Interpretations of quantum mechanics

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments. However, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.

While some variation of the Copenhagen interpretation is commonly presented in textbooks, many other interpretations have been developed. Despite nearly a century of debate and experiment, no consensus has been reached among physicists and philosophers of physics concerning which interpretation best "represents" reality.

History

Erwin Schrödinger

 

Max Born

 

Niels Bohr

The definition of quantum theorists' terms, such as wave function and matrix mechanics, progressed through many stages. For instance, Erwin Schrödinger originally viewed the electron's wave function as its charge density smeared across space, but Max Born reinterpreted the absolute square value of the wave function as the electron's probability density distributed across space; the Born rule, as it is now called, matched experiment, whereas Schrödinger's charge density view did not.

The views of several early pioneers of quantum mechanics, such as Niels Bohr and Werner Heisenberg, are often grouped together as the "Copenhagen interpretation", though physicists and historians of physics have argued that this terminology obscures differences between the views so designated. Copenhagen-type ideas were never universally embraced, and challenges to a perceived Copenhagen orthodoxy gained increasing attention in the 1950s with the pilot-wave interpretation of David Bohm and the many-worlds interpretation of Hugh Everett III.

The physicist N. David Mermin once quipped, "New interpretations appear every year. None ever disappear." As a rough guide to development of the mainstream view during the 1990s and 2000s, a "snapshot" of opinions was collected in a poll by Schlosshauer et al. at the "Quantum Physics and the Nature of Reality" conference of July 2011. The authors reference a similarly informal poll carried out by Max Tegmark at the "Fundamental Problems in Quantum Theory" conference in August 1997. The main conclusion of the authors is that "the Copenhagen interpretation still reigns supreme", receiving the most votes in their poll (42%), besides the rise to mainstream notability of the many-worlds interpretations: "The Copenhagen interpretation still reigns supreme here, especially if we lump it together with intellectual offsprings such as information-based interpretations and the quantum Bayesian interpretation. In Tegmark's poll, the Everett interpretation received 17% of the vote, which is similar to the number of votes (18%) in our poll."

Some concepts originating from studies of interpretations have found more practical application in quantum information science.

Nature

More or less, all interpretations of quantum mechanics share two qualities:

1. They interpret a formalism—a set of equations and principles to generate predictions via input of initial conditions
2. They interpret a phenomenology—a set of observations, including those obtained by empirical research and those obtained informally, such as humans' experience of an unequivocal world

Two qualities vary among interpretations:

1. Epistemology—claims about the possibility, scope, and means toward relevant knowledge of the world
2. Ontology—claims about what things, such as categories and entities, exist in the world

In the philosophy of science, the distinction between knowledge and reality is termed epistemic versus ontic. A general law can be seen as a generalisation of the regularity of outcomes (epistemic), whereas a causal mechanism may be thought of as determining or regulating outcomes (ontic). A phenomenon can be interpreted either as ontic or as epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception (epistemic), or may be explained as intrinsic physical randomness (ontic). Confusing the epistemic with the ontic—if for example one were to presume that a general law actually "governs" outcomes, and that the statement of a regularity has the role of a causal mechanism—is a category mistake.

In a broad sense, scientific theory can be viewed as offering an approximately true description or explanation of the natural world (scientific realism) or as providing nothing more than an account of our knowledge of the natural world (antirealism). A realist stance sees the epistemic as giving us a window onto the ontic, whereas an antirealist stance sees the epistemic as providing only a logically consistent picture of the ontic. In the first half of the 20th Century, a key antirealist philosophy was logical positivism, which sought to exclude unobservable aspects of reality from scientific theory.

Since the 1950s antirealism has adopted a more modest approach, often in the form of instrumentalism, permitting talk of unobservables but ultimately discarding the very question of realism and positing scientific theory as a tool to help us make predictions, not to attain a deep metaphysical understanding of the world. The instrumentalist view is typified by David Mermin's famous slogan: "Shut up and calculate" (which is often misattributed to Richard Feynman).

Interpretive challenges

1. Abstract, mathematical nature of quantum field theories: the mathematical structure of quantum mechanics is abstract and does not result in a single, clear interpretation of its quantities.
2. Apparent indeterministic and irreversible processes: in classical field theory, a physical property at a given location in the field is readily derived. In most mathematical formulations of quantum mechanics, measurement (understood as an interaction with a given state) has a special role in the theory, as it is the sole process that can cause a nonunitary, irreversible evolution of the state.
3. Role of the observer in determining outcomes. Copenhagen-type interpretations imply that the wavefunction is a calculational tool, and represents reality only immediately after a measurement performed by an observer. Everettian interpretations grant that all possible outcomes are real, and that measurement-type interactions cause a branching process in which each possibility is realised.
4. Classically unexpected correlations between remote objects: entangled quantum systems, as illustrated in the EPR paradox, obey statistics that seem to violate principles of local causality by action at a distance.
5. Complementarity of proffered descriptions: complementarity holds that no set of classical physical concepts can simultaneously refer to all properties of a quantum system. For instance, wave description A and particulate description B can each describe quantum system S, but not simultaneously. This implies the composition of physical properties of S does not obey the rules of classical propositional logic when using propositional connectives (see "Quantum logic"). Like contextuality, the "origin of complementarity lies in the non-commutativity of operators" that describe quantum objects.
6. Rapidly rising intricacy, far exceeding humans' present calculational capacity, as a system's size increases: since the state space of a quantum system is exponential in the number of subsystems, it is difficult to derive classical approximations.
7. Contextual behaviour of systems locally: Quantum contextuality demonstrates that classical intuitions, in which properties of a system hold definite values independent of the manner of their measurement, fail even for local systems. Also, physical principles such as Leibniz's Principle of the identity of indiscernibles no longer apply in the quantum domain, signaling that most classical intuitions may be incorrect about the quantum world.

Influential interpretations

Copenhagen interpretation

Main article: Copenhagen interpretation

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest attitudes towards quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught. There is no definitive historical statement of what is the Copenhagen interpretation, and there were in particular fundamental disagreements between the views of Bohr and Heisenberg. For example, Heisenberg emphasized a sharp "cut" between the observer (or the instrument) and the system being observed, while Bohr offered an interpretation that is independent of a subjective observer or measurement or collapse, which relies on an "irreversible" or effectively irreversible process that imparts the classical behavior of "observation" or "measurement".

Features common to Copenhagen-type interpretations include the idea that quantum mechanics is intrinsically indeterministic, with probabilities calculated using the Born rule, and the principle of complementarity, which states certain pairs of complementary properties cannot all be observed or measured simultaneously. Moreover, properties only result from the act of "observing" or "measuring"; the theory avoids assuming definite values from unperformed experiments. Copenhagen-type interpretations hold that quantum descriptions are objective, in that they are independent of physicists' mental arbitrariness. The statistical interpretation of wavefunctions due to Max Born differs sharply from Schrödinger's original intent, which was to have a theory with continuous time evolution and in which wavefunctions directly described physical reality.

Many worlds

Main article: Many-worlds interpretation

The many-worlds interpretation is an interpretation of quantum mechanics in which a universal wavefunction obeys the same deterministic, reversible laws at all times; in particular there is no (indeterministic and irreversible) wavefunction collapse associated with measurement. The phenomena associated with measurement are claimed to be explained by decoherence, which occurs when states interact with the environment. More precisely, the parts of the wavefunction describing observers become increasingly entangled with the parts of the wavefunction describing their experiments. Although all possible outcomes of experiments continue to lie in the wavefunction's support, the times at which they become correlated with observers effectively "split" the universe into mutually unobservable alternate histories.

Quantum information theories

Quantum informational approaches have attracted growing support. They subdivide into two kinds.

• Information ontologies, such as J. A. Wheeler's "it from bit". These approaches have been described as a revival of immaterialism.
• Interpretations where quantum mechanics is said to describe an observer's knowledge of the world, rather than the world itself. This approach has some similarity with Bohr's thinking. Collapse (also known as reduction) is often interpreted as an observer acquiring information from a measurement, rather than as an objective event. These approaches have been appraised as similar to instrumentalism. James Hartle writes,

The state is not an objective property of an individual system but is that information, obtained from a knowledge of how a system was prepared, which can be used for making predictions about future measurements. ... A quantum mechanical state being a summary of the observer's information about an individual physical system changes both by dynamical laws, and whenever the observer acquires new information about the system through the process of measurement. The existence of two laws for the evolution of the state vector ... becomes problematical only if it is believed that the state vector is an objective property of the system ... The "reduction of the wavepacket" does take place in the consciousness of the observer, not because of any unique physical process which takes place there, but only because the state is a construct of the observer and not an objective property of the physical system.

Relational quantum mechanics

Main article: Relational quantum mechanics

The essential idea behind relational quantum mechanics, following the precedent of special relativity, is that different observers may give different accounts of the same series of events: for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may be in a superposition of two or more states. Consequently, if quantum mechanics is to be a complete theory, relational quantum mechanics argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the observed system. However, it is held by relational quantum mechanics that this applies to all physical objects, whether or not they are conscious or macroscopic. Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. Thus the physical content of the theory has to do not with objects themselves, but the relations between them.

QBism

Main article: Quantum Bayesianism

QBism, which originally stood for "quantum Bayesianism", is an interpretation of quantum mechanics that takes an agent's actions and experiences as the central concerns of the theory. This interpretation is distinguished by its use of a subjective Bayesian account of probabilities to understand the quantum mechanical Born rule as a normative addition to good decision-making. QBism draws from the fields of quantum information and Bayesian probability and aims to eliminate the interpretational conundrums that have beset quantum theory.

QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, quantum measurement, and entanglement. According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the degrees of belief an agent has about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism. The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it.

Consistent histories

Main article: Consistent histories

The consistent histories interpretation generalizes the conventional Copenhagen interpretation and attempts to provide a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that allows the history of a system to be described so that the probabilities for each history obey the additive rules of classical probability. It is claimed to be consistent with the Schrödinger equation.

According to this interpretation, the purpose of a quantum-mechanical theory is to predict the relative probabilities of various alternative histories (for example, of a particle).

Ensemble interpretation

Main article: Ensemble interpretation

The ensemble interpretation, also called the statistical interpretation, can be viewed as a minimalist interpretation. That is, it claims to make the fewest assumptions associated with the standard mathematics. It takes the statistical interpretation of Born to the fullest extent. The interpretation states that the wave function does not apply to an individual system – for example, a single particle – but is an abstract statistical quantity that only applies to an ensemble (a vast multitude) of similarly prepared systems or particles. In the words of Einstein:

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

— Einstein in Albert Einstein: Philosopher-Scientist, ed. P.A. Schilpp (Harper & Row, New York)

The most prominent current advocate of the ensemble interpretation is Leslie E. Ballentine, professor at Simon Fraser University, author of the text book Quantum Mechanics, A Modern Development.

De Broglie–Bohm theory

Main article: De Broglie–Bohm theory

The de Broglie–Bohm theory of quantum mechanics (also known as the pilot wave theory) is a theory by Louis de Broglie and extended later by David Bohm to include measurements. Particles, which always have positions, are guided by the wavefunction. The wavefunction evolves according to the Schrödinger wave equation, and the wavefunction never collapses. The theory takes place in a single spacetime, is non-local, and is deterministic. The simultaneous determination of a particle's position and velocity is subject to the usual uncertainty principle constraint. The theory is considered to be a hidden-variable theory, and by embracing non-locality it satisfies Bell's inequality. The measurement problem is resolved, since the particles have definite positions at all times. Collapse is explained as phenomenological.

Transactional interpretation

Main article: Transactional interpretation

The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer is an interpretation of quantum mechanics inspired by the Wheeler–Feynman absorber theory.^[42] It describes the collapse of the wave function as resulting from a time-symmetric transaction between a possibility wave from the source to the receiver (the wave function) and a possibility wave from the receiver to source (the complex conjugate of the wave function). This interpretation of quantum mechanics is unique in that it not only views the wave function as a real entity, but the complex conjugate of the wave function, which appears in the Born rule for calculating the expected value for an observable, as also real.

Von Neumann–Wigner interpretation

Main article: Von Neumann–Wigner interpretation

In his treatise The Mathematical Foundations of Quantum Mechanics, John von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the Schrödinger equation (the universal wave function). He also described how measurement could cause a collapse of the wave function. This point of view was prominently expanded on by Eugene Wigner, who argued that human experimenter consciousness (or maybe even dog consciousness) was critical for the collapse, but he later abandoned this interpretation.

However, consciousness remains a mystery. The origin and place in nature of consciousness are not well understood. Some specific proposals for consciousness caused wave-function collapse have been shown to be unfalsifiable.

Quantum logic

Main article: Quantum logic

Quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann, who attempted to reconcile some of the apparent inconsistencies of classical Boolean logic with the facts related to measurement and observation in quantum mechanics.

Modal interpretations of quantum theory

Modal interpretations of quantum mechanics were first conceived of in 1972 by Bas van Fraassen, in his paper "A formal approach to the philosophy of science". Van Fraassen introduced a distinction between a dynamical state, which describes what might be true about a system and which always evolves according to the Schrödinger equation, and a value state, which indicates what is actually true about a system at a given time. The term "modal interpretation" now is used to describe a larger set of models that grew out of this approach. The Stanford Encyclopedia of Philosophy describes several versions, including proposals by Kochen, Dieks, Clifton, Dickson, and Bub. According to Michel Bitbol, Schrödinger's views on how to interpret quantum mechanics progressed through as many as four stages, ending with a non-collapse view that in respects resembles the interpretations of Everett and van Fraassen. Because Schrödinger subscribed to a kind of post-Machian neutral monism, in which "matter" and "mind" are only different aspects or arrangements of the same common elements, treating the wavefunction as ontic and treating it as epistemic became interchangeable.

Time-symmetric theories

Time-symmetric interpretations of quantum mechanics were first suggested by Walter Schottky in 1921. Several theories have been proposed that modify the equations of quantum mechanics to be symmetric with respect to time reversal. (See Wheeler–Feynman time-symmetric theory.) This creates retrocausality: events in the future can affect ones in the past, exactly as events in the past can affect ones in the future. In these theories, a single measurement cannot fully determine the state of a system (making them a type of hidden-variables theory), but given two measurements performed at different times, it is possible to calculate the exact state of the system at all intermediate times. The collapse of the wavefunction is therefore not a physical change to the system, just a change in our knowledge of it due to the second measurement. Similarly, they explain entanglement as not being a true physical state but just an illusion created by ignoring retrocausality. The point where two particles appear to "become entangled" is simply a point where each particle is being influenced by events that occur to the other particle in the future.

Not all advocates of time-symmetric causality favour modifying the unitary dynamics of standard quantum mechanics. Thus a leading exponent of the two-state vector formalism, Lev Vaidman, states that the two-state vector formalism dovetails well with Hugh Everett's many-worlds interpretation.

Other interpretations

Main article: Minority interpretations of quantum mechanics

As well as the mainstream interpretations discussed above, a number of other interpretations have been proposed that have not made a significant scientific impact for whatever reason. These range from proposals by mainstream physicists to the more occult ideas of quantum mysticism.

Related concepts

Some ideas are discussed in the context of interpreting quantum mechanics but are not necessarily regarded as interpretations themselves.

Quantum Darwinism

Main article: Quantum Darwinism

Quantum Darwinism is a theory meant to explain the emergence of the classical world from the quantum world as due to a process of Darwinian natural selection induced by the environment interacting with the quantum system; where the many possible quantum states are selected against in favor of a stable pointer state. It was proposed in 2003 by Wojciech Zurek and a group of collaborators including Ollivier, Poulin, Paz and Blume-Kohout. The development of the theory is due to the integration of a number of Zurek's research topics pursued over the course of twenty-five years including pointer states, einselection and decoherence.

Objective-collapse theories

Main article: Objective-collapse theory

Objective-collapse theories differ from the Copenhagen interpretation by regarding both the wave function and the process of collapse as ontologically objective (meaning these exist and occur independent of the observer). In objective theories, collapse occurs either randomly ("spontaneous localization") or when some physical threshold is reached, with observers having no special role. Thus, objective-collapse theories are realistic, indeterministic, no-hidden-variables theories. Standard quantum mechanics does not specify any mechanism of collapse; quantum mechanics would need to be extended if objective collapse is correct. The requirement for an extension means that objective-collapse theories are alternatives to quantum mechanics rather than interpretations of it. Examples include

• the Ghirardi–Rimini–Weber theory
• the continuous spontaneous localization model
• the Penrose interpretation

Comparisons

The most common interpretations are summarized in the table below. The values shown in the cells of the table are not without controversy, for the precise meanings of some of the concepts involved are unclear and, in fact, are themselves at the center of the controversy surrounding the given interpretation. For another table comparing interpretations of quantum theory, see reference.

No experimental evidence exists that distinguishes among these interpretations. To that extent, the physical theory stands, and is consistent with itself and with reality. Nevertheless, designing experiments that would test the various interpretations is the subject of active research.

Most of these interpretations have variants. For example, it is difficult to get a precise definition of the Copenhagen interpretation as it was developed and argued by many people.

Schrödinger equation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations.

The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. Other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".

The equation given by Schrödinger is nonrelativistic because it contains a first derivative in time and a second derivative in space, and therefore space and time are not on equal footing. Paul Dirac incorporated special relativity and quantum mechanics into a single formulation that simplifies to the Schrödinger equation in the non-relativistic limit. This is the Dirac equation, which contains a single derivative in both space and time. Another partial differential equation, the Klein–Gordon equation, led to a problem with probability density even though it was a relativistic wave equation. The probability density could be negative, which is physically unviable. This was fixed by Dirac by taking the so-called square root of the Klein–Gordon operator and in turn introducing Dirac matrices. In a modern context, the Klein–Gordon equation describes spin-less particles, while the Dirac equation describes spin-1/2 particles.

Definition

Preliminaries

Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only the concepts and notations of basic calculus, particularly derivatives with respect to space and time. A special case of the Schrödinger equation that admits a statement in those terms is the position-space Schrödinger equation for a single nonrelativistic particle in one dimension: $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(x,t)=\left[-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+V(x,t)\right]\mathrm{\Psi }(x,t).}$$ Here, ${\displaystyle \mathrm{\Psi }(x,t)}$ is a wave function, a function that assigns a complex number to each point ${\displaystyle x}$ at each time ${\displaystyle t}$. The parameter ${\displaystyle m}$ is the mass of the particle, and ${\displaystyle V(x,t)}$ is the potential that represents the environment in which the particle exists. The constant ${\displaystyle i}$ is the imaginary unit, and ${\displaystyle \hslash }$ is the reduced Planck constant, which has units of action (energy multiplied by time).

Complex plot of a wave function that satisfies the nonrelativistic free Schrödinger equation with V = 0. For more details see wave packet

Broadening beyond this simple case, the mathematical formulation of quantum mechanics developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl defines the state of a quantum mechanical system to be a vector ${\displaystyle |\psi ⟩}$ belonging to a separable complex Hilbert space ${\displaystyle \mathcal{H}}$. This vector is postulated to be normalized under the Hilbert space's inner product, that is, in Dirac notation it obeys ${\displaystyle ⟨\psi |\psi ⟩=1}$. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of square-integrable functions ${\displaystyle L^2}$, while the Hilbert space for the spin of a single proton is the two-dimensional complex vector space ${\displaystyle {\mathbb{C}}^2}$ with the usual inner product.

Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space. A wave function can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue ${\displaystyle \lambda }$ is non-degenerate and the probability is given by ${\displaystyle |⟨\lambda |\psi ⟩{|}^2}$, where ${\displaystyle |\lambda ⟩}$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ${\displaystyle ⟨\psi |P_{\lambda }|\psi ⟩}$, where ${\displaystyle P_{\lambda }}$ is the projector onto its associated eigenspace.

A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes regard these eigenstates, composed of elements outside the Hilbert space, as "generalized eigenvectors". These are used for calculational convenience and do not represent physical states. Thus, a position-space wave function ${\displaystyle \mathrm{\Psi }(x,t)}$ as used above can be written as the inner product of a time-dependent state vector ${\displaystyle |\mathrm{\Psi }(t)⟩}$ with unphysical but convenient "position eigenstates" ${\displaystyle |x⟩}$: $${\displaystyle \mathrm{\Psi }(x,t)=⟨x|\mathrm{\Psi }(t)⟩.}$$

Time-dependent equation

Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state.

The form of the Schrödinger equation depends on the physical situation. The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:

Time-dependent Schrödinger equation (general)

${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }(t)⟩=\hat{H}|\mathrm{\Psi }(t)⟩}$

where ${\displaystyle t}$ is time, ${\displaystyle |\mathrm{\Psi }(t)⟩}$ is the state vector of the quantum system (${\displaystyle \mathrm{\Psi }}$ being the Greek letter psi), and ${\displaystyle \hat{H}}$ is an observable, the Hamiltonian operator.

The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is an approximation that yields accurate results in many situations, but only to a certain extent (see relativistic quantum mechanics and relativistic quantum field theory).

To apply the Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system, then insert it into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system. In practice, the square of the absolute value of the wave function at each point is taken to define a probability density function. For example, given a wave function in position space ${\displaystyle \mathrm{\Psi }(x,t)}$ as above, we have $${\displaystyle Pr(x,t)=|\mathrm{\Psi }(x,t){|}^2.}$$

Time-independent equation

The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states. These states are particularly important as their individual study later simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation.

Time-independent Schrödinger equation (general)

${\displaystyle \hat{\mathrm{H}}|\mathrm{\Psi }⟩=E|\mathrm{\Psi }⟩}$

where ${\displaystyle E}$ is the energy of the system. This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function is dependent on time as explained in the section on linearity below. In the language of linear algebra, this equation is an eigenvalue equation. Therefore, the wave function is an eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) ${\displaystyle E}$.

Properties

Linearity

The Schrödinger equation is a linear differential equation, meaning that if two state vectors ${\displaystyle |{\psi }_1⟩}$ and ${\displaystyle |{\psi }_2⟩}$ are solutions, then so is any linear combination $${\displaystyle |\psi ⟩=a|{\psi }_1⟩+b|{\psi }_2⟩}$$ of the two state vectors where a and b are any complex numbers. Moreover, the sum can be extended for any number of state vectors. This property allows superpositions of quantum states to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over a basis of states. A choice often employed is the basis of energy eigenstates, which are solutions of the time-independent Schrödinger equation. In this basis, a time-dependent state vector ${\displaystyle |\mathrm{\Psi }(t)⟩}$ can be written as the linear combination $${\displaystyle |\mathrm{\Psi }(t)⟩=\sum _n{A}_n{e}^{-i{E}_{n}t/\hslash }|{\psi }_{E_n}⟩,}$$ where ${\displaystyle A_n}$ are complex numbers and the vectors ${\displaystyle |{\psi }_{E_n}⟩}$ are solutions of the time-independent equation ${\displaystyle \hat{H}|{\psi }_{E_n}⟩=E_n|{\psi }_{E_n}⟩}$.

Unitarity

Further information: Wigner's theorem and Stone's theorem

Holding the Hamiltonian ${\displaystyle \hat{H}}$ constant, the Schrödinger equation has the solution $${\displaystyle |\mathrm{\Psi }(t)⟩=e^{-i\hat{H}t/\hslash }|\mathrm{\Psi }(0)⟩.}$$ The operator ${\displaystyle \hat{U}(t)=e^{-i\hat{H}t/\hslash }}$ is known as the time-evolution operator, and it is unitary: it preserves the inner product between vectors in the Hilbert space. Unitarity is a general feature of time evolution under the Schrödinger equation. If the initial state is ${\displaystyle |\mathrm{\Psi }(0)⟩}$, then the state at a later time ${\displaystyle t}$ will be given by $${\displaystyle |\mathrm{\Psi }(t)⟩=\hat{U}(t)|\mathrm{\Psi }(0)⟩}$$ for some unitary operator ${\displaystyle \hat{U}(t)}$. Conversely, suppose that ${\displaystyle \hat{U}(t)}$ is a continuous family of unitary operators parameterized by ${\displaystyle t}$. Without loss of generality, the parameterization can be chosen so that ${\displaystyle \hat{U}(0)}$ is the identity operator and that ${\displaystyle \hat{U}(t/N{)}^N=\hat{U}(t)}$ for any ${\displaystyle N>0}$. Then ${\displaystyle \hat{U}(t)}$ depends upon the parameter ${\displaystyle t}$ in such a way that $${\displaystyle \hat{U}(t)=e^{-i\hat{G}t}}$$ for some self-adjoint operator ${\displaystyle \hat{G}}$, called the generator of the family ${\displaystyle \hat{U}(t)}$. A Hamiltonian is just such a generator (up to the factor of the Planck constant that would be set to 1 in natural units). To see that the generator is Hermitian, note that with ${\displaystyle \hat{U}(\delta t)\approx \hat{U}(0)-i\hat{G}\delta t}$, we have $${\displaystyle \hat{U}(\delta t{)}^{†}\hat{U}(\delta t)\approx (\hat{U}(0{)}^{†}+i{\hat{G}}^{†}\delta t)(\hat{U}(0)-i\hat{G}\delta t)=I+i\delta t({\hat{G}}^{†}-\hat{G})+O(\delta t^2),}$$ so ${\displaystyle \hat{U}(t)}$ is unitary only if, to first order, its derivative is Hermitian.

Changes of basis

The Schrödinger equation is often presented using quantities varying as functions of position, but as a vector-operator equation it has a valid representation in any arbitrary complete basis of kets in Hilbert space. As mentioned above, "bases" that lie outside the physical Hilbert space are also employed for calculational purposes. This is illustrated by the position-space and momentum-space Schrödinger equations for a nonrelativistic, spinless particle. The Hilbert space for such a particle is the space of complex square-integrable functions on three-dimensional Euclidean space, and its Hamiltonian is the sum of a kinetic-energy term that is quadratic in the momentum operator and a potential-energy term: $${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }(t)⟩=\left(\frac{1}{2m}{\hat{p}}^2+\hat{V}\right)|\mathrm{\Psi }(t)⟩.}$$ Writing ${\displaystyle \mathbf{r}}$ for a three-dimensional position vector and ${\displaystyle \mathbf{p}}$ for a three-dimensional momentum vector, the position-space Schrödinger equation is $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(\mathbf{r},t)=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }(\mathbf{r},t)+V(\mathbf{r})\mathrm{\Psi }(\mathbf{r},t).}$$ The momentum-space counterpart involves the Fourier transforms of the wave function and the potential: $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\tilde{\mathrm{\Psi }}(\mathbf{p},t)=\frac{{\mathbf{p}}^2}{2m}\tilde{\mathrm{\Psi }}(\mathbf{p},t)+(2\pi \hslash {)}^{-3/2}\int d^3{\mathbf{p}}^{\prime }\tilde{V}(\mathbf{p}-{\mathbf{p}}^{\prime })\tilde{\mathrm{\Psi }}({\mathbf{p}}^{\prime },t).}$$ The functions ${\displaystyle \mathrm{\Psi }(\mathbf{r},t)}$ and ${\displaystyle \tilde{\mathrm{\Psi }}(\mathbf{p},t)}$ are derived from ${\displaystyle |\mathrm{\Psi }(t)⟩}$ by $${\displaystyle \mathrm{\Psi }(\mathbf{r},t)=⟨\mathbf{r}|\mathrm{\Psi }(t)⟩,}$$ $${\displaystyle \tilde{\mathrm{\Psi }}(\mathbf{p},t)=⟨\mathbf{p}|\mathrm{\Psi }(t)⟩,}$$ where ${\displaystyle |\mathbf{r}⟩}$ and ${\displaystyle |\mathbf{p}⟩}$ do not belong to the Hilbert space itself, but have well-defined inner products with all elements of that space.

When restricted from three dimensions to one, the position-space equation is just the first form of the Schrödinger equation given above. The relation between position and momentum in quantum mechanics can be appreciated in a single dimension. In canonical quantization, the classical variables ${\displaystyle x}$ and ${\displaystyle p}$ are promoted to self-adjoint operators ${\displaystyle \hat{x}}$ and ${\displaystyle \hat{p}}$ that satisfy the canonical commutation relation $${\displaystyle [\hat{x},\hat{p}]=i\hslash .}$$ This implies that $${\displaystyle ⟨x|\hat{p}|\mathrm{\Psi }⟩=-i\hslash \frac{d}{dx}\mathrm{\Psi }(x),}$$ so the action of the momentum operator ${\displaystyle \hat{p}}$ in the position-space representation is $-i\hslash \frac{d}{dx}$. Thus, ${\displaystyle {\hat{p}}^2}$ becomes a second derivative, and in three dimensions, the second derivative becomes the Laplacian ${\displaystyle {\mathrm{\nabla }}^2}$.

The canonical commutation relation also implies that the position and momentum operators are Fourier conjugates of each other. Consequently, functions originally defined in terms of their position dependence can be converted to functions of momentum using the Fourier transform. In solid-state physics, the Schrödinger equation is often written for functions of momentum, as Bloch's theorem ensures the periodic crystal lattice potential couples ${\displaystyle \tilde{\mathrm{\Psi }}(p)}$ with ${\displaystyle \tilde{\mathrm{\Psi }}(p+\hslash K)}$ for only discrete reciprocal lattice vectors ${\displaystyle K}$. This makes it convenient to solve the momentum-space Schrödinger equation at each point in the Brillouin zone independently of the other points in the Brillouin zone.

Probability current

Main articles: Probability current and Continuity equation

The Schrödinger equation is consistent with local probability conservation. It also ensures that a normalized wavefunction remains normalized after time evolution. In matrix mechanics, this means that the time evolution operator is a unitary operator. In contrast to, for example, the Klein Gordon equation, although a redefined inner product of a wavefunction can be time independent, the total volume integral of modulus square of the wavefunction need not be time independent.

The continuity equation for probability in non relativistic quantum mechanics is stated as: $${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\rho \left(\mathbf{r},t\right)+\mathrm{\nabla }\cdot \mathbf{j}=0,}$$where $${\displaystyle \mathbf{j}=\frac{1}{2m}\left({\mathrm{\Psi }}^{\ast }\hat{\mathbf{p}}\mathrm{\Psi }-\mathrm{\Psi }\hat{\mathbf{p}}{\mathrm{\Psi }}^{\ast }\right)=-\frac{i\hslash }{2m}({\psi }^{\ast }\mathrm{\nabla }\psi -\psi \mathrm{\nabla }{\psi }^{\ast })=\frac{\hslash }{m}\mathrm{Im}({\psi }^{\ast }\mathrm{\nabla }\psi )}$$ is the probability current or probability flux (flow per unit area).

If the wavefunction is represented as $\psi (\mathbf{x},t)=\sqrt{\rho (\mathbf{x},t)}\exp \left(\frac{iS(\mathbf{x},t)}{\hslash }\right),$ where ${\displaystyle S(\mathbf{x},t)}$ is a real function which represents the complex phase of the wavefunction, then the probability flux is calculated as:$${\displaystyle \mathbf{j}=\frac{\rho \mathrm{\nabla }S}{m}}$$Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. Although the $\frac{\mathrm{\nabla }S}{m}$ term appears to play the role of velocity, it does not represent velocity at a point since simultaneous measurement of position and velocity violates uncertainty principle.

Separation of variables

If the Hamiltonian is not an explicit function of time, Schrödinger's equation reads: $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(\mathbf{r},t)=\left[-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(\mathbf{r})\right]\mathrm{\Psi }(\mathbf{r},t).}$$ The operator on the left side depends only on time; the one on the right side depends only on space. Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts $${\displaystyle \mathrm{\Psi }(\mathbf{r},t)=\psi (\mathbf{r})\tau (t),}$$ where ${\displaystyle \psi (\mathbf{r})}$ is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and ${\displaystyle \tau (t)}$ is a function of time only. Substituting this expression for ${\displaystyle \mathrm{\Psi }}$ into the time dependent left hand side shows that ${\displaystyle \tau (t)}$ is a phase factor: $${\displaystyle \mathrm{\Psi }(\mathbf{r},t)=\psi (\mathbf{r})e^{-iEt/\hslash }.}$$ A solution of this type is called stationary, since the only time dependence is a phase factor that cancels when the probability density is calculated via the Born rule.

The spatial part of the full wave function solves the equation $${\displaystyle {\mathrm{\nabla }}^2\psi (\mathbf{r})+\frac{2m}{{\hslash }^2}\left[E-V(\mathbf{r})\right]\psi (\mathbf{r})=0,}$$ where the energy ${\displaystyle E}$ appears in the phase factor.

This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the standing wave solutions of the time-independent equation are the states with definite energy, instead of a probability distribution of different energies. In physics, these standing waves are called "stationary states" or "energy eigenstates"; in chemistry they are called "atomic orbitals" or "molecular orbitals". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels. The energy eigenstates form a basis: any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is an example of the spectral theorem, and in a finite-dimensional state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.

Separation of variables can also be a useful method for the time-independent Schrödinger equation. For example, depending on the symmetry of the problem, the Cartesian axes might be separated, as in $${\displaystyle \psi (\mathbf{r})={\psi }_x(x){\psi }_y(y){\psi }_z(z),}$$ or radial and angular coordinates might be separated: $${\displaystyle \psi (\mathbf{r})={\psi }_r(r){\psi }_{\theta }(\theta ){\psi }_{\varphi }(\varphi ).}$$

Examples

See also: List of quantum-mechanical systems with analytical solutions

Particle in a box

1-dimensional potential energy box (or infinite potential well)

Main article: Particle in a box

The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy inside a certain region and infinite potential energy outside. For the one-dimensional case in the ${\displaystyle x}$ direction, the time-independent Schrödinger equation may be written $${\displaystyle -\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi .}$$

With the differential operator defined by $${\displaystyle {\hat{p}}_x=-i\hslash \frac{d}{dx}}$$ the previous equation is evocative of the classic kinetic energy analogue, $${\displaystyle \frac{1}{2m}{\hat{p}}_x^2=E,}$$ with state ${\displaystyle \psi }$ in this case having energy ${\displaystyle E}$ coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are $${\displaystyle \psi (x)=A{e}^{ikx}+B{e}^{-ikx}E=\frac{{\hslash }^2{k}^2}{2m}}$$ or, from Euler's formula, $${\displaystyle \psi (x)=C\sin (kx)+D\cos (kx).}$$

The infinite potential walls of the box determine the values of ${\displaystyle C,D,}$ and ${\displaystyle k}$ at ${\displaystyle x=0}$ and ${\displaystyle x=L}$ where ${\displaystyle \psi }$ must be zero. Thus, at ${\displaystyle x=0}$, $${\displaystyle \psi (0)=0=C\sin (0)+D\cos (0)=D}$$ and ${\displaystyle D=0}$. At ${\displaystyle x=L}$, $${\displaystyle \psi (L)=0=C\sin (kL),}$$ in which ${\displaystyle C}$ cannot be zero as this would conflict with the postulate that ${\displaystyle \psi }$ has norm 1. Therefore, since ${\displaystyle \sin (kL)=0}$, ${\displaystyle kL}$ must be an integer multiple of ${\displaystyle \pi }$, $${\displaystyle k=\frac{n\pi }{L}n=1,2,3,\dots .}$$

This constraint on ${\displaystyle k}$ implies a constraint on the energy levels, yielding $${\displaystyle E_n=\frac{{\hslash }^2{\pi }^2{n}^2}{2m{L}^2}=\frac{n^2{h}^2}{8m{L}^2}.}$$

A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy.

Harmonic oscillator

A harmonic oscillator in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a spring, oscillates back and forth. (C–H) are six solutions to the Schrödinger Equation for this situation. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the wave function. Stationary states, or energy eigenstates, which are solutions to the time-independent Schrödinger equation, are shown in C, D, E, F, but not G or H.

Main article: Quantum harmonic oscillator

The Schrödinger equation for this situation is $${\displaystyle E\psi =-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}\psi +\frac{1}{2}m{\omega }^2{x}^2\psi ,}$$ where ${\displaystyle x}$ is the displacement and ${\displaystyle \omega }$ the angular frequency. Furthermore, it can be used to describe approximately a wide variety of other systems, including vibrating atoms, molecules, and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics.

The solutions in position space are $${\displaystyle {\psi }_n(x)=\sqrt{\frac{1}{2^{n}n!}}{\left(\frac{m\omega }{\pi \hslash }\right)}^{1/4}{e}^{-\frac{m\omega x^2}{2\hslash }}{\mathcal{H}}_n\left(\sqrt{\frac{m\omega }{\hslash }}x\right),}$$ where ${\displaystyle n\in \{0,1,2,\dots \}}$, and the functions ${\displaystyle {\mathcal{H}}_n}$ are the Hermite polynomials of order ${\displaystyle n}$. The solution set may be generated by $${\displaystyle {\psi }_n(x)=\frac{1}{\sqrt{n!}}{\left(\sqrt{\frac{m\omega }{2\hslash }}\right)}^n{\left(x-\frac{\hslash }{m\omega }\frac{d}{dx}\right)}^n{\left(\frac{m\omega }{\pi \hslash }\right)}^{\frac{1}{4}}{e}^{\frac{-m\omega x^2}{2\hslash }}.}$$

The eigenvalues are $${\displaystyle E_n=\left(n+\frac{1}{2}\right)\hslash \omega .}$$

The case ${\displaystyle n=0}$ is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian.

The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized.

Hydrogen atom

Wave functions of the electron in a hydrogen atom at different energy levels. They are plotted according to solutions of the Schrödinger equation.

The Schrödinger equation for the electron in a hydrogen atom (or a hydrogen-like atom) is $${\displaystyle E\psi =-\frac{{\hslash }^2}{2\mu }{\mathrm{\nabla }}^2\psi -\frac{q^2}{4\pi {\epsilon }_{0}r}\psi }$$ where ${\displaystyle q}$ is the electron charge, ${\displaystyle \mathbf{r}}$ is the position of the electron relative to the nucleus, ${\displaystyle r=|\mathbf{r}|}$ is the magnitude of the relative position, the potential term is due to the Coulomb interaction, wherein ${\displaystyle {\epsilon }_0}$ is the permittivity of free space and $${\displaystyle \mu =\frac{m_q{m}_p}{m_q+m_p}}$$ is the 2-body reduced mass of the hydrogen nucleus (just a proton) of mass ${\displaystyle m_p}$ and the electron of mass ${\displaystyle m_q}$. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common center of mass, and constitute a two-body problem to solve. The motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.

The Schrödinger equation for a hydrogen atom can be solved by separation of variables. In this case, spherical polar coordinates are the most convenient. Thus, $${\displaystyle \psi (r,\theta ,\phi )=R(r)Y_{\ell }^m(\theta ,\phi )=R(r)\mathrm{\Theta }(\theta )\mathrm{\Phi }(\phi ),}$$ where R are radial functions and ${\displaystyle Y_l^m(\theta ,\phi )}$ are spherical harmonics of degree ${\displaystyle \ell }$ and order ${\displaystyle m}$. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are: $${\displaystyle {\psi }_{n\ell m}(r,\theta ,\phi )=\sqrt{{\left(\frac{2}{n{a}_0}\right)}^3\frac{(n-\ell -1)!}{2n[(n+\ell )!]}}{e}^{-r/n{a}_0}{\left(\frac{2r}{n{a}_0}\right)}^{\ell }{L}_{n-\ell -1}^{2\ell +1}\left(\frac{2r}{n{a}_0}\right)\cdot Y_{\ell }^m(\theta ,\phi )}$$ where

• ${\displaystyle a_0=\frac{4\pi {\epsilon }_0{\hslash }^2}{m_q{q}^2}}$ is the Bohr radius,
• ${\displaystyle L_{n-\ell -1}^{2\ell +1}(\cdots )}$ are the generalized Laguerre polynomials of degree ${\displaystyle n-\ell -1}$,
• ${\displaystyle n,\ell ,m}$ are the principal, azimuthal, and magnetic quantum numbers respectively, which take the values ${\displaystyle n=1,2,3,\dots ,}$ ${\displaystyle \ell =0,1,2,\dots ,n-1,}$ ${\displaystyle m=-\ell ,\dots ,\ell .}$

Approximate solutions

It is typically not possible to solve the Schrödinger equation exactly for situations of physical interest. Accordingly, approximate solutions are obtained using techniques like variational methods and WKB approximation. It is also common to treat a problem of interest as a small modification to a problem that can be solved exactly, a method known as perturbation theory.

Semiclassical limit

One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential ${\displaystyle V}$, the Ehrenfest theorem says $${\displaystyle m\frac{d}{dt}⟨x⟩=⟨p⟩;\frac{d}{dt}⟨p⟩=-⟨V^{\prime }(X)⟩.}$$ Although the first of these equations is consistent with the classical behavior, the second is not: If the pair ${\displaystyle (⟨X⟩,⟨P⟩)}$ were to satisfy Newton's second law, the right-hand side of the second equation would have to be $${\displaystyle -V^{\prime }\left(⟨X⟩\right)}$$ which is typically not the same as ${\displaystyle -⟨V^{\prime }(X)⟩}$. For a general ${\displaystyle V^{\prime }}$, therefore, quantum mechanics can lead to predictions where expectation values do not mimic the classical behavior. In the case of the quantum harmonic oscillator, however, ${\displaystyle V^{\prime }}$ is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories.

For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point ${\displaystyle x_0}$, then ${\displaystyle V^{\prime }\left(⟨X⟩\right)}$ and ${\displaystyle ⟨V^{\prime }(X)⟩}$ will be almost the same, since both will be approximately equal to ${\displaystyle V^{\prime }(x_0)}$. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position.

The Schrödinger equation in its general form $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }\left(\mathbf{r},t\right)=\hat{H}\mathrm{\Psi }\left(\mathbf{r},t\right)}$$ is closely related to the Hamilton–Jacobi equation (HJE) $${\displaystyle -\frac{\mathrm{\partial }}{\mathrm{\partial }t}S(q_i,t)=H\left(q_i,\frac{\mathrm{\partial }S}{\mathrm{\partial }{q}_i},t\right)}$$ where ${\displaystyle S}$ is the classical action and ${\displaystyle H}$ is the Hamiltonian function (not operator). Here the generalized coordinates ${\displaystyle q_i}$ for ${\displaystyle i=1,2,3}$ (used in the context of the HJE) can be set to the position in Cartesian coordinates as ${\displaystyle \mathbf{r}=(q_1,q_2,q_3)=(x,y,z)}$.

Substituting $${\displaystyle \mathrm{\Psi }=\sqrt{\rho (\mathbf{r},t)}{e}^{iS(\mathbf{r},t)/\hslash }}$$ where ${\displaystyle \rho }$ is the probability density, into the Schrödinger equation and then taking the limit ${\displaystyle \hslash \to 0}$ in the resulting equation yield the Hamilton–Jacobi equation.

Density matrices

Main article: Density matrix

Wave functions are not always the most convenient way to describe quantum systems and their behavior. When the preparation of a system is only imperfectly known, or when the system under investigation is a part of a larger whole, density matrices may be used instead. A density matrix is a positive semi-definite operator whose trace is equal to 1. (The term "density operator" is also used, particularly when the underlying Hilbert space is infinite-dimensional.) The set of all density matrices is convex, and the extreme points are the operators that project onto vectors in the Hilbert space. These are the density-matrix representations of wave functions; in Dirac notation, they are written $${\displaystyle \hat{\rho }=|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|.}$$

The density-matrix analogue of the Schrödinger equation for wave functions is $${\displaystyle i\hslash \frac{\mathrm{\partial }\hat{\rho }}{\mathrm{\partial }t}=[\hat{H},\hat{\rho }],}$$ where the brackets denote a commutator. This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices. If the Hamiltonian is time-independent, this equation can be easily solved to yield $${\displaystyle \hat{\rho }(t)=e^{-i\hat{H}t/\hslash }\hat{\rho }(0)e^{i\hat{H}t/\hslash }.}$$

More generally, if the unitary operator ${\displaystyle \hat{U}(t)}$ describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by $${\displaystyle \hat{\rho }(t)=\hat{U}(t)\hat{\rho }(0)\hat{U}(t{)}^{†}.}$$

Unitary evolution of a density matrix conserves its von Neumann entropy.

Relativistic quantum physics and quantum field theory

The one-particle Schrödinger equation described above is valid essentially in the nonrelativistic domain. For one reason, it is essentially invariant under Galilean transformations, which form the symmetry group of Newtonian dynamics. Moreover, processes that change particle number are natural in relativity, and so an equation for one particle (or any fixed number thereof) can only be of limited use. A more general form of the Schrödinger equation that also applies in relativistic situations can be formulated within quantum field theory (QFT), a framework that allows the combination of quantum mechanics with special relativity. The region in which both simultaneously apply may be described by relativistic quantum mechanics. Such descriptions may use time evolution generated by a Hamiltonian operator, as in the Schrödinger functional method.

Klein–Gordon and Dirac equations

Attempts to combine quantum physics with special relativity began with building relativistic wave equations from the relativistic energy–momentum relation $${\displaystyle E^2=(pc{)}^2+{\left(m_0{c}^2\right)}^2,}$$ instead of nonrelativistic energy equations. The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation, $${\displaystyle -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\psi +{\mathrm{\nabla }}^2\psi =\frac{m^2{c}^2}{{\hslash }^2}\psi ,}$$ was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for a relativistic theory. Taking the "square root" of the left-hand side of the Klein–Gordon equation in this way required factorizing it into a product of two operators, which Dirac wrote using 4 × 4 matrices ${\displaystyle {\alpha }_1,{\alpha }_2,{\alpha }_3,\beta }$. Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read $${\displaystyle \left(\beta m{c}^2+c\left(\sum _{n=1}^3{\alpha }_n{p}_n\right)\right)\psi =i\hslash \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}.}$$

This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function. Including influences upon the particle requires modifying the Hamiltonian operator. For example, the Dirac Hamiltonian for a particle of mass m and electric charge q in an electromagnetic field (described by the electromagnetic potentials φ and A) is: $${\displaystyle {\hat{H}}_{\text{Dirac}}={\gamma }^0\left[c\mathit{\gamma }\cdot \left(\hat{\mathbf{p}}-q\mathbf{A}\right)+m{c}^2+{\gamma }^{0}q\phi \right],}$$ in which the γ = (γ¹, γ², γ³) and γ⁰ are the Dirac gamma matrices related to the spin of the particle. The Dirac equation is true for all spin-1⁄2 particles, and the solutions to the equation are 4-component spinor fields with two components corresponding to the particle and the other two for the antiparticle.

For the Klein–Gordon equation, the general form of the Schrödinger equation is inconvenient to use, and in practice the Hamiltonian is not expressed in an analogous way to the Dirac Hamiltonian. The equations for relativistic quantum fields, of which the Klein–Gordon and Dirac equations are two examples, can be obtained in other ways, such as starting from a Lagrangian density and using the Euler–Lagrange equations for fields, or using the representation theory of the Lorentz group in which certain representations can be used to fix the equation for a free particle of given spin (and mass).

In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of the position and momentum operators (and possibly time), but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin s, are complex-valued 2(2s + 1)-component spinor fields.

Fock space

As originally formulated, the Dirac equation is an equation for a single quantum particle, just like the single-particle Schrödinger equation with wave function ${\displaystyle \mathrm{\Psi }(x,t)}$. This is of limited use in relativistic quantum mechanics, where particle number is not fixed. Heuristically, this complication can be motivated by noting that mass–energy equivalence implies material particles can be created from energy. A common way to address this in QFT is to introduce a Hilbert space where the basis states are labeled by particle number, a so-called Fock space. The Schrödinger equation can then be formulated for quantum states on this Hilbert space. However, because the Schrödinger equation picks out a preferred time axis, the Lorentz invariance of the theory is no longer manifest, and accordingly, the theory is often formulated in other ways.

History

Erwin Schrödinger

Following Max Planck's quantization of light (see black-body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wave number in special relativity, it followed that the momentum ${\displaystyle p}$ of a photon is inversely proportional to its wavelength ${\displaystyle \lambda }$, or proportional to its wave number ${\displaystyle k}$: $${\displaystyle p=\frac{h}{\lambda }=\hslash k,}$$ where ${\displaystyle h}$ is the Planck constant and ${\displaystyle \hslash =h/2\pi }$ is the reduced Planck constant. Louis de Broglie hypothesized that this is true for all particles, even particles which have mass such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed. These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum ${\displaystyle L}$ according to $${\displaystyle L=n\frac{h}{2\pi }=n\hslash .}$$ According to de Broglie, the electron is described by a wave, and a whole number of wavelengths must fit along the circumference of the electron's orbit: $${\displaystyle n\lambda =2\pi r.}$$

This approach essentially confined the electron wave in one dimension, along a circular orbit of radius ${\displaystyle r}$.

In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum 4-vector to derive what we now call the de Broglie relation. Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation and solve for its energy eigenvalues for the hydrogen atom; the paper was rejected by the Physical Review, according to Kamen.

Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William Rowan Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system—the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.

Schrödinger's equation inscribed on the gravestone of Annemarie and Erwin Schrödinger. (Newton's dot notation for the time derivative is used.)

The equation he found is $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(\mathbf{r},t)=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }(\mathbf{r},t)+V(\mathbf{r})\mathrm{\Psi }(\mathbf{r},t).}$$

By that time Arnold Sommerfeld had refined the Bohr model with relativistic corrections. Schrödinger used the relativistic energy–momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential (in natural units): $${\displaystyle {\left(E+\frac{e^2}{r}\right)}^2\psi (x)=-{\mathrm{\nabla }}^2\psi (x)+m^2\psi (x).}$$

He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.

While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926. Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave ${\displaystyle \mathrm{\Psi }(\mathbf{x},t)}$, moving in a potential well ${\displaystyle V}$, created by the proton. This computation accurately reproduced the energy levels of the Bohr model.

The Schrödinger equation details the behavior of ${\displaystyle \mathrm{\Psi }}$ but says nothing of its nature. Schrödinger tried to interpret the real part of ${\displaystyle \mathrm{\Psi }\frac{\mathrm{\partial }{\mathrm{\Psi }}^{\ast }}{\mathrm{\partial }t}}$ as a charge density, and then revised this proposal, saying in his next paper that the modulus squared of ${\displaystyle \mathrm{\Psi }}$ is a charge density. This approach was, however, unsuccessful. In 1926, just a few days after this paper was published, Max Born successfully interpreted ${\displaystyle \mathrm{\Psi }}$ as the probability amplitude, whose modulus squared is equal to probability density. Later, Schrödinger himself explained this interpretation as follows:

The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog.

— Erwin Schrödinger

Interpretation

Main article: Interpretations of quantum mechanics

The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reality depends upon the interpretation of quantum mechanics that one adopts.

In the views often grouped together as the Copenhagen interpretation, a system's wave function is a collection of statistical information about that system. The Schrödinger equation relates information about the system at one time to information about it at another. While the time-evolution process represented by the Schrödinger equation is continuous and deterministic, in that knowing the wave function at one instant is in principle sufficient to calculate it for all future times, wave functions can also change discontinuously and stochastically during a measurement. The wave function changes, according to this school of thought, because new information is available. The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule.^{[26][51][note 4]} Other, more recent interpretations of quantum mechanics, such as relational quantum mechanics and QBism also give the Schrödinger equation a status of this sort.

Schrödinger himself suggested in 1952 that the different terms of a superposition evolving under the Schrödinger equation are "not alternatives but all really happen simultaneously". This has been interpreted as an early version of Everett's many-worlds interpretation. This interpretation, formulated independently in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes. This interpretation removes the axiom of wave function collapse, leaving only continuous evolution under the Schrödinger equation, and so all possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Why we should assign probabilities at all to outcomes that are certain to occur in some worlds, and why should the probabilities be given by the Born rule? Several ways to answer these questions in the many-worlds framework have been proposed, but there is no consensus on whether they are successful.

Bohmian mechanics reformulates quantum mechanics to make it deterministic, at the price of adding a force due to a "quantum potential". It attributes to each physical system not only a wave function but in addition a real position that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation.