Diabetic ketoacidosis

From Wikipedia, the free encyclopedia

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

Diabetic ketoacidosis
Dehydration may be severe in diabetic ketoacidosis, and intravenous fluids are usually needed as part of its treatment.
Specialty	Endocrinology
Symptoms	Vomiting, abdominal pain, deep gasping breathing, increased urination, confusion, a specific smell
Complications	Cerebral edema
Usual onset	Relatively rapid
Causes	Shortage of insulin
Risk factors	Usually type 1 diabetes, less often other types
Diagnostic method	High blood sugar, low blood pH, ketoacids
Differential diagnosis	Hyperosmolar nonketotic state, alcoholic ketoacidosis, uremia, salicylate toxicity
Treatment	Intravenous fluids, insulin, potassium
Frequency	4–25% of people with type 1 diabetes per year

Diabetic ketoacidosis (DKA) is a potentially life-threatening complication of diabetes mellitus. Signs and symptoms may include vomiting, abdominal pain, deep gasping breathing, increased urination, weakness, confusion and occasionally loss of consciousness. A person's breath may develop a specific "fruity" smell. Onset of symptoms is usually rapid. People without a previous diagnosis of diabetes may develop DKA as the first obvious symptom.

DKA happens most often in those with type 1 diabetes but can also occur in those with other types of diabetes under certain circumstances. Triggers may include infection, not taking insulin correctly, stroke and certain medications such as steroids. DKA results from a shortage of insulin; in response, the body switches to burning fatty acids, which produces acidic ketone bodies. DKA is typically diagnosed when testing finds high blood sugar, low blood pH and ketoacids in either the blood or urine.

The primary treatment of DKA is with intravenous fluids and insulin. Depending on the severity, insulin may be given intravenously or by injection under the skin. Usually, potassium is also needed to prevent the development of low blood potassium. Throughout treatment, blood sugar and potassium levels should be regularly checked. Underlying causes for the DKA should be identified. In those with severely low blood pH who are critically ill, sodium bicarbonate may be given; however, its use is of unclear benefit and typically not recommended.

Rates of DKA vary around the world. In the United Kingdom, about 4% of people with type 1 diabetes develop DKA each year, while in Malaysia the condition affects about 25% of type 1 diabetics a year. DKA was first described in 1886, and until the introduction of insulin therapy in the 1920s, it was almost universally fatal. The risk of death with adequate and timely treatment is between <1% and 5%.

Signs and symptoms

The symptoms of an episode of diabetic ketoacidosis usually evolve over a period of about 24 hours. Predominant symptoms are nausea and vomiting, pronounced thirst, excessive urine production and abdominal pain that may be severe. In severe DKA, breathing becomes rapid and of a deep, gasping character, called "Kussmaul breathing". The abdomen may be tender to the point that a serious abdominal condition may be suspected, such as acute pancreatitis, appendicitis or gastrointestinal perforation. Vomiting altered blood that resembles coffee grounds occurs in a minority of people and tends to originate from erosion of the esophagus. In severe DKA, there may be confusion or a marked decrease in alertness, including coma.

On physical examination there is usually clinical evidence of dehydration, such as a dry mouth and decreased skin turgor. If the dehydration is profound enough to cause a decrease in the circulating blood volume, a rapid heart rate and low blood pressure may be observed. Often, a "ketotic" odor is present, which is often described as "fruity" or like "pear drops". The smell is due to the presence of acetone. If Kussmaul respiration is present, this is reflected in an increased respiratory rate.

Small children with DKA are relatively prone to brain swelling, also called cerebral edema, which may cause headache, coma, loss of the pupillary light reflex, and can progress to death. It occurs in about 1 out of 100 children with DKA and more rarely occurs in adults.

Cause

DKA most frequently occurs in those who know they have diabetes, but it may also be the first presentation in someone who had not previously been known to be diabetic. There is often a particular underlying problem that has led to the DKA episode; this may be intercurrent illness (pneumonia, influenza, gastroenteritis, a urinary tract infection), pregnancy, inadequate insulin administration (e.g. defective insulin pen device), myocardial infarction (heart attack), stroke or the use of cocaine. Young people with recurrent episodes of DKA may have an underlying eating disorder, or may be using insufficient insulin for fear that it will cause weight gain.

Diabetic ketoacidosis may occur in those previously known to have diabetes mellitus type 2 or in those who on further investigations turn out to have features of type 2 diabetes (e.g. obesity, strong family history); this is more common in African, African-American and Hispanic people. Their condition is then labeled "ketosis-prone type 2 diabetes".

Drugs in the gliflozin class (SGLT2 inhibitors), which are generally used for type 2 diabetes, have been associated with cases of diabetic ketoacidosis where the blood sugars may not be significantly elevated ("euglycemic DKA"). While this is a relatively uncommon adverse event, it is thought to be more common if someone receiving an SGLT2 inhibitor who is also receiving insulin has reduced or missed insulin doses. Furthermore it can be triggered by severe acute illness, dehydration, extensive exercise, surgery, low-carbohydrate diets, or excessive alcohol intake. SGLT2 inhibitors should be stopped before surgery and only recommenced when it is safe to do so. SGLT2 inhibitors may be used in people with type 1 diabetes, but the possibility of ketoacidosis requires specific risk management. Specifically, they should not be used if someone is also using a low carbohydrate or ketogenic diet.

Mechanism

Diabetic ketoacidosis arises because of a lack of insulin in the body. The lack of insulin and corresponding elevation of glucagon leads to increased release of glucose by the liver (a process that is normally suppressed by insulin) from glycogen via glycogenolysis and also through gluconeogenesis. High glucose levels spill over into the urine, taking water and solutes (such as sodium and potassium) along with it in a process known as osmotic diuresis. This leads to polyuria, dehydration, and polydipsia. The absence of insulin also leads to the release of free fatty acids from adipose tissue (lipolysis), which the liver converts into acetyl CoA through a process called beta oxidation. Acetyl CoA is metabolised into ketone bodies under severe states of energy deficiency, like starvation, through a process called ketogenesis, whose final products are aceto-acetate and β-Hydroxybutyrate. These ketone bodies can serve as an energy source in the absence of insulin-mediated glucose delivery, and is a protective mechanism in case of starvation. The ketone bodies, however, have a low pKa and therefore turn the blood acidic (metabolic acidosis). The body initially buffers the change with the bicarbonate buffering system, but this system is quickly overwhelmed and other mechanisms must work to compensate for the acidosis. One such mechanism is hyperventilation to lower the blood carbon dioxide levels (a form of compensatory respiratory alkalosis). This hyperventilation, in its extreme form, may be observed as Kussmaul respiration.

In various situations such as infection, insulin demands rise but are not matched by the failing pancreas. Blood sugars rise, dehydration ensues, and resistance to the normal effects of insulin increases further by way of a vicious circle.

As a result of the above mechanisms, the average adult with DKA has a total body water shortage of about 6 liters (or 100 mL/kg), in addition to substantial shortages in sodium, potassium, chloride, phosphate, magnesium and calcium. Glucose levels usually exceed 13.8 mmol/L or 250 mg/dL.

β-hydroxybutyrate (the conjugate base of β-hydroxybutyric acid, drawn above) despite chemically containing a carboxylate group instead of a ketone, is the principal "ketone body" in diabetic ketoacidosis.

DKA is common in type 1 diabetes as this form of diabetes is associated with an absolute lack of insulin production by the islets of Langerhans. In type 2 diabetes, insulin production is present but is insufficient to meet the body's requirements as a result of end-organ insulin resistance. Usually, these amounts of insulin are sufficient to suppress ketogenesis. If DKA occurs in someone with type 2 diabetes, their condition is called "ketosis-prone type 2 diabetes". The exact mechanism for this phenomenon is unclear, but there is evidence both of impaired insulin secretion and insulin action. Once the condition has been treated, insulin production resumes and often the person may be able to resume diet or tablet treatment as normally recommended in type 2 diabetes.

The clinical state of DKA is associated, in addition to the above, with the release of various counterregulatory hormones such as glucagon and adrenaline as well as cytokines, the latter of which leads to increased markers of inflammation, even in the absence of infection.

Cerebral edema, which is the most dangerous DKA complication, is probably the result of a number of factors. Some authorities suggest that it is the result from overvigorous fluid replacement, but the complication may develop before treatment has been commenced. It is more likely in those with more severe DKA, and in the first episode of DKA. Likely factors in the development of cerebral edema are dehydration, acidosis and low carbon dioxide levels; in addition, the increased level of inflammation and coagulation may, together with these factors, lead to decreased blood flow to parts of the brain, which then swells up once fluid replacement has been commenced. The swelling of brain tissue leads to raised intracranial pressure ultimately leading to death.

Diagnosis

Investigations

Diabetic ketoacidosis may be diagnosed when the combination of hyperglycemia (high blood sugars), ketones in the blood or on urinalysis and acidosis are demonstrated. In about 10% of cases the blood sugar is not significantly elevated ("euglycemic diabetic ketoacidosis").

A pH measurement is performed to detect acidosis. Blood from a vein is adequate, as there is little difference between the arterial and the venous pH; arterial samples are only required if there are concerns about oxygen levels. Ketones can be measured in the urine (acetoacetate) and blood (β-hydroxybutyrate). When compared with urine acetoacetate testing, capillary blood β-hydroxybutyrate determination can reduce the need for admission, shorten the duration of hospital admission and potentially reduce the costs of hospital care. At very high levels, capillary blood ketone measurement becomes imprecise.

In addition to the above, blood samples are usually taken to measure urea and creatinine (measures of kidney function, which may be impaired in DKA as a result of dehydration) and electrolytes. Furthermore, markers of infection (complete blood count, C-reactive protein) and acute pancreatitis (amylase and lipase) may be measured. Given the need to exclude infection, chest radiography and urinalysis are usually performed.

If cerebral edema is suspected because of confusion, recurrent vomiting or other symptoms, computed tomography may be performed to assess its severity and to exclude other causes such as stroke.

Criteria

Diabetic ketoacidosis is distinguished from other diabetic emergencies by the presence of large amounts of ketones in blood and urine, and marked metabolic acidosis. Hyperosmolar hyperglycemic state (HHS, sometimes labeled "hyperosmolar non-ketotic state" or HONK) is much more common in type 2 diabetes and features increased plasma osmolarity (above 320 mosm/kg) due to profound dehydration and concentration of the blood; mild acidosis and ketonemia may occur in this state, but not to the extent observed in DKA. There is a degree of overlap between DKA and HHS, as in DKA the osmolarity may also be increased.

Ketoacidosis is not always the result of diabetes. It may also result from alcohol excess and from starvation; in both states the glucose level is normal or low. Metabolic acidosis may occur in people with diabetes for other reasons, such as poisoning with ethylene glycol or paraldehyde.

The American Diabetes Association categorizes DKA in adults into one of three stages of severity:

• Mild: blood pH mildly decreased to between 7.25 and 7.30 (normal 7.35–7.45); serum bicarbonate decreased to 15–18 mmol/l (normal above 20); the person is alert
• Moderate: pH 7.00–7.25, bicarbonate 10–15, mild drowsiness may be present
• Severe: pH below 7.00, bicarbonate below 10, stupor or coma may occur

A 2004 statement by the European Society for Paediatric Endocrinology and the Lawson Wilkins Pediatric Endocrine Society (for children) uses slightly different cutoffs, where mild DKA is defined by pH 7.20–7.30 (bicarbonate 10–15 mmol/l), moderate DKA by pH 7.1–7.2 (bicarbonate 5–10) and severe DKA by pH<7.1 (bicarbonate below 5).

Prevention

Attacks of DKA can be prevented in those known to have diabetes to an extent by adherence to "sick day rules"; these are clear-cut instructions to person on how to treat themselves when unwell. Instructions include advice on how much extra insulin to take when sugar levels appear uncontrolled, an easily digestible diet rich in salt and carbohydrates, means to suppress fever and treat infection, and recommendations when to call for medical help.

People with diabetes can monitor their own ketone levels when unwell and seek help if they are elevated.

Management

The main aims in the treatment of diabetic ketoacidosis are replacing the lost fluids and electrolytes while suppressing the high blood sugars and ketone production with insulin. Admission to an intensive care unit (ICU) or similar high-dependency area or ward for close observation may be necessary.

Fluid replacement

The amount of fluid replaced depends on the estimated degree of dehydration. If dehydration is so severe as to cause shock (severely decreased blood pressure with insufficient blood supply to the body's organs), or a depressed level of consciousness, rapid infusion of saline (1 liter for adults, 10 ml/kg in repeated doses for children) is recommended to restore circulating volume. Slower rehydration based on calculated water and sodium shortage may be possible if the dehydration is moderate, and again saline is the recommended fluid. Very mild ketoacidosis with no associated vomiting and mild dehydration may be treated with oral rehydration and subcutaneous rather than intravenous insulin under observation for signs of deterioration.

Normal saline (0.9% saline) has generally been the fluid of choice. There have been a few small trials looking at balanced fluids with few differences.

A special but unusual consideration is cardiogenic shock, where the blood pressure is decreased not due to dehydration but due to inability of the heart to pump blood through the blood vessels. This situation requires ICU admission, monitoring of the central venous pressure (which requires the insertion of a central venous catheter in a large upper body vein), and the administration of medication that increases the heart pumping action and blood pressure.

Insulin

Some guidelines recommend a bolus (initial large dose) of insulin of 0.1 unit of insulin per kilogram of body weight. This can be administered immediately after the potassium level is known to be higher than 3.3 mmol/l; if the level is any lower, administering insulin could lead to a dangerously low potassium level (see below). Other guidelines recommend a bolus given intramuscularly if there is a delay in commencing an intravenous infusion of insulin, whereas guidelines for the management of pediatric DKA recommend delaying the initiation of insulin until fluids have been administered. It is possible to use rapid acting insulin analogs injections under the skin for mild or moderate cases.

In general, insulin is given at 0.1 unit/kg per hour to reduce the blood sugars and suppress ketone production. Guidelines differ as to which dose to use when blood sugar levels start falling; American guidelines recommend reducing the dose of insulin once glucose falls below 16.6 mmol/l (300 mg/dl) and UK guidelines at 14 mmol/l (253 mg/dl). Others recommend infusing glucose in addition to saline to allow for ongoing infusion of higher doses of insulin.

Potassium

Potassium levels can fluctuate severely during the treatment of DKA, because insulin decreases potassium levels in the blood by redistributing it into cells via increased sodium-potassium pump activity. A large part of the shifted extracellular potassium would have been lost in urine because of osmotic diuresis. Hypokalemia (low blood potassium concentration) often follows treatment. This increases the risk of dangerous irregularities in the heart rate. Therefore, continuous observation of the heart rate is recommended, as well as repeated measurement of the potassium levels and addition of potassium to the intravenous fluids once levels fall below 5.3 mmol/l. If potassium levels fall below 3.3 mmol/l, insulin administration may need to be interrupted to allow correction of the hypokalemia.

Sodium bicarbonate

The administration of sodium bicarbonate solution to rapidly improve the acid levels in the blood is controversial. There is little evidence that it improves outcomes beyond standard therapy, and indeed some evidence that while it may improve the acidity of the blood, it may actually worsen acidity inside the body's cells and increase the risk of certain complications. Its use is therefore discouraged, although some guidelines recommend it for extreme acidosis (pH<6.9), and smaller amounts for severe acidosis (pH 6.9–7.0).

Cerebral edema

Cerebral edema, if associated with coma, often necessitates admission to intensive care, artificial ventilation, and close observation. The administration of fluids is slowed. The ideal treatment of cerebral edema in DKA is not established, but intravenous mannitol and hypertonic saline (3%) are used—as in some other forms of cerebral edema—in an attempt to reduce the swelling. Cerebral edema is unusual in adults.

Resolution

Resolution of DKA is defined as general improvement in the symptoms, such as the ability to tolerate oral nutrition and fluids, normalization of blood acidity (pH>7.3), and absence of ketones in blood (<1 mmol/l) or urine. Once this has been achieved, insulin may be switched to the usual subcutaneously administered regimen, one hour after which the intravenous administration can be discontinued.

In people with suspected ketosis-prone type 2 diabetes, determination of antibodies against glutamic acid decarboxylase and islet cells may aid in the decision whether to continue insulin administration long-term (if antibodies are detected), or whether to withdraw insulin and attempt treatment with oral medication as in type 2 diabetes. Generally speaking, routine measurement of C-peptide as a measure of insulin production is not recommended unless there is genuine doubt as to whether someone has type 1 or type 2 diabetes.

Epidemiology

Diabetic ketoacidosis occurs in 4.6–8.0 per 1000 people with diabetes annually. Rates among those with type 1 diabetes are higher with about 4% in United Kingdom developing DKA a year while in Malaysia the condition affects about 25% a year. In the United States, 135,000 hospital admissions occur annually as a result of DKA, at an estimated cost of $2.4 billion or a quarter to a half the total cost of caring for people with type 1 diabetes. There has been a documented increasing trend to hospital admissions. The risk is increased in those with an ongoing risk factor, such as an eating disorder, and those who cannot afford insulin. About 30% of children with type 1 diabetes receive their diagnosis after an episode of DKA. Lower socio‐economic status and higher area‐level deprivation are associated with an increased risk of diabetic ketoacidosis in people with diabetes mellitus type 1.

Previously considered universally fatal, the risk of death with adequate and timely treatment is between <1% and 5%. Up to 1% of children with DKA develop a complication known as cerebral edema. Rates of cerebral edema in US children with DKA has risen from 0.4% in 2002 to 0.7% in 2012. Between 2 and 5 out of 10 children who develop brain swelling will die as a result.

History

The first full description of diabetic ketoacidosis is attributed to Julius Dreschfeld, a German pathologist working in Manchester, United Kingdom. In his description, which he gave in an 1886 lecture at the Royal College of Physicians in London, he drew on reports by Adolph Kussmaul as well as describing the main ketones, acetoacetate and β-hydroxybutyrate, and their chemical determination. The condition remained almost universally fatal until the discovery of insulin in the 1920s; by the 1930s, mortality had fallen to 29 percent, and by the 1950s it had become less than 10 percent. The entity of cerebral edema due to DKA was described in 1936 by a team of doctors from Philadelphia.

Numerous research studies since the 1950s have focused on the ideal treatment for diabetic ketoacidosis. A significant proportion of these studies have been conducted at the University of Tennessee Health Science Center and Emory University School of Medicine. Treatment options studied have included high- or low-dose intravenous, subcutaneous or intramuscular (e.g. the "Alberti regime") insulin, phosphate supplementation, need for a loading dose of insulin, and appropriateness of using bicarbonate therapy in moderate DKA. Various questions remain unanswered, such as whether bicarbonate administration in severe DKA makes any real difference to the clinical course, and whether an insulin loading dose is needed in adults.

The entity of ketosis-prone type 2 diabetes was first fully described in 1987 after several preceding case reports. It was initially thought to be a form of maturity onset diabetes of the young, and went through several other descriptive names (such as "idiopathic type 1 diabetes", "Flatbush diabetes", "atypical diabetes" and "type 1.5 diabetes") before the current terminology of "ketosis-prone type 2 diabetes" was adopted.

Wave function collapse

From Wikipedia, the free encyclopedia

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

 

Part of a series of articles about
Quantum mechanics
${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }$ Schrödinger equation
Introduction Glossary History

In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment.

Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation throughout this apparent collapse. More importantly, this is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate.

Historically Werner Heisenberg was the first to use the idea of wave function reduction to explain quantum measurement.

Mathematical description

Before collapsing, the wave function may be any square-integrable function, and is therefore associated with the probability density of a quantum mechanical–system. This function is expressible as a linear combination of the eigenstates of any observable. Observables represent classical dynamical variables, and when one is measured by a classical observer, the wave function is projected onto a random eigenstate of that observable. The observer simultaneously measures the classical value of that observable to be the eigenvalue of the final state.

Mathematical background

For an explanation of the notation used, see bra–ket notation. For details on this formalism, see quantum state.

The quantum state of a physical system is described by a wave function (in turn—an element of a projective Hilbert space). This can be expressed as a vector using Dirac or bra–ket notation :

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

The kets ${\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots }$ specify the different quantum "alternatives" available—a particular quantum state. They form an orthonormal eigenvector basis, formally

${\displaystyle \langle \phi _{i}|\phi _{j}\rangle =\delta _{ij}}$

where ${\displaystyle \delta _{ij}}$ represents the Kronecker delta.

An observable (i.e. measurable parameter of the system) is associated with each eigenbasis, with each quantum alternative having a specific value or eigenvalue, ${\displaystyle e_{i}}$, of the observable. A "measurable parameter of the system" could be the usual position ${\displaystyle r}$ and the momentum ${\displaystyle p}$ of (say) a particle, but also its energy ${\displaystyle E}$, ${\displaystyle z}$ components of spin (${\displaystyle s_{z}}$), orbital (${\displaystyle L_{z}}$) and total angular (${\displaystyle J_{z}}$) momenta, etc. In the basis representation these are respectively ${\displaystyle |\mathbf {r} ,t\rangle =|x,t\rangle +|y,t\rangle +|z,t\rangle ,|\mathbf {p} ,t\rangle =|p_{x},t\rangle +|p_{y},t\rangle +|p_{z},t\rangle ,|E\rangle ,|s_{z}\rangle ,|L_{z}\rangle ,|J_{z}\rangle ,\dots }$.

The coefficients ${\displaystyle c_{1},c_{2},c_{3}...}$ are the probability amplitudes corresponding to each basis ${\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots }$. These are complex numbers. The moduli square of ${\displaystyle c_{i}}$, that is ${\displaystyle |c_{i}|^{2}=c_{i}*c_{i}}$ (where ${\displaystyle *}$ denotes complex conjugate), is the probability of measuring the system to be in the state ${\displaystyle |\phi _{i}\rangle }$.

For simplicity in the following, all wave functions are assumed to be normalized; the total probability of measuring all possible states is one:

${\displaystyle \langle \psi |\psi \rangle =\sum _{i}|c_{i}|^{2}=1.}$

The process of collapse

With these definitions it is easy to describe the process of collapse. For any observable, the wave function is initially some linear combination of the eigenbasis ${\displaystyle \{|\phi _{i}\rangle \}}$ of that observable. When an external agency (an observer, experimenter) measures the observable associated with the eigenbasis ${\displaystyle \{|\phi _{i}\rangle \}}$, the wave function collapses from the full ${\displaystyle |\psi \rangle }$ to just one of the basis eigenstates, ${\displaystyle |\phi _{i}\rangle }$, that is:

${\displaystyle |\psi \rangle \rightarrow |\phi _{i}\rangle .}$

The probability of collapsing to a given eigenstate ${\displaystyle |\phi _{k}\rangle }$ is the Born probability, ${\displaystyle P_{k}=|c_{k}|^{2}}$. Immediately post-measurement, other elements of the wave function vector, ${\displaystyle c_{i\neq k}}$, have "collapsed" to zero, and ${\displaystyle |c_{i}|^{2}=1}$.

More generally, collapse is defined for an operator ${\displaystyle {\hat {Q}}}$ with eigenbasis ${\displaystyle \{|\phi _{i}\rangle \}}$. If the system is in state ${\displaystyle |\psi \rangle }$, and ${\displaystyle {\hat {Q}}}$ is measured, the probability of collapsing the system to eigenstate ${\displaystyle |\phi _{i}\rangle }$ and measuring the eigenvalue ${\displaystyle q_{i}}$ of ${\displaystyle |\phi _{i}\rangle }$ with respect to ${\displaystyle {\hat {Q}}}$ would be ${\displaystyle |\langle \psi |\phi _{i}\rangle |^{2}}$. Note that this is not the probability that the particle is in state ${\displaystyle |\phi _{i}\rangle }$; it is in state ${\displaystyle |\psi \rangle }$ until cast to an eigenstate of ${\displaystyle {\hat {Q}}}$.

However, we never observe collapse to a single eigenstate of a continuous-spectrum operator (e.g. position, momentum, or a scattering Hamiltonian), because such eigenfunctions are non-normalizable. In these cases, the wave function will partially collapse to a linear combination of "close" eigenstates (necessarily involving a spread in eigenvalues) that embodies the imprecision of the measurement apparatus. The more precise the measurement, the tighter the range. Calculation of probability proceeds identically, except with an integral over the expansion coefficient ${\displaystyle c(q,t)dq}$. This phenomenon is unrelated to the uncertainty principle, although increasingly precise measurements of one operator (e.g. position) will naturally homogenize the expansion coefficient of wave function with respect to another, incompatible operator (e.g. momentum), lowering the probability of measuring any particular value of the latter.

Quantum decoherence

Main article: Quantum decoherence

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.

History and context

The concept of wavefunction collapse was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", and incorporated into the mathematical formulation of quantum mechanics by John von Neumann, in his 1932 treatise Mathematische Grundlagen der Quantenmechanik. Heisenberg did not try to specify exactly what the collapse of the wavefunction meant. However, he emphasized that it should not be understood as a physical process. Niels Bohr also repeatedly cautioned that we must give up a "pictorial representation", and perhaps also interpreted collapse as a formal, not physical, process.

Consistent with Heisenberg, von Neumann postulated that there were two processes of wave function change:

1. The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement, as outlined above.
2. The deterministic, unitary, continuous time evolution of an isolated system that obeys the Schrödinger equation (or a relativistic equivalent, i.e. the Dirac equation).

In general, quantum systems exist in superpositions of those basis states that most closely correspond to classical descriptions, and, in the absence of measurement, evolve according to the Schrödinger equation. However, when a measurement is made, the wave function collapses—from an observer's perspective—to just one of the basis states, and the property being measured uniquely acquires the eigenvalue of that particular state, ${\displaystyle \lambda _{i}}$. After the collapse, the system again evolves according to the Schrödinger equation.

By explicitly dealing with the interaction of object and measuring instrument, von Neumann has attempted to create consistency of the two processes of wave function change.

He was able to prove the possibility of a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the necessity of such a collapse. Although von Neumann's projection postulate is often presented as a normative description of quantum measurement, it was conceived by taking into account experimental evidence available during the 1930s (in particular the Compton-Simon experiment was paradigmatic), but many important present-day measurement procedures do not satisfy it (so-called measurements of the second kind).

The existence of the wave function collapse is required in

• the Copenhagen interpretation
• the objective collapse interpretations
• the transactional interpretation
• the von Neumann interpretation in which consciousness causes collapse.

On the other hand, the collapse is considered a redundant or optional approximation in

• the consistent histories approach, self-dubbed "Copenhagen done right"
• the Bohm interpretation
• the many-worlds interpretation
• the ensemble interpretation

The cluster of phenomena described by the expression wave function collapse is a fundamental problem in the interpretation of quantum mechanics, and is known as the measurement problem.

In the Copenhagen Interpretation collapse is postulated to be a special characteristic of interaction with classical systems (of which measurements are a special case). Mathematically it can be shown that collapse is equivalent to interaction with a classical system modeled within quantum theory as systems with Boolean algebras of observables and equivalent to a conditional expectation value.

Everett's many-worlds interpretation deals with it by discarding the collapse-process, thus reformulating the relation between measurement apparatus and system in such a way that the linear laws of quantum mechanics are universally valid; that is, the only process according to which a quantum system evolves is governed by the Schrödinger equation or some relativistic equivalent.

A general description of the evolution of quantum mechanical systems is possible by using density operators and quantum operations. In this formalism (which is closely related to the C*-algebraic formalism) the collapse of the wave function corresponds to a non-unitary quantum operation. Within the C* formalism this non-unitary process is equivalent to the algebra gaining a non-trivial centre or centre of its centralizer corresponding to classical observables.

The significance ascribed to the wave function varies from interpretation to interpretation, and varies even within an interpretation (such as the Copenhagen Interpretation). If the wave function merely encodes an observer's knowledge of the universe then the wave function collapse corresponds to the receipt of new information. This is somewhat analogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey a wave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of the wave function is also seen as a real process, to the same extent.