The Schrödinger equation is a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
partial differential equation that governs the
wave function of a quantum-mechanical system.
It is a key result in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and its discovery was a significant landmark in the development of the subject. The equation is named after
Erwin Schrödinger, 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
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 1933.
[
]
Conceptually, the Schrödinger equation is the quantum counterpart of
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in mo ...
in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. 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 a
wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
, and must therefore be generated by the exponential of a
self-adjoint operator, which is the quantum
Hamiltonian.
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include
matrix mechanics, introduced by
Werner Heisenberg, and the
path integral formulation, developed chiefly by
Richard Feynman.
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
incorporated matrix mechanics and the Schrödinger equation into a single formulation. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".
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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, particularly
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s 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:
Here,
is a wave function, a function that assigns a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
to each point
at each time
. The parameter
is the mass of the particle, and
is the ''
potential'' that represents the environment in which the particle exists.
The constant
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, and
is the reduced
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, which has units of
action (energy multiplied by time).
Broadening beyond this simple case, the
mathematical formulation of quantum mechanics developed by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
,
David Hilbert,
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
, and
Hermann Weyl defines the state of a quantum mechanical system to be a vector
belonging to a (
separable)
Hilbert space . This vector is postulated to be normalized under the Hilbert space's inner product, that is, in
Dirac notation it obeys
. 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 complex
square-integrable functions
, while the Hilbert space for the
spin of a single proton is simply the space of two-dimensional complex vectors
with the usual inner product.
Physical quantities of interest – position, momentum, energy, spin – are represented by "observables", which are
Hermitian (more precisely,
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a st ...
) linear
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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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
is non-degenerate and the probability is given by
, where
is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by
, where
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 introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.
Thus, a position-space wave function
as used above can be written as the inner product of a time-dependent state vector
with unphysical but convenient "position eigenstates"
:
Time-dependent equation
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:
where
is time,
is the state vector of the quantum system (
being the Greek letter
psi), and
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
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
).
To apply the Schrödinger equation, write down the
Hamiltonian for the system, accounting for the
kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to its motion
Art and ent ...
and
potential energies of the particles constituting the system, then insert it into the Schrödinger equation. The resulting partial
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
.
For example, given a wave function in position space
as above, we have
Time-independent equation
The time-dependent Schrödinger equation described above predicts that wave functions can form
standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s, 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.
where
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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, this equation is an
eigenvalue equation. Therefore, the wave function is an
eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s)
.
Properties
Linearity
The Schrödinger equation is a
linear differential equation, meaning that if two state vectors
and
are solutions, then so is any
linear combination
of the two where and 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
can be written as the linear combination
where
are complex numbers and the vectors
are solutions of the time-independent equation
.
Unitarity
Holding the Hamiltonian
constant, the Schrödinger equation has the solution
The operator
is known as the time-evolution operator, and it is
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
: 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
, then the state at a later time
will be given by
for some unitary operator
. Conversely, suppose that
is a continuous family of unitary operators parameterized by
.
Without loss of generality, the parameterization can be chosen so that
is the identity operator and that
for any
. Then
depends upon the parameter
in such a way that
for some self-adjoint operator
, called the ''generator'' of the family
. A Hamiltonian is just such a generator (up to the factor of Planck's constant that would be set to 1 in
natural units).
To see that the generator is Hermitian, note that with
, we have
so
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:
Writing
for a three-dimensional position vector and
for a three-dimensional momentum vector, the position-space Schrödinger equation is
The momentum-space counterpart involves the
Fourier transforms of the wave function and the potential:
The functions
and
are derived from
by
where
and
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
and
are promoted to self-adjoint operators
and
that satisfy the
canonical commutation relation
This implies that
so the action of the momentum operator
in the position-space representation is
. Thus,
becomes a
second derivative, and in three dimensions, the second derivative becomes the
Laplacian .
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
with
for only discrete
reciprocal lattice vectors
. 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
The Schrödinger equation is consistent with
local probability conservation.
Multiplying the Schrödinger equation on the right by the complex conjugate wave function, and multiplying the wave function to the left of the complex conjugate of the Schrödinger equation, and subtracting, gives the
continuity equation for probability:
where
is the
probability density (probability per unit volume, denotes
complex conjugate), and
is the
probability current (flow per unit area).
Separation of variables
If the Hamiltonian is not an explicit function of time, the equation is
separable into a product of spatial and temporal parts. Solving the equation by separation of variables means seeking a solution of the form
where
is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and
is a function of time only. Substituting this expression for
into the Schrödinger equation yields
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.
[
This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the ]standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
solutions of the time-dependent 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 eigenstate
A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ...
s"; in chemistry they are called "atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
s" or "molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of find ...
s". 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 the spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
in mathematics, 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,
or radial and angular coordinates might be separated:
Examples
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 direction, the time-independent Schrödinger equation may be written
With the differential operator defined by
the previous equation is evocative of the classic kinetic energy analogue,
with state in this case having energy coincident with the kinetic energy of the particle.
The general solutions of the Schrödinger equation for the particle in a box are
or, from Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
,
The infinite potential walls of the box determine the values of and at and where must be zero. Thus, at ,
and . At ,
in which cannot be zero as this would conflict with the postulate that has norm 1. Therefore, since , must be an integer multiple of ,
This constraint on implies a constraint on the energy levels, yielding
A finite potential well The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike ...
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
In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. ...
, 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
The Schrödinger equation for this situation is
where is the displacement and 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
where , and the functions are the Hermite polynomials of order . The solution set may be generated by
The eigenvalues are
The case is called the ground state, its energy is called the zero-point energy
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...
, and the wave function is a Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
.
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
The Schrödinger equation for the electron in a hydrogen atom (or a hydrogen-like atom) is
where is the electron charge, is the position of the electron relative to the nucleus, is the magnitude of the relative position, the potential term is due to the Coulomb interaction
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, wherein is the permittivity of free space and
is the 2-body reduced mass of the hydrogen nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
* Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucl ...
(just a proton) of mass and the electron of mass . 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 centre 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,
where are radial functions and are spherical harmonics of degree and order . 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:
where
* is the Bohr radius,
* are the generalized Laguerre polynomials of degree ,
* are the principal, azimuthal, and magnetic quantum numbers respectively, which take the values
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
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.
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
The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the ...
. For a one-dimensional quantum particle moving in a potential , the Ehrenfest theorem says
Although the first of these equations is consistent with the classical behavior, the second is not: If the pair were to satisfy Newton's second law, the right-hand side of the second equation would have to be
which is typically not the same as . For a general , 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, 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 , then and will be ''almost'' the same, since both will be approximately equal to . 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
is closely related to the Hamilton–Jacobi equation (HJE)
where is the classical action and is the Hamiltonian function (not operator). Here the generalized coordinates for (used in the context of the HJE) can be set to the position in Cartesian coordinates as .
Substituting
where is the probability density, into the Schrödinger equation and then taking the limit in the resulting equation yield the Hamilton–Jacobi equation.
Density matrices
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
The density-matrix analogue of the Schrödinger equation for wave functions is
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
More generally, if the unitary operator describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by
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 comprise 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
instead of nonrelativistic energy equations. The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation,
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 . Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read
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 and electric charge in an electromagnetic field (described by the electromagnetic potentials and ) is:
in which the and are the Dirac gamma matrices related to the spin of the particle. The Dirac equation is true for all particles, and the solutions to the equation are spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
s 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 , are complex-valued spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
s.
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 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
Following Max Planck's quantization of light (see black-body radiation), Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
interpreted Planck's quanta to be photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s, 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
Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...
. Since energy and momentum are related in the same way as frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and wave number in special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
, it followed that the momentum of a photon is inversely proportional to its wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, or proportional to its wave number :
where is Planck's constant and 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 wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s, 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 according to
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:
This approach essentially confined the electron wave in one dimension, along a circular orbit of radius .
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. Unfortunately the paper was rejected by the ''Physical Review'', as recounted by Kamen.
Following up on de Broglie's ideas, physicist Peter Debye
Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry.
Biography
Early life
Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherland ...
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
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
's analogy between mechanics and optics,[See the Hamilton–Jacobi equation.] 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.
The equation he found is
However, 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):
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
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
as a wave , moving in a potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is ca ...
, created by the proton. This computation accurately reproduced the energy levels of the Bohr model.
The Schrödinger equation details the behavior of but says nothing of its ''nature''. Schrödinger tried to interpret the real part of as a charge density, and then revised this proposal, saying in his next paper that the modulus squared of is a charge density. This approach was, however, unsuccessful. In 1926, just a few days after this paper was published, Max Born successfully interpreted as the probability amplitude, whose modulus squared is equal to probability density.[ Later, Schrödinger himself explained this interpretation as follows:
]
Interpretation
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
An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraor ...
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. Other, more recent interpretations of quantum mechanics, such as relational quantum mechanics
:''This article is intended for those already familiar with quantum mechanics and its attendant interpretational difficulties. Readers who are new to the subject may first want to read the introduction to quantum mechanics.''
Relational quantum m ...
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 don't 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 making it explicitly nonlocal (a price exacted by Bell's theorem). 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.
See also
* Eckhaus equation
* Pauli equation
* Fokker–Planck equation
* List of things named after Erwin Schrödinger
* Logarithmic Schrödinger equation
* Nonlinear Schrödinger equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in non ...
* Quantum channel
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information ...
*
* Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which ma ...
* Wigner quasiprobability distribution
Notes
References
External links
*
Quantum Cook Book
an
PHYS 201: Fundamentals of Physics II
by Ramamurti Shankar
Ramamurti Shankar (born April 28, 1947) is the Josiah Willard Gibbs professor of Physics at Yale University, in New Haven, Connecticut.
Education
He received his B. Tech in electrical engineering from the Indian Institute of Technology in Ma ...
, Yale OpenCourseware
The Modern Revolution in Physics
– an online textbook.
Quantum Physics I
at MIT OpenCourseWare
{{DEFAULTSORT:Schrodinger Equation
Differential equations
Partial differential equations
Wave mechanics
Functions of space and time