Levelized cost of energy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Levelized_cost_of_energy 

The levelized cost of energy (LCOE), or levelized cost of electricity, is a measure of the average net present cost of electricity generation for a generating plant over its lifetime. It is used for investment planning and to compare different methods of electricity generation on a consistent basis. The LCOE "represents the average revenue per unit of electricity generated that would be required to recover the costs of building and operating a generating plant during an assumed financial life and duty cycle", and is calculated as the ratio between all the discounted costs over the lifetime of an electricity generating plant divided by a discounted sum of the actual energy amounts delivered. Inputs to LCOE are chosen by the estimator. They can include cost of capital, decommissioning, "fuel costs, fixed and variable operations and maintenance costs, financing costs, and an assumed utilization rate."

Calculation

The LCOE is calculated as:

${\displaystyle \mathrm {LCOE} ={\frac {\text{sum of costs over lifetime}}{\text{sum of electrical energy produced over lifetime}}}={\frac {\sum _{t=1}^{n}{\frac {I_{t}+M_{t}+F_{t}}{\left({1+r}\right)^{t}}}}{\sum _{t=1}^{n}{\frac {E_{t}}{\left({1+r}\right)^{t}}}}}}$

It	:	investment expenditures in the year t
Mt	:	operations and maintenance expenditures in the year t
Ft	:	fuel expenditures in the year t
Et	:	electrical energy generated in the year t
r	:	discount rate
n	:	expected lifetime of system or power station

Note: caution must be taken when using formulas for the levelized cost, as they often embody unseen assumptions, neglect effects like taxes, and may be specified in real or nominal levelized cost. For example, other versions of the above formula do not discount the electricity stream.

Typically the LCOE is calculated over the design lifetime of a plant and given in currency per energy unit, for example EUR per kilowatt-hour or AUD per megawatt-hour.

LCOE does not represent cost of electricity for consumer and is most meaningful from the investor point of view. Care should be taken in comparing different LCOE studies and the sources of the information as the LCOE for a given energy source is highly dependent on the assumptions, financing terms and technological deployment analyzed.

Thus, a key requirement for the analysis is a clear statement of the applicability of the analysis based on justified assumptions. In particular, for LCOE to be usable for rank-ordering energy-generation alternatives, caution must be taken to calculate it in "real" terms, i.e. including adjustment for expected inflation.

Considerations

There are potential limits to some levelized cost of electricity metrics for comparing energy generating sources. One of the most important potential limitations of LCOE is that it may not control for time effects associated with matching electricity production to demand. This can happen at two levels:

• Dispatchability, the ability of a generating system to come online, go offline, or ramp up or down, quickly as demand swings.
• The extent to which the availability profile matches or conflicts with the market demand profile.

In particular, if matching grid energy storage is not included in models for variable renewable energy sources such as solar and wind that are otherwise not dispatchable, they may produce electricity when it is not needed in the grid without storage. The value of this electricity may be lower than if it was produced at another time, or even negative. At the same time, intermittent sources can be competitive if they are available to produce when demand and prices are highest, such as solar during summertime mid-day peaks seen in hot countries where air conditioning is a major consumer. Some dispatchable technologies, such as most coal power plants, are incapable of fast ramping. Excess generation when not needed may force curtailments, thus reducing the revenue of a energy provider.

Another potential limitation of LCOE is that some analyses may not adequately consider indirect costs of generation. These can include environmental externalities or grid upgrades requirements. Intermittent power sources, such as wind and solar, may incur extra costs associated with needing to have storage or backup generation available.

The LCOE of energy efficiency and conservation (EEC) efforts can be calculated, and included alongside LCOE numbers of other options such as generation infrastructure for comparison. If this is omitted or incomplete, LCOE may not give a comprehensive picture of potential options available for meeting energy needs, and of any opportunity costs. Considering the LCOE only for utility scale plants will tend to maximize generation and risks overestimating required generation due to efficiency, thus "lowballing" their LCOE. For solar systems installed at the point of end use, it is more economical to invest in EEC first, then solar. This results in a smaller required solar system than what would be needed without the EEC measures. However, designing a solar system on the basis of its LCOE without considering that of EEC would cause the smaller system LCOE to increase, as the energy generation drops faster than the system cost. Every option should be considered, not just the LCOE of the energy source. LCOE is not as relevant to end-users than other financial considerations such as income, cashflow, mortgage, leases, rent, and electricity bills. Comparing solar investments in relation to these can make it easier for end-users to make a decision, or using cost-benefit calculations "and/or an asset’s capacity value or contribution to peak on a system or circuit level".

Capacity factor

Assumption of capacity factor has significant impact on the calculation of LCOE as it determines the actual amount of energy produced by specific installed power. Formulas that output cost per unit of energy ($/MWh) already account for the capacity factor, while formulas that output cost per unit of power ($/MW) do not.

Discount rate

Cost of capital expressed as the discount rate is one of the most controversial inputs into the LCOE equation, as it significantly impacts the outcome and a number of comparisons assume arbitrary discount rate values with little transparency of why specific value was selected. Comparisons that assume public funding, subsidies and social cost of capital (see below) tend to choose low discount rates (3%), while comparisons prepared by private investment banks tend to assume high discount rate (7-15%) associated with commercial for-profit funding.

The differences in outcomes for different assumed discount rates are dramatic — for example, NEA LCOE calculation for residential PV at 3% discount rate produces $150/MWh, while at 10% it produces $250/MWh. LCOE estimate prepared by Lazard (2020) for nuclear power based on unspecified methodology produced $164/MWh, while LCOE calculated by the investor for an actual Olkiluoto Nuclear Power Plant in Finland came out to be below 30 EUR/MWh.

A choice of 10% discount rate results in the energy production in 20 years being assigned accounting value of just 15%, which nearly triples the LCOE price. This approach, which is considered prudent from today's private financial investor's perspective, is being criticised as inappropriate for assessment of public infrastructure that mitigates climate change as it ignores social cost of the CO2 emissions for future generations and focuses on short-term investment perspective only. The approach has been criticised equally by proponents of nuclear and renewable technologies, which require high initial investment but then have low operational cost and, most importantly, are low-carbon. According to Social Cost of Carbon methodology, the discount rate for low-carbon technologies should be 1-3%.

Levelized avoided cost of energy

The metric levelized avoided cost of energy (LACE) addresses some of the shortcomings of LCOE by considering the economic value that the source provides to the grid. The economic value takes into account the dispatchability of a resource, as well as the existing energy mix in a region.

Levelized cost of storage

The levelized cost of storage (LCOS) is the analogous of LCOE applied to electricity storage technologies, such as batteries. Distinction between the two metrics can be blurred when the LCOE of systems incorporating both generation and storage are considered.

Superconductivity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Superconductivity 

 

A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. Persistent electric current flows on the surface of the superconductor, acting to exclude the magnetic field of the magnet (Faraday's law of induction). This current effectively forms an electromagnet that repels the magnet.

 

Play media

Video of the Meissner effect in a high-temperature superconductor (black pellet) with a NdFeB magnet (metallic)

 

A high-temperature superconductor levitating above a magnet

Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source.

The superconductivity phenomenon was discovered in 1911 by Dutch physicist Heike Kamerlingh Onnes. Like ferromagnetism and atomic spectral lines, superconductivity is a phenomenon which can only be explained by quantum mechanics. It is characterized by the Meissner effect, the complete ejection of magnetic field lines from the interior of the superconductor during its transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics.

In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 90 K (−183 °C). Such a high transition temperature is theoretically impossible for a conventional superconductor, leading the materials to be termed high-temperature superconductors. The cheaply available coolant liquid nitrogen boils at 77 K, and thus the existence of superconductivity at higher temperatures than this facilitates many experiments and applications that are less practical at lower temperatures.

Classification

There are many criteria by which superconductors are classified. The most common are:

Response to a magnetic field

A superconductor can be Type I, meaning it has a single critical field, above which all superconductivity is lost and below which the magnetic field is completely expelled from the superconductor; or Type II, meaning it has two critical fields, between which it allows partial penetration of the magnetic field through isolated points. These points are called vortices. Furthermore, in multicomponent superconductors it is possible to have a combination of the two behaviours. In that case the superconductor is of Type-1.5.

By theory of operation

It is conventional if it can be explained by the BCS theory or its derivatives, or unconventional, otherwise. Alternatively, a superconductor is called unconventional if the superconducting order parameter transforms according to a non-trivial irreducible representation of the point group or space group of the system.

By critical temperature

A superconductor is generally considered high-temperature if it reaches a superconducting state above a temperature of 30 K (−243.15 °C); as in the initial discovery by Georg Bednorz and K. Alex Müller. It may also reference materials that transition to superconductivity when cooled using liquid nitrogen – that is, at only T_c > 77 K, although this is generally used only to emphasize that liquid nitrogen coolant is sufficient. Low temperature superconductors refer to materials with a critical temperature below 30 K. One exception to this rule is the iron pnictide group of superconductors which display behaviour and properties typical of high-temperature superconductors, yet some of the group have critical temperatures below 30 K.

By material

"Top: Periodic table of superconducting elemental solids and their experimental critical temperature (T). Bottom: Periodic table of superconducting binary hydrides (0–300 GPa). Theoretical predictions indicated in blue and experimental results in red."

Superconductor material classes include chemical elements (e.g. mercury or lead), alloys (such as niobium–titanium, germanium–niobium, and niobium nitride), ceramics (YBCO and magnesium diboride), superconducting pnictides (like fluorine-doped LaOFeAs) or organic superconductors (fullerenes and carbon nanotubes; though perhaps these examples should be included among the chemical elements, as they are composed entirely of carbon).

Elementary properties of superconductors

Several physical properties of superconductors vary from material to material, such as the critical temperature, the value of the superconducting gap, the critical magnetic field, and the critical current density at which superconductivity is destroyed. On the other hand, there is a class of properties that are independent of the underlying material. The Meissner effect, the quantization of the magnetic flux or permanent currents, i.e. the state of zero resistance are the most important examples. The existence of these "universal" properties is rooted in the nature of the broken symmetry of the superconductor and the emergence of off-diagonal long range order. Superconductivity is a thermodynamic phase, and thus possesses certain distinguishing properties which are largely independent of microscopic details.

Off diagonal long range order is closely connected to the formation of Cooper pairs. An article by V.F. Weisskopf presents simple physical explanations for the formation of Cooper pairs, for the origin of the attractive force causing the binding of the pairs, for the finite energy gap, and for the existence of permanent currents.

Zero electrical DC resistance

Electric cables for accelerators at CERN. Both the massive and slim cables are rated for 12,500 A. Top: regular cables for LEP; bottom: superconductor-based cables for the LHC

 

Cross section of a preform superconductor rod from abandoned Texas Superconducting Super Collider (SSC).

The simplest method to measure the electrical resistance of a sample of some material is to place it in an electrical circuit in series with a current source I and measure the resulting voltage V across the sample. The resistance of the sample is given by Ohm's law as R = V / I. If the voltage is zero, this means that the resistance is zero.

Superconductors are also able to maintain a current with no applied voltage whatsoever, a property exploited in superconducting electromagnets such as those found in MRI machines. Experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a current lifetime of at least 100,000 years. Theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature. In practice, currents injected in superconducting coils have persisted for more than 25 years (as of August 4, 2020) in superconducting gravimeters. In such instruments, the measurement principle is based on the monitoring of the levitation of a superconducting niobium sphere with a mass of 4 grams.

In a normal conductor, an electric current may be visualized as a fluid of electrons moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into heat, which is essentially the vibrational kinetic energy of the lattice ions. As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance and Joule heating.

The situation is different in a superconductor. In a conventional superconductor, the electronic fluid cannot be resolved into individual electrons. Instead, it consists of bound pairs of electrons known as Cooper pairs. This pairing is caused by an attractive force between electrons from the exchange of phonons. Due to quantum mechanics, the energy spectrum of this Cooper pair fluid possesses an energy gap, meaning there is a minimum amount of energy ΔE that must be supplied in order to excite the fluid. Therefore, if ΔE is larger than the thermal energy of the lattice, given by kT, where k is Boltzmann's constant and T is the temperature, the fluid will not be scattered by the lattice. The Cooper pair fluid is thus a superfluid, meaning it can flow without energy dissipation.

In a class of superconductors known as type II superconductors, including all known high-temperature superconductors, an extremely low but nonzero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of magnetic vortices in the electronic superfluid, which dissipates some of the energy carried by the current. If the current is sufficiently small, the vortices are stationary, and the resistivity vanishes. The resistance due to this effect is tiny compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen into a disordered but stationary phase known as a "vortex glass". Below this vortex glass transition temperature, the resistance of the material becomes truly zero.

Phase transition

Behavior of heat capacity (c_v, blue) and resistivity (ρ, green) at the superconducting phase transition

In superconducting materials, the characteristics of superconductivity appear when the temperature T is lowered below a critical temperature T_c. The value of this critical temperature varies from material to material. Conventional superconductors usually have critical temperatures ranging from around 20 K to less than 1 K. Solid mercury, for example, has a critical temperature of 4.2 K. As of 2015, the highest critical temperature found for a conventional superconductor is 203K for H₂S, although high pressures of approximately 90 gigapascals were required. Cuprate superconductors can have much higher critical temperatures: YBa₂Cu₃O₇, one of the first cuprate superconductors to be discovered, has a critical temperature above 90 K, and mercury-based cuprates have been found with critical temperatures in excess of 130 K. The basic physical mechanism responsible for the high critical temperature is not yet clear. However, it is clear that a two-electron pairing is involved, although the nature of the pairing (${\displaystyle s}$ wave vs. ${\displaystyle d}$ wave) remains controversial.

Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external magnetic field is applied which is greater than the critical magnetic field. This is because the Gibbs free energy of the superconducting phase increases quadratically with the magnetic field while the free energy of the normal phase is roughly independent of the magnetic field. If the material superconducts in the absence of a field, then the superconducting phase free energy is lower than that of the normal phase and so for some finite value of the magnetic field (proportional to the square root of the difference of the free energies at zero magnetic field) the two free energies will be equal and a phase transition to the normal phase will occur. More generally, a higher temperature and a stronger magnetic field lead to a smaller fraction of electrons that are superconducting and consequently to a longer London penetration depth of external magnetic fields and currents. The penetration depth becomes infinite at the phase transition.

The onset of superconductivity is accompanied by abrupt changes in various physical properties, which is the hallmark of a phase transition. For example, the electronic heat capacity is proportional to the temperature in the normal (non-superconducting) regime. At the superconducting transition, it suffers a discontinuous jump and thereafter ceases to be linear. At low temperatures, it varies instead as e^−α/T for some constant, α. This exponential behavior is one of the pieces of evidence for the existence of the energy gap.

The order of the superconducting phase transition was long a matter of debate. Experiments indicate that the transition is second-order, meaning there is no latent heat. However, in the presence of an external magnetic field there is latent heat, because the superconducting phase has a lower entropy below the critical temperature than the normal phase. It has been experimentally demonstrated that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material.

Calculations in the 1970s suggested that it may actually be weakly first-order due to the effect of long-range fluctuations in the electromagnetic field. In the 1980s it was shown theoretically with the help of a disorder field theory, in which the vortex lines of the superconductor play a major role, that the transition is of second order within the type II regime and of first order (i.e., latent heat) within the type I regime, and that the two regions are separated by a tricritical point. The results were strongly supported by Monte Carlo computer simulations.

Meissner effect

When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. The Meissner effect does not cause the field to be completely ejected but instead the field penetrates the superconductor but only to a very small distance, characterized by a parameter λ, called the London penetration depth, decaying exponentially to zero within the bulk of the material. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100 nm.

The Meissner effect is sometimes confused with the kind of diamagnetism one would expect in a perfect electrical conductor: according to Lenz's law, when a changing magnetic field is applied to a conductor, it will induce an electric current in the conductor that creates an opposing magnetic field. In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic field exactly cancels the applied field.

The Meissner effect is distinct from this—it is the spontaneous expulsion which occurs during transition to superconductivity. Suppose we have a material in its normal state, containing a constant internal magnetic field. When the material is cooled below the critical temperature, we would observe the abrupt expulsion of the internal magnetic field, which we would not expect based on Lenz's law.

The Meissner effect was given a phenomenological explanation by the brothers Fritz and Heinz London, who showed that the electromagnetic free energy in a superconductor is minimized provided

${\displaystyle \nabla ^{2}\mathbf {H} =\lambda ^{-2}\mathbf {H} \,}$

where H is the magnetic field and λ is the London penetration depth.

This equation, which is known as the London equation, predicts that the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface.

A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H_c. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H_c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H_c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.

London moment

Conversely, a spinning superconductor generates a magnetic field, precisely aligned with the spin axis. The effect, the London moment, was put to good use in Gravity Probe B. This experiment measured the magnetic fields of four superconducting gyroscopes to determine their spin axes. This was critical to the experiment since it is one of the few ways to accurately determine the spin axis of an otherwise featureless sphere.

History of superconductivity

Heike Kamerlingh Onnes (right), the discoverer of superconductivity. Paul Ehrenfest, Hendrik Lorentz, Niels Bohr stand to his left.

Superconductivity was discovered on April 8, 1911 by Heike Kamerlingh Onnes, who was studying the resistance of solid mercury at cryogenic temperatures using the recently produced liquid helium as a refrigerant. At the temperature of 4.2 K, he observed that the resistance abruptly disappeared. In the same experiment, he also observed the superfluid transition of helium at 2.2 K, without recognizing its significance. The precise date and circumstances of the discovery were only reconstructed a century later, when Onnes's notebook was found. In subsequent decades, superconductivity was observed in several other materials. In 1913, lead was found to superconduct at 7 K, and in 1941 niobium nitride was found to superconduct at 16 K.

Great efforts have been devoted to finding out how and why superconductivity works; the important step occurred in 1933, when Meissner and Ochsenfeld discovered that superconductors expelled applied magnetic fields, a phenomenon which has come to be known as the Meissner effect. In 1935, Fritz and Heinz London showed that the Meissner effect was a consequence of the minimization of the electromagnetic free energy carried by superconducting current.

London constitutive equations

The theoretical model that was first conceived for superconductivity was completely classical: it is summarized by London constitutive equations. It was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. A major triumph of the equations of this theory is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold. By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface.

The two constitutive equations for a superconductor by London are:

${\displaystyle {\frac {\partial \mathbf {j} }{\partial t}}={\frac {ne^{2}}{m}}\mathbf {E} ,\qquad \mathbf {\nabla } \times \mathbf {j} =-{\frac {ne^{2}}{m}}\mathbf {B} .}$

The first equation follows from Newton's second law for superconducting electrons.

Conventional theories (1950s)

During the 1950s, theoretical condensed matter physicists arrived at an understanding of "conventional" superconductivity, through a pair of remarkable and important theories: the phenomenological Ginzburg–Landau theory (1950) and the microscopic BCS theory (1957).

In 1950, the phenomenological Ginzburg–Landau theory of superconductivity was devised by Landau and Ginzburg. This theory, which combined Landau's theory of second-order phase transitions with a Schrödinger-like wave equation, had great success in explaining the macroscopic properties of superconductors. In particular, Abrikosov showed that Ginzburg–Landau theory predicts the division of superconductors into the two categories now referred to as Type I and Type II. Abrikosov and Ginzburg were awarded the 2003 Nobel Prize for their work (Landau had received the 1962 Nobel Prize for other work, and died in 1968). The four-dimensional extension of the Ginzburg–Landau theory, the Coleman-Weinberg model, is important in quantum field theory and cosmology.

Also in 1950, Maxwell and Reynolds et al. found that the critical temperature of a superconductor depends on the isotopic mass of the constituent element. This important discovery pointed to the electron-phonon interaction as the microscopic mechanism responsible for superconductivity.

The complete microscopic theory of superconductivity was finally proposed in 1957 by Bardeen, Cooper and Schrieffer. This BCS theory explained the superconducting current as a superfluid of Cooper pairs, pairs of electrons interacting through the exchange of phonons. For this work, the authors were awarded the Nobel Prize in 1972.

The BCS theory was set on a firmer footing in 1958, when N. N. Bogolyubov showed that the BCS wavefunction, which had originally been derived from a variational argument, could be obtained using a canonical transformation of the electronic Hamiltonian. In 1959, Lev Gor'kov showed that the BCS theory reduced to the Ginzburg–Landau theory close to the critical temperature.

Generalizations of BCS theory for conventional superconductors form the basis for understanding of the phenomenon of superfluidity, because they fall into the lambda transition universality class. The extent to which such generalizations can be applied to unconventional superconductors is still controversial.

Further history

The first practical application of superconductivity was developed in 1954 with Dudley Allen Buck's invention of the cryotron. Two superconductors with greatly different values of critical magnetic field are combined to produce a fast, simple switch for computer elements.

Soon after discovering superconductivity in 1911, Kamerlingh Onnes attempted to make an electromagnet with superconducting windings but found that relatively low magnetic fields destroyed superconductivity in the materials he investigated. Much later, in 1955, G. B. Yntema  succeeded in constructing a small 0.7-tesla iron-core electromagnet with superconducting niobium wire windings. Then, in 1961, J. E. Kunzler, E. Buehler, F. S. L. Hsu, and J. H. Wernick  made the startling discovery that, at 4.2 kelvin niobium–tin, a compound consisting of three parts niobium and one part tin, was capable of supporting a current density of more than 100,000 amperes per square centimeter in a magnetic field of 8.8 tesla. Despite being brittle and difficult to fabricate, niobium–tin has since proved extremely useful in supermagnets generating magnetic fields as high as 20 tesla. In 1962 T. G. Berlincourt and R. R. Hake discovered that more ductile alloys of niobium and titanium are suitable for applications up to 10 tesla. Promptly thereafter, commercial production of niobium–titanium supermagnet wire commenced at Westinghouse Electric Corporation and at Wah Chang Corporation. Although niobium–titanium boasts less-impressive superconducting properties than those of niobium–tin, niobium–titanium has, nevertheless, become the most widely used "workhorse" supermagnet material, in large measure a consequence of its very high ductility and ease of fabrication. However, both niobium–tin and niobium–titanium find wide application in MRI medical imagers, bending and focusing magnets for enormous high-energy-particle accelerators, and a host of other applications. Conectus, a European superconductivity consortium, estimated that in 2014, global economic activity for which superconductivity was indispensable amounted to about five billion euros, with MRI systems accounting for about 80% of that total.

In 1962, Josephson made the important theoretical prediction that a supercurrent can flow between two pieces of superconductor separated by a thin layer of insulator. This phenomenon, now called the Josephson effect, is exploited by superconducting devices such as SQUIDs. It is used in the most accurate available measurements of the magnetic flux quantum Φ₀ = h/(2e), where h is the Planck constant. Coupled with the quantum Hall resistivity, this leads to a precise measurement of the Planck constant. Josephson was awarded the Nobel Prize for this work in 1973.

In 2008, it was proposed that the same mechanism that produces superconductivity could produce a superinsulator state in some materials, with almost infinite electrical resistance. The first development and study of superconducting Bose–Einstein condensate (BEC) in 2020 suggests that there is a "smooth transition between" BEC and Bardeen-Cooper-Shrieffer regimes.

High-temperature superconductivity

Timeline of superconducting materials. Colors represent different classes of materials:

•   BCS (dark green circle)
•   Heavy-fermions-based (light green star)
•   Cuprate (blue diamond)
•   Buckminsterfullerene-based (purple inverted triangle)
•   Carbon-allotrope (red triangle)
•   Iron-pnictogen-based (orange square)

Main article: High-temperature superconductivity

Until 1986, physicists had believed that BCS theory forbade superconductivity at temperatures above about 30 K. In that year, Bednorz and Müller discovered superconductivity in lanthanum barium copper oxide (LBCO), a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987). It was soon found that replacing the lanthanum with yttrium (i.e., making YBCO) raised the critical temperature above 90 K.

This temperature jump is particularly significant, since it allows liquid nitrogen as a refrigerant, replacing liquid helium. This can be important commercially because liquid nitrogen can be produced relatively cheaply, even on-site. Also, the higher temperatures help avoid some of the problems that arise at liquid helium temperatures, such as the formation of plugs of frozen air that can block cryogenic lines and cause unanticipated and potentially hazardous pressure buildup.

Many other cuprate superconductors have since been discovered, and the theory of superconductivity in these materials is one of the major outstanding challenges of theoretical condensed matter physics. There are currently two main hypotheses – the resonating-valence-bond theory, and spin fluctuation which has the most support in the research community. The second hypothesis proposed that electron pairing in high-temperature superconductors is mediated by short-range spin waves known as paramagnons.

In 2008, holographic superconductivity, which uses holographic duality or AdS/CFT correspondence theory, was proposed by Gubser, Hartnoll, Herzog, and Horowitz, as a possible explanation of high-temperature superconductivity in certain materials.

From about 1993, the highest-temperature superconductor known was a ceramic material consisting of mercury, barium, calcium, copper and oxygen (HgBa₂Ca₂Cu₃O_8+δ) with T_c = 133–138 K.

In February 2008, an iron-based family of high-temperature superconductors was discovered. Hideo Hosono, of the Tokyo Institute of Technology, and colleagues found lanthanum oxygen fluorine iron arsenide (LaO_1−xF_xFeAs), an oxypnictide that superconducts below 26 K. Replacing the lanthanum in LaO_1−xF_xFeAs with samarium leads to superconductors that work at 55 K.

In 2014 and 2015, hydrogen sulfide (H
₂S) at extremely high pressures (around 150 gigapascals) was first predicted and then confirmed to be a high-temperature superconductor with a transition temperature of 80 K. Additionally, in 2019 it was discovered that lanthanum hydride (LaH
₁₀) becomes a superconductor at 250 K under a pressure of 170 gigapascals.

In 2018, a research team from the Department of Physics, Massachusetts Institute of Technology, discovered superconductivity in bilayer graphene with one layer twisted at an angle of approximately 1.1 degrees with cooling and applying a small electric charge. Even if the experiments were not carried out in a high-temperature environment, the results are correlated less to classical but high temperature superconductors, given that no foreign atoms need to be introduced. The superconductivity effect came about as a result of electrons twisted into a vortex between the graphene layers, called "skyrmions". These act as a single particle and can pair up across the graphene's layers, leading to the basic conditions required for superconductivity.

In 2020, a room-temperature superconductor made from hydrogen, carbon and sulfur under pressures of around 270 gigapascals was described in a paper in Nature. This is currently the highest temperature at which any material has shown superconductivity.

Applications

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Video of superconducting levitation of YBCO

Superconducting magnets are some of the most powerful electromagnets known. They are used in MRI/NMR machines, mass spectrometers, the beam-steering magnets used in particle accelerators and plasma confining magnets in some tokamaks. They can also be used for magnetic separation, where weakly magnetic particles are extracted from a background of less or non-magnetic particles, as in the pigment industries. They can also be used in large wind turbines to overcome the restrictions imposed by high electrical currents, with an industrial grade 3.6 megawatt superconducting windmill generator having been tested successfully in Denmark.

In the 1950s and 1960s, superconductors were used to build experimental digital computers using cryotron switches. More recently, superconductors have been used to make digital circuits based on rapid single flux quantum technology and RF and microwave filters for mobile phone base stations.

Superconductors are used to build Josephson junctions which are the building blocks of SQUIDs (superconducting quantum interference devices), the most sensitive magnetometers known. SQUIDs are used in scanning SQUID microscopes and magnetoencephalography. Series of Josephson devices are used to realize the SI volt. Depending on the particular mode of operation, a superconductor–insulator–superconductor Josephson junction can be used as a photon detector or as a mixer. The large resistance change at the transition from the normal- to the superconducting state is used to build thermometers in cryogenic micro-calorimeter photon detectors. The same effect is used in ultrasensitive bolometers made from superconducting materials.

Other early markets are arising where the relative efficiency, size and weight advantages of devices based on high-temperature superconductivity outweigh the additional costs involved. For example, in wind turbines the lower weight and volume of superconducting generators could lead to savings in construction and tower costs, offsetting the higher costs for the generator and lowering the total levelized cost of electricity (LCOE).

Promising future applications include high-performance smart grid, electric power transmission, transformers, power storage devices, electric motors (e.g. for vehicle propulsion, as in vactrains or maglev trains), magnetic levitation devices, fault current limiters, enhancing spintronic devices with superconducting materials, and superconducting magnetic refrigeration. However, superconductivity is sensitive to moving magnetic fields, so applications that use alternating current (e.g. transformers) will be more difficult to develop than those that rely upon direct current. Compared to traditional power lines, superconducting transmission lines are more efficient and require only a fraction of the space, which would not only lead to a better environmental performance but could also improve public acceptance for expansion of the electric grid.

 

Quantum state

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_state
Quantum mechanics
${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }$ Schrödinger equation

In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.

Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component s_z. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional complex vector ${\displaystyle (\alpha ,\beta )}$, with a length of one; that is, with

${\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}$

where ${\displaystyle |\alpha |}$ and ${\displaystyle |\beta |}$ are the absolute values of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. A mixed state, in this case, has the structure of a ${\displaystyle 2\times 2}$ matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:

${\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\big (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\big )},}$

which involves superposition of joint spin states for two particles with spin 1⁄2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.

A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.

Conceptual description

Pure states

Probability densities for the electron of a hydrogen atom in different quantum states.

In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a Hilbert space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.

On the other hand, a system in a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as:

${\displaystyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle .}$

The coefficient which corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the time evolution operator. The symbols ${\displaystyle |}$ and ${\displaystyle \rangle }$^[a] surrounding the ${\displaystyle \Psi }$ are part of bra–ket notation.

Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states ${\displaystyle \Phi _{n}}$. A number ${\displaystyle P_{n}}$ represents the probability of a randomly selected system being in the state ${\displaystyle \Phi _{n}}$. Unlike the linear combination case each system is in a definite eigenstate.

The expectation value ${\displaystyle \langle A\rangle _{\sigma }}$ of an observable A is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.

There is no state which is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values.^[b] This is the content of the Heisenberg uncertainty relation.

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state.^[c] More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time,^[d] then they will produce the same results. This has some strange consequences, however, as follows.

Consider two incompatible observables, A and B, where A corresponds to a measurement earlier in time than B.^[e] Suppose that the system is in an eigenstate of B at the experiment's beginning. If we measure only B, all runs of the experiment will yield the same result. If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B are statistical. Thus: Quantum mechanical measurements influence one another, and the order in which they are performed is important.

Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.

Schrödinger picture vs. Heisenberg picture

One can take the observables to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables P(t), Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state ${\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }$.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.

Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture.

Formalism in quantum physics

Pure states as rays in a complex Hilbert space

Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.

Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space ${\displaystyle H}$ can be obtained from another vector by multiplying by some non-zero complex number, the two vectors are said to correspond to the same "ray" in ${\displaystyle H}$ and also to the same point in the projective Hilbert space of ${\displaystyle H}$.

Bra–ket notation

Calculations in quantum mechanics make frequent use of linear operators, scalar products, dual spaces and Hermitian conjugation. In order to make such calculations flow smoothly, and to make it unnecessary (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra–ket notation. Although the details of this are beyond the scope of this article, some consequences of this are:

• The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form ${\displaystyle |\psi \rangle }$ (where the "${\displaystyle \psi }$" can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual mathematical notation, where vectors are usually lower-case latin letters, and it is clear from the context that they are indeed vectors.
• Dirac defined two kinds of vector, bra and ket, dual to each other.^[f]
• Each ket ${\displaystyle |\psi \rangle }$ is uniquely associated with a so-called bra, denoted ${\displaystyle \langle \psi |}$, which corresponds to the same physical quantum state. Technically, the bra is the adjoint of the ket. It is an element of the dual space, and related to the ket by the Riesz representation theorem. In a finite-dimensional space with a chosen basis, writing ${\displaystyle |\psi \rangle }$ as a column vector, ${\displaystyle \langle \psi |}$ is a row vector; to obtain it just take the transpose and entry-wise complex conjugate of ${\displaystyle |\psi \rangle }$.
• Scalar products^[g][h] (also called brackets) are written so as to look like a bra and ket next to each other: ${\displaystyle \langle \psi _{1}|\psi _{2}\rangle }$. (The phrase "bra-ket" is supposed to resemble "bracket".)

Spin

The angular momentum has the same dimension (M·L²·T⁻¹) as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number S that, in units of Planck's reduced constant ħ, is either an integer (0, 1, 2 ...) or a half-integer (1/2, 3/2, 5/2 ...). For a massive particle with spin S, its spin quantum number m always assumes one of the 2S + 1 possible values in the set

${\displaystyle \{-S,-S+1,\ldots +S-1,+S\}}$

As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C^2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).

Many-body states and particle statistics

The quantum state of a system of N particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3 spatial coordinates and spin, e.g.

${\displaystyle |\psi (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\rangle .}$

Here, the spin variables m_ν assume values from the set

${\displaystyle \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,+S_{\nu }-1,\,+S_{\nu }\}}$

where ${\displaystyle S_{\nu }}$ is the spin of νth particle. ${\displaystyle S_{\nu }=0}$ for a particle that does not exhibit spin.

The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all N particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic).

Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödinger mechanics).

When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.

Basis states of one-particle systems

As with any Hilbert space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets ${\displaystyle |{k_{i}}\rangle }$, any ket ${\displaystyle |\psi \rangle }$ can be written

${\displaystyle |\psi \rangle =\sum _{i}c_{i}|{k_{i}}\rangle }$

where c_i are complex numbers. In physical terms, this is described by saying that ${\displaystyle |\psi \rangle }$ has been expressed as a quantum superposition of the states ${\displaystyle |{k_{i}}\rangle }$. If the basis kets are chosen to be orthonormal (as is often the case), then ${\displaystyle c_{i}=\langle {k_{i}}|\psi \rangle }$.

One property worth noting is that the normalized states ${\displaystyle |\psi \rangle }$ are characterized by

${\displaystyle \langle \psi |\psi \rangle =1,}$

and for orthonormal basis this translates to

${\displaystyle \sum _{i}\left|c_{i}\right|^{2}=1.}$

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the ${\displaystyle |{k_{i}}\rangle }$ are eigenstates (with eigenvalues k_i) of an observable, and that observable is measured on the normalized state ${\displaystyle |\psi \rangle }$, then the probability that the result of the measurement is k_i is |c_i|². (The normalization condition above mandates that the total sum of probabilities is equal to one.)

A particularly important example is the position basis, which is the basis consisting of eigenstates ${\displaystyle |\mathbf {r} \rangle }$ with eigenvalues ${\displaystyle \mathbf {r} }$ of the observable which corresponds to measuring position.^[i] If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket ${\displaystyle |\psi \rangle }$ is associated with a complex-valued function of three-dimensional space

${\displaystyle \psi (\mathbf {r} )\equiv \langle \mathbf {r} |\psi \rangle .}$^[k]

This function is called the wave function corresponding to ${\displaystyle |\psi \rangle }$. Similarly to the discrete case above, the probability density of the particle being found at position ${\displaystyle \mathbf {r} }$ is ${\displaystyle |\psi (\mathbf {r} )|^{2}}$ and the normalized states have

${\displaystyle \int \mathrm {d} ^{3}\mathbf {r} \,|\psi (\mathbf {r} )|^{2}=1}$.

In terms of the continuous set of position basis ${\displaystyle |\mathbf {r} \rangle }$, the state ${\displaystyle |\psi \rangle }$ is:

${\displaystyle |\psi \rangle =\int \mathrm {d} ^{3}\mathbf {r} \,\psi (\mathbf {r} )|\mathbf {r} \rangle }$.

Superposition of pure states

As mentioned above, quantum states may be superposed. If ${\displaystyle |\alpha \rangle }$ and ${\displaystyle |\beta \rangle }$ are two kets corresponding to quantum states, the ket

${\displaystyle c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }$

is a different quantum state (possibly not normalized). Note that both the amplitudes and phases (arguments) of ${\displaystyle c_{\alpha }}$ and ${\displaystyle c_{\beta }}$ will influence the resulting quantum state. In other words, for example, even though ${\displaystyle |\psi \rangle }$ and ${\displaystyle e^{i\theta }|\psi \rangle }$ (for real θ) correspond to the same physical quantum state, they are not interchangeable, since ${\displaystyle |\phi \rangle +|\psi \rangle }$ and ${\displaystyle |\phi \rangle +e^{i\theta }|\psi \rangle }$ will not correspond to the same physical state for all choices of ${\displaystyle |\phi \rangle }$. However, ${\displaystyle |\phi \rangle +|\psi \rangle }$ and ${\displaystyle e^{i\theta }(|\phi \rangle +|\psi \rangle )}$ will correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important.

One practical example of superposition is the double-slit experiment, in which superposition leads to quantum interference. The photon state is a superposition of two different states, one corresponding to the photon travel through the left slit, and the other corresponding to travel through the right slit. The relative phase of those two states depends on the difference of the distances from the two slits. Depending on that phase, the interference is constructive at some locations and destructive in others, creating the interference pattern. We may say that superposed states are in coherent superposition, by analogy with coherence in other wave phenomena.

Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.

Mixed states

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). Mixed states inevitably arise from pure states when, for a composite quantum system ${\displaystyle H_{1}\otimes H_{2}}$ with an entangled state on it, the part ${\displaystyle H_{2}}$ is inaccessible to the observer. The state of the part ${\displaystyle H_{1}}$ is expressed then as the partial trace over ${\displaystyle H_{2}}$.

A mixed state cannot be described with a single ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space ${\displaystyle H}$ can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system ${\displaystyle H\otimes K}$ for a sufficiently large Hilbert space ${\displaystyle K}$.

The density matrix describing a mixed state is defined to be an operator of the form

${\displaystyle \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}$

where ${\displaystyle p_{s}}$ is the fraction of the ensemble in each pure state ${\displaystyle |\psi _{s}\rangle .}$ The density matrix can be thought of as a way of using the one-particle formalism to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in.

A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ² is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.

The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by

${\displaystyle \langle A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}$

where ${\displaystyle |\alpha _{i}\rangle ,\;a_{i}}$ are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace. It is important to note that two types of averaging are occurring, one being a weighted quantum superposition over the basis kets ${\displaystyle |\psi _{s}\rangle }$ of the pure states, and the other being a statistical (said incoherent) average with the probabilities p_s of those states.

According to Eugene Wigner, the concept of mixture was put forward by Lev Landau.

Mathematical generalizations

States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables.