac{1}{2}|\mathrm{H}⟩⟨\mathrm{H}|+\frac{1}{2}|\mathrm{V}⟩⟨\mathrm{V}|=\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).}$

There are also other ways to generate unpolarized light: one possibility is to introduce uncertainty in the preparation of the photon, for example, passing it through a birefringent crystal with a rough surface, so that slightly different parts of the light beam acquire different polarizations. Another possibility is using entangled states: a radioactive decay can emit two photons traveling in opposite directions, in the quantum state ${\displaystyle (|\mathrm{R},\mathrm{L}⟩+|\mathrm{L},\mathrm{R}⟩)/\sqrt{2}}$. The joint state of the two photons together is pure, but the density matrix for each photon individually, found by taking the partial trace of the joint density matrix, is completely mixed.

Equivalent ensembles and purifications

Main article: Schrödinger–HJW theorem

A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix. Those cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized: let ${\displaystyle \{p_j,|{\psi }_j⟩\}}$ be an ensemble. Then for any complex matrix ${\displaystyle U}$ such that ${\displaystyle U^{†}U=I}$ (a partial isometry), the ensemble ${\displaystyle \{q_i,|{\phi }_i⟩\}}$ defined by

${\displaystyle \sqrt{q_i}|{\phi }_i⟩=\sum _j{U}_{ij}\sqrt{p_j}|{\psi }_j⟩}$

will give rise to the same density operator, and all equivalent ensembles are of this form.

A closely related fact is that a given density operator has infinitely many different purifications, which are pure states that generate the density operator when a partial trace is taken. Let

${\displaystyle \rho =\sum _j{p}_j|{\psi }_j⟩⟨{\psi }_j|}$

be the density operator generated by the ensemble ${\displaystyle \{p_j,|{\psi }_j⟩\}}$, with states ${\displaystyle |{\psi }_j⟩}$ not necessarily orthogonal. Then for all partial isometries ${\displaystyle U}$ we have that

${\displaystyle |\mathrm{\Psi }⟩=\sum _j\sqrt{p_j}|{\psi }_j⟩U|a_j⟩}$

is a purification of ${\displaystyle \rho }$, where ${\displaystyle |a_j⟩}$ is an orthogonal basis, and furthermore all purifications of ${\displaystyle \rho }$ are of this form.

Measurement

Let ${\displaystyle A}$ be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states ${\displaystyle |{\psi }_j⟩}$ occurs with probability ${\displaystyle p_j}$. Then the corresponding density operator equals

${\displaystyle \rho =\sum _j{p}_j|{\psi }_j⟩⟨{\psi }_j|.}$

The expectation value of the measurement can be calculated by extending from the case of pure states:

${\displaystyle ⟨A⟩=\sum _j{p}_j⟨{\psi }_j|A|{\psi }_j⟩=\sum _j{p}_j\mathrm{tr}\left(|{\psi }_j⟩⟨{\psi }_j|A\right)=\mathrm{tr}\left(\sum _j{p}_j|{\psi }_j⟩⟨{\psi }_j|A\right)=\mathrm{tr}(\rho A),}$

where ${\displaystyle \mathrm{tr}}$ denotes trace. Thus, the familiar expression ${\displaystyle ⟨A⟩=⟨\psi |A|\psi ⟩}$ for pure states is replaced by

${\displaystyle ⟨A⟩=\mathrm{tr}(\rho A)}$

for mixed states.

Moreover, if ${\displaystyle A}$ has spectral resolution

${\displaystyle A=\sum _i{a}_i{P}_i,}$

where ${\displaystyle P_i}$ is the projection operator into the eigenspace corresponding to eigenvalue ${\displaystyle a_i}$, the post-measurement density operator is given by

${\displaystyle {\rho }_i^{\prime }=\frac{P_i\rho P_i}{\mathrm{tr}\left[\rho P_i\right]}}$

when outcome i is obtained. In the case where the measurement result is not known the ensemble is instead described by

${\displaystyle {\rho }^{\prime }=\sum _i{P}_i\rho P_i.}$

If one assumes that the probabilities of measurement outcomes are linear functions of the projectors ${\displaystyle P_i}$, then they must be given by the trace of the projector with a density operator. Gleason's theorem shows that in Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption of non-contextuality. This restriction on the dimension can be removed by assuming non-contextuality for POVMs as well, but this has been criticized as physically unmotivated.

Entropy

The von Neumann entropy ${\displaystyle S}$ of a mixture can be expressed in terms of the eigenvalues of ${\displaystyle \rho }$ or in terms of the trace and logarithm of the density operator ${\displaystyle \rho }$. Since ${\displaystyle \rho }$ is a positive semi-definite operator, it has a spectral decomposition such that ${\displaystyle \rho =\sum _i{\lambda }_i|{\phi }_i⟩⟨{\phi }_i|}$, where ${\displaystyle |{\phi }_i⟩}$ are orthonormal vectors, ${\displaystyle {\lambda }_i\ge 0}$, and ${\displaystyle \sum {\lambda }_i=1}$. Then the entropy of a quantum system with density matrix ${\displaystyle \rho }$ is

${\displaystyle S=-\sum _i{\lambda }_i\ln {\lambda }_i=-\mathrm{tr}(\rho \ln \rho ).}$

This definition implies that the von Neumann entropy of any pure state is zero. If ${\displaystyle {\rho }_i}$ are states that have support on orthogonal subspaces, then the von Neumann entropy of a convex combination of these states,

${\displaystyle \rho =\sum _i{p}_i{\rho }_i,}$

is given by the von Neumann entropies of the states ${\displaystyle {\rho }_i}$ and the Shannon entropy of the probability distribution ${\displaystyle p_i}$:

${\displaystyle S(\rho )=H(p_i)+\sum _i{p}_{i}S({\rho }_i).}$

When the states ${\displaystyle {\rho }_i}$ do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination ${\displaystyle \rho }$.

Given a density operator ${\displaystyle \rho }$ and a projective measurement as in the previous section, the state ${\displaystyle {\rho }^{\prime }}$ defined by the convex combination

${\displaystyle {\rho }^{\prime }=\sum _i{P}_i\rho P_i,}$

which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred, has a von Neumann entropy larger than that of ${\displaystyle \rho }$, except if ${\displaystyle \rho ={\rho }^{\prime }}$. It is however possible for the ${\displaystyle {\rho }^{\prime }}$ produced by a generalized measurement, or POVM, to have a lower von Neumann entropy than ${\displaystyle \rho }$.

Von Neumann equation for time evolution

See also: Liouville's theorem (Hamiltonian) § Quantum Liouville equation

Just as the Schrödinger equation describes how pure states evolve in time, the von Neumann equation (also known as the Liouville–von Neumann equation) describes how a density operator evolves in time. The von Neumann equation dictates that

${\displaystyle i\hslash \frac{d}{dt}\rho =[H,\rho ],}$

where the brackets denote a commutator.

This equation only holds when the density operator is taken to be in the Schrödinger picture, even though this equation seems at first look to emulate the Heisenberg equation of motion in the Heisenberg picture, with a crucial sign difference:

${\displaystyle i\hslash \frac{d}{dt}{A}_{\text{H}}=-[H_{\text{H}},A_{\text{H}}],}$

where ${\displaystyle A_{\text{H}}(t)}$ is some Heisenberg picture operator; but in this picture the density matrix is not time-dependent, and the relative sign ensures that the time derivative of the expected value ${\displaystyle ⟨A⟩}$ comes out the same as in the Schrödinger picture.

If the Hamiltonian is time-independent, the von Neumann equation can be easily solved to yield

${\displaystyle \rho (t)=e^{-iHt/\hslash }\rho (0)e^{iHt/\hslash }.}$

For a more general Hamiltonian, if ${\displaystyle G(t)}$ is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by

${\displaystyle \rho (t)=G(t)\rho (0)G(t{)}^{†}.}$

If one enters the interaction picture, choosing to focus on some component ${\displaystyle H_1}$ of the Hamiltonian ${\displaystyle H=H_0+H_1}$, the equation for the evolution of the interaction-picture density operator ${\displaystyle {\rho }_{\mathrm{I}}(t)}$ possesses identical structure to the von Neumann equation, except the Hamiltonian must also be transformed into the new picture:

${\displaystyle {\displaystyle i\hslash \frac{d}{dt}{\rho }_{\text{I}}(t)=[H_{1,\text{I}}(t),{\rho }_{\text{I}}(t)],}}$

where ${\displaystyle {\displaystyle H_{1,\text{I}}(t)=e^{i{H}_{0}t/\hslash }{H}_1{e}^{-i{H}_{0}t/\hslash }}}$.

Wigner functions and classical analogies

Main article: Phase-space formulation

The density matrix operator may also be realized in phase space. Under the Wigner map, the density matrix transforms into the equivalent Wigner function,

${\displaystyle W(x,p)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\pi \hslash }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\psi }^{\ast }(x+y)\psi (x-y)e^{2ipy/\hslash }dy.}$

The equation for the time evolution of the Wigner function, known as Moyal equation, is then the Wigner-transform of the above von Neumann equation,

${\displaystyle \frac{\mathrm{\partial }W(x,p,t)}{\mathrm{\partial }t}=-\{\{W(x,p,t),H(x,p)\}\},}$

where ${\displaystyle H(x,p)}$ is the Hamiltonian, and ${\displaystyle \{\{\cdot ,\cdot \}\}}$ is the Moyal bracket, the transform of the quantum commutator.

The evolution equation for the Wigner function is then analogous to that of its classical limit, the Liouville equation of classical physics. In the limit of a vanishing Planck constant ${\displaystyle \hslash }$, ${\displaystyle W(x,p,t)}$ reduces to the classical Liouville probability density function in phase space.

Example applications

Density matrices are a basic tool of quantum mechanics, and appear at least occasionally in almost any type of quantum-mechanical calculation. Some specific examples where density matrices are especially helpful and common are as follows:

• Statistical mechanics uses density matrices, most prominently to express the idea that a system is prepared at a nonzero temperature. Constructing a density matrix using a canonical ensemble gives a result of the form ${\displaystyle \rho =\exp (-\beta H)/Z(\beta )}$, where ${\displaystyle \beta }$ is the inverse temperature ${\displaystyle (k_{\mathrm{B}}T{)}^{-1}}$ and ${\displaystyle H}$ is the system's Hamiltonian. The normalization condition that the trace of ${\displaystyle \rho }$ be equal to 1 defines the partition function to be ${\displaystyle Z(\beta )=\mathrm{t}\mathrm{r}\exp (-\beta H)}$. If the number of particles involved in the system is itself not certain, then a grand canonical ensemble can be applied, where the states summed over to make the density matrix are drawn from a Fock space.
• Quantum decoherence theory typically involves non-isolated quantum systems developing entanglement with other systems, including measurement apparatuses. Density matrices make it much easier to describe the process and calculate its consequences. Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives. This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the classical limit of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.
• Similarly, in quantum computation, quantum information theory, open quantum systems, and other fields where state preparation is noisy and decoherence can occur, density matrices are frequently used. Noise is often modelled via a depolarizing channel or an amplitude damping channel. Quantum tomography is a process by which, given a set of data representing the results of quantum measurements, a density matrix consistent with those measurement results is computed.
• When analyzing a system with many electrons, such as an atom or molecule, an imperfect but useful first approximation is to treat the electrons as uncorrelated or each having an independent single-particle wavefunction. This is the usual starting point when building the Slater determinant in the Hartree–Fock method. If there are ${\displaystyle N}$ electrons filling the ${\displaystyle N}$ single-particle wavefunctions ${\displaystyle |{\psi }_i⟩}$ and if only single-particle observables are considered, then their expectation values for the ${\displaystyle N}$-electron system can be computed using the density matrix $\sum _{i=1}^N|{\psi }_i⟩⟨{\psi }_i|$ (the one-particle density matrix of the ${\displaystyle N}$-electron system).

C*-algebraic formulation of states

It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. For this reason, observables are identified with elements of an abstract C*-algebra A (that is one without a distinguished representation as an algebra of operators) and states are positive linear functionals on A. However, by using the GNS construction, we can recover Hilbert spaces that realize A as a subalgebra of operators.

Geometrically, a pure state on a C*-algebra A is a state that is an extreme point of the set of all states on A. By properties of the GNS construction these states correspond to irreducible representations of A.

The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators, and therefore the pure states of K(H) are exactly the pure states in the sense of quantum mechanics.

The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures.

History

The formalism of density operators and matrices was introduced in 1927 by John von Neumann and independently, but less systematically, by Lev Landau and later in 1946 by Felix Bloch. Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The name density matrix itself relates to its classical correspondence to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics, which was introduced by Eugene Wigner in 1932.

In contrast, the motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.