Schrödinger equation

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In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger.
In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler–Lagrange equations and Hamilton's equations. All of these formulations are used to solve for the motion of a mechanical system and mathematically predict what the system will do at any time beyond the initial settings and configuration of the system.
In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but (in general) a linear partial differential equation. The differential equation describes the wave function of the system, also called the quantum state or state vector.
In the standard interpretation of quantum mechanics, the wave function is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe.^[1]
Like Newton's Second law, the Schrödinger equation can be mathematically transformed into other formulations such as Werner Heisenberg's matrix mechanics, and Richard Feynman's path integral formulation. Also like Newton's Second law, the Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem that is not as severe in matrix mechanics and completely absent in the path integral formulation. The equation is derived by partially differentiating the standard wave equation and substituting the relation between the momentum of the particle and the wavelength of the wave associated with the particle in De Broglie's hypothesis.

Equation

Time-dependent equation

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:^[2]

Time-dependent Schrödinger equation (general) $i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi$

where i is the imaginary unit, ħ is the reduced Planck constant, Ψ is the wave function of the quantum system, and $\hat{H}$ is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation).

A wave function that satisfies the non-relativistic Schrödinger equation with V=0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here.

The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field; see the Pauli equation):

Time-dependent Schrödinger equation (single non-relativistic particle) $i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)$

where m is the particle's mass, V is its potential energy, ∇² is the Laplacian, and Ψ is the wave function (more precisely, in this context, it is called the "position-space wave function"). In plain language, it means "total energy equals kinetic energy plus potential energy", but the terms take unfamiliar forms for reasons explained below.
Given the particular differential operators involved, this is a linear partial differential equation. It is also a diffusion equation.
The term "Schrödinger equation" can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in various complicated expressions for the Hamiltonian. The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see relativistic quantum mechanics).
To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.

Time-independent 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 an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".

The time-dependent Schrödinger equation predicts that wave functions can form standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals). These states are important in their own right, and moreover if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state. The time-independent Schrödinger equation is the equation describing stationary states. (It is only used when the Hamiltonian itself is not dependent on time.)

Time-independent Schrödinger equation (general) $E\Psi=\hat H \Psi$

In words, the equation states:

When the Hamiltonian operator acts on the wave function Ψ, the result might be proportional to the same wave function Ψ. If it is, then Ψ is a stationary state, and the proportionality constant, E, is the energy of the state Ψ.

The time-independent Schrödinger equation is discussed further below. In linear algebra terminology, this equation is an eigenvalue equation.
As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):

Time-independent Schrödinger equation (single non-relativistic particle) $E \Psi(\mathbf{r}) = \frac{-\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r}) + V(\mathbf{r}) \Psi(\mathbf{r})$

with definitions as above.

Implications

The Schrödinger equation, and its solutions, introduced a breakthrough in thinking about physics. Schrödinger's equation was the first of its type, and solutions led to very unusual and unexpected consequences for the time.

Total, kinetic, and potential energy

The overall form of the equation is not unusual or unexpected as it uses the principle of the conservation of energy. The terms of the nonrelativistic Schrödinger equation can be interpreted as total energy of the system, equal to the system kinetic energy plus the system potential energy. In this respect, it is just the same as in classical physics.

Quantization

The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur. One example is energy quantization: the energy of an electron in an atom is always one of the quantized energy levels, a fact discovered via atomic spectroscopy. (Energy quantization is discussed below.) Another example is quantization of angular momentum. This was an assumption in the earlier Bohr model of the atom, but it is a prediction of the Schrödinger equation.
Not every measurement gives a quantized result in quantum mechanics. For example, position, momentum, time, and (in some situations) energy can have any value across a continuous range.^{[citation needed]}

Measurement and uncertainty

Main articles: Measurement in quantum mechanics, Heisenberg uncertainty principle, and Interpretations of quantum mechanics

In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change deterministically as the particle moves according to Newton's laws. In quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a probability distribution. The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.
The Heisenberg uncertainty principle is the statement of the inherent measurement uncertainty in quantum mechanics. It states that the more precisely a particle's position is known, the less precisely its momentum is known, and vice versa.
The Schrödinger equation describes the (deterministic) evolution of the wave function of a particle. However, even if the wave function is known exactly, the result of a specific measurement on the wave function is uncertain.

Quantum tunneling

Main article: Quantum tunneling

In classical physics, when a ball is rolled slowly up a large hill, it will come to a stop and roll back, because it doesn't have enough energy to get over the top of the hill to the other side. However, the Schrödinger equation predicts that there is a small probability that the ball will get to the other side of the hill, even if it has too little energy to reach the top. This is called quantum tunneling. It is related to the uncertainty principle: Although the ball seems to be on one side of the hill, its position is uncertain so there is a chance of finding it on the other side.

The solution to Schrödinger's equation for a 1-d step potential system (dotted line), shown in slices of constant time, the opacity corresponds to the probability density |Ψ|² of the particle (blue circles) at the location shown. The probability of the particle passing through the barrier (transmission) is greater than it reflecting from the barrier, since the total energy E of the particle exceeds the potential energy V.^[3]

Quantum tunneling through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side.

The fuzziness of a particle's position illustrated, which is not definite in quantum mechanics.

Particles as waves

Main articles: Matter wave, Wave–particle duality, and Double-slit experiment

A double slit experiment showing the accumulation of electrons on a screen as time passes.

The nonrelativistic Schrödinger equation is a type of partial differential equation called a wave equation. Therefore it is often said particles can exhibit behavior usually attributed to waves. In most modern interpretations this description is reversed – the quantum state, i.e. wave, is the only genuine physical reality, and under the appropriate conditions it can show features of particle-like behavior.
Two-slit diffraction is a famous example of the strange behaviors that waves regularly display, that are not intuitively associated with particles. The overlapping waves from the two slits cancel each other out in some locations, and reinforce each other in other locations, causing a complex pattern to emerge. Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.
However, since the Schrödinger equation is a wave equation, a single particle fired through a double-slit does show this same pattern (figure on left). Note: The experiment must be repeated many times for the complex pattern to emerge. The appearance of the pattern proves that each electron passes through both slits simultaneously.^[4]^[5]^[6] Although this is counterintuitive, the prediction is correct; in particular, electron diffraction and neutron diffraction are well understood and widely used in science and engineering.
Related to diffraction, particles also display superposition and interference.
The superposition property allows the particle to be in a quantum superposition of two or more states with different classical properties at the same time. For example, a particle can have several different energies at the same time, and can be in several different locations at the same time. In the above example, a particle can pass through two slits at the same time. This superposition is still a single quantum state, as shown by the interference effects, even though that conflicts with classical intuition.

Plane wave

Wave packet

Propagation of de Broglie waves in 1d – real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the color opacity) of finding the particle at a given point x is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the curvature reverses sign, so the amplitude begins decrease again, and vice versa – the result is an alternating amplitude: a wave.

Interpretation of the wave function

Main article: Interpretations of quantum mechanics

The Schrödinger equation provides a way to calculate the possible wave functions of a system and how they dynamically change in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. Interpretations of quantum mechanics address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.
An important aspect is the relationship between the Schrödinger equation and wavefunction collapse. In the oldest Copenhagen interpretation, particles follow the Schrödinger equation except during wavefunction collapse, during which they behave entirely differently. The advent of quantum decoherence theory allowed alternative approaches (such as the Everett many-worlds interpretation and consistent histories), wherein the Schrödinger equation is always satisfied, and wavefunction collapse should be explained as a consequence of the Schrödinger equation.

Historical background and development

Main article: Theoretical and experimental justification for the Schrödinger equation

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 wavenumber in special relativity, it followed that the momentum p of a photon is proportional to its wavenumber k.

$p = \frac{h}{\lambda} = \hbar k$

Louis de Broglie hypothesized that this is true for all particles, even particles 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.^[7] 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:

$L = n{h \over 2\pi} = n\hbar.$

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:

$n \lambda = 2 \pi r.\,$

This approach essentially confined the electron wave in one dimension, along a circular orbit.
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^[8] 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.^[9]
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 R. 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.^[10] A modern version of his reasoning is reproduced below. The equation he found is:^[11]

$i\hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t).$

However, by that time, Arnold Sommerfeld had refined the Bohr model with relativistic corrections.^[12]^[13] 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):

$\left(E + {e^2\over r} \right)^2 \psi(x) = - \nabla^2\psi(x) + m^2 \psi(x).$

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 in an isolated mountain cabin with a lover, in December 1925.^[14]
While at the cabin, Schrödinger decided that his earlier non-relativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite difficulties solving the differential equation for hydrogen (he had later help from his friend the mathematician Hermann Weyl) Schrödinger showed that his non-relativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.^[15] In the equation, Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave Ψ(x, t), moving in a potential well V, created by the proton. This computation accurately reproduced the energy levels of the Bohr model. In a paper, Schrödinger himself explained this equation 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^[16]

This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's matrix mechanics, which he considered overly formal.^[17]
The Schrödinger equation details the behavior of ψ but says nothing of its nature. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful.^[18] In 1926, just a few days after Schrödinger's fourth and final paper was published, Max Born successfully interpreted ψ as the probability amplitude, whose absolute square is equal to probability density.^[19] Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory— and never reconciled with the Copenhagen interpretation.^[20]
Louis de Broglie in his later years has proposed a real valued wave function connected to the complex wave function by a propotionality constant and developed the De Broglie–Bohm theory.

The wave equation for particles

Main article: wave–particle duality

The Schrödinger equation was developed principally from the De Broglie hypothesis, a wave equation that would describe particles,^[21] and can be constructed in the following way.^[22] For a more rigorous mathematical derivation of Schrödinger's equation, see also.^[23]

Assumptions

Energy conservation: The total energy E of a particle is the sum of kinetic energy T and potential energy V, this sum is also the frequent expression for the Hamiltonian H in classical mechanics:

$E = T + V =H \,\!$

Explicitly, for a particle in one dimension with position x, mass m and momentum p, and potential energy V which generally varies with position and time t:

$E = \frac{p^2}{2m}+V(x,t)=H.$

For three dimensions, the position vector r and momentum vector p must be used:

$E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H$

This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian. However, there can be interactions between the particles (an N-body problem), so the potential energy V can change as the spatial configuration of particles changes, and possibly with time. The potential energy, in general, is not the sum of the separate potential energies for each particle, it is a function of all the spatial positions of the particles. Explicitly:

$E=\sum_{n=1}^N \frac{\mathbf{p}_n\cdot\mathbf{p}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2\cdots\mathbf{r}_N,t) = H \,\!$

De Broglie relations: Einstein's light quanta hypothesis (1905) states that the energy E of a photon is proportional to the frequency ν (or angular frequency, ω = 2πν) of the corresponding quantum wavepacket of light:

$E = h\nu = \hbar \omega \,\!$

Likewise De Broglie's hypothesis (1924) states that any particle can be associated with a wave, and that the momentum p of the particle is related to the wavelength λ of such a wave, in one dimension, by:

$p = \frac{h}{\lambda} = \hbar k\;,$

in three dimensions:

$\mathbf{p} = \frac{h}{\lambda} = \hbar \mathbf{k}\;.$

where k is the wavevector (wavelength is related to the magnitude of k)
Linearity: The previous assumptions only allow one to derive the equation for plane waves, corresponding to free particles. In general, physical situations are not purely described by plane waves, so for generality the superposition principle is required; any wave can be made by superposition of sinusoidal plane waves. So if the equation is linear, a linear combination of plane waves is also an allowed solution. Hence a necessary and separate requirement is that the Schrödinger equation is a linear differential equation.
Taken together, these attributes mean it should be possible to structure an equation based on the energies of the particles – their possible kinetic and potential energies the system constrains them to have, in terms of some function of the state of the system – the wave function (denoted Ψ). The wave function summarizes the quantum state of the particles in the system, limited by the constraints on the system: the probability the particles are in some spatial configuration at some instant of time. Solving it for the wave function can be used to predict how the particles will behave under the influence of the specified potential and with each other.

Solution to equation

The Schrödinger equation is mathematically a wave equation, since the solutions are functions which describe wave-like motions. Normally wave equations in physics can be derived from other physical laws – the wave equation for mechanical vibrations on strings and in matter can be derived from Newton's laws – where the analogue wave function is the displacement of matter, and electromagnetic waves from Maxwell's equations, where the wave functions are electric and magnetic fields. On the contrary, the basis for Schrödinger's equation is the energy of the particle, and a separate postulate of quantum mechanics: the wave function is a description of the system.^[24] The SE is therefore a new concept in itself; as Feynman put it:

“	Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.	”
—Richard Feynman^[25]

The particle-like behavior of the electron wave, for example, follows from the Schrödinger equation under the appropriate circumstances, as presented below. This behavior is often called wave-particle duality.
The Planck–Einstein and de Broglie relations

$E=\hbar\omega, \quad \mathbf{p}=\hbar\mathbf{k}$

illuminate the deep connections between energy with time, and space with momentum. This is more apparent using natural units, setting ħ = 1 makes these equations into identities:

$E=\omega, \quad \mathbf{p}=\mathbf{k}$

Energy and angular frequency both have the same dimensions as the reciprocal of time, and momentum and wavenumber both have the dimensions of inverse length. In practice they are used interchangeably; to prevent duplication of quantities and reduce the number of dimensions of related quantities. For familiarity SI units are still used in this article.
Schrödinger's insight, late in 1925, was to express the phase of a plane wave as a complex phase factor using these relations:

$\Psi = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar}$

and to realize that the first order partial derivatives with respect to space

$\nabla\Psi = \dfrac{i}{\hbar}\mathbf{p}Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = \dfrac{i}{\hbar}\mathbf{p}\Psi$

and time

$\dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = -\dfrac{i E}{\hbar} \Psi$

imply the derivatives

$\begin{matrix} -i\hbar\nabla\Psi = \mathbf{p}\Psi & \rightarrow & -\dfrac{\hbar^2}{2m}\nabla^2\Psi = \dfrac{1}{2m}\mathbf{p}\cdot\mathbf{p}\Psi \\ \dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} \Psi & \rightarrow & i\hbar\dfrac{\partial \Psi}{\partial t} = E \Psi \\ \end{matrix}$

Another postulate of quantum mechanics is that all observables are represented by operators which act on the wavefunction, and the eigenvalues of the operator are the values the observable takes. The previous derivatives lead to the energy operator, corresponding to the time derivative,

$\hat{E}= i\hbar\dfrac{\partial}{\partial t}$

and the momentum operator, corresponding to the spatial derivatives (the gradient),

$\hat{\mathbf{p}}= -i\hbar\nabla$

where the circumflexes ("hats") denote these observables are operators. These are differential operators, except for potential energy V which is just a multiplicative factor. An interesting point is that energy is also a symmetry with respect to time, and momentum is a symmetry with respect to space, and these are the reasons why energy and momentum are conserved – see Noether's theorem.
Multiplying the energy equation by Ψ, and substitution of the energy and momentum operators:

$E= \dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V \rightarrow \hat{E}\Psi= \dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}\Psi+V\Psi$

immediately led Schrödinger to his equation:

$i\hbar\dfrac{\partial \Psi}{\partial t}= -\dfrac{\hbar^2}{2m}\nabla^2\Psi +V\Psi$

These equations allow wave–particle duality can be assessed as follows. The kinetic energy T is related to the square of momentum p. As the particle's momentum increases, the kinetic energy increases more rapidly, but since the wavenumber k increases the wavelength $\scriptstyle \lambda$ decreases:

$\mathbf{p}\cdot\mathbf{p} \propto \mathbf{k}\cdot\mathbf{k} \propto T \propto \dfrac{1}{\lambda^2}$

and since the kinetic energy is also proportional to the second spatial derivatives, it is also proportional to the magnitude of the curvature of the wave:

$\hat{T} = \frac{-\hbar^2}{2m}\nabla\cdot\nabla \propto \nabla^2 \,.$

As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.^[21]

Solutions to equation

The general solutions of the equation can easily be seen as follows. The plane wave is definitely a solution because this was used to construct the equation, so due to linearity any linear combination of plane waves is also a solution. For discrete k the sum is a superposition of plane waves:

$\Psi(\mathbf{r},t) = \sum_{n=1}^\infty A_n e^{i(\mathbf{k}_n\cdot\mathbf{r}-\omega_n t)} \,\!$

and for continuous k the sum becomes an integral, the Fourier transform of a momentum space wavefunction:^[26]

$\Psi(\mathbf{r},t) = \frac{1}{(\sqrt{2\pi})^3}\int\Phi(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}d^3\mathbf{k} \,\!$

where d³k = dk_xdk_ydk_z is the differential volume element in k-space, and the integrals are taken over all k-space. The momentum wavefunction Φ(k) arises in the integrand since the position and momentum space wavefunctions are Fourier transforms of each other. Since these satisfy the Schrödinger equation—the solutions to Schrödinger's equation for a given situation will not only be the plane waves used to obtain it, but any wavefunctions which satisfy the Schrödinger's equation prescribed by the system, in addition to the relevant boundary conditions—it can be concluded the Schrödinger equation is true for any (non-relativistic) situation.
To summarize, the Schrödinger equation is a differential equation of wave–particle duality, and particles can behave like waves because their corresponding wavefunction satisfies the equation.

Wave and particle motion

Increasing levels of wavepacket localization, meaning the particle has a more localized position.

In the limit ħ → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.

Schrödinger required that a wave packet solution near position r with wavevector near k will move along the trajectory determined by classical mechanics for times short enough for the spread in k (and hence in velocity) not to substantially increase the spread in r . Since, for a given spread in k, the spread in velocity is proportional to Planck's constant ħ, it is sometimes said that in the limit as ħ approaches zero, the equations of classical mechanics are restored from quantum mechanics.^[27] Great care is required in how that limit is taken, and in what cases.
The limiting short-wavelength is equivalent to ħ tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the Heisenberg uncertainty principle for position and momentum, the products of uncertainty in position and momentum become zero as ħ → 0:

$\sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!$

where σ denotes the (root mean square) measurement uncertainty in x and p_x (and similarly for the y and z directions) which implies the position and momentum can only be known to arbitrary precision in this limit.
The Schrödinger equation in its general form

$i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right) \,\!$

is closely related to the Hamilton–Jacobi equation (HJE)

$\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\!$

where S is action and H is the Hamiltonian function (not operator). Here the generalized coordinates q_i for i = 1, 2, 3 (used in the context of the HJE) can be set to the position in Cartesian coordinates as r = (q₁, q₂, q₃) = (x, y, z).^[27]
Substituting

$\Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}\,\!$

where ρ is the probability density, into the Schrödinger equation and then taking the limit ħ → 0 in the resulting equation, yields the Hamilton–Jacobi equation.
The implications are:

The motion of a particle, described by a (short-wavelength) wave packet solution to the Schrödinger equation, is also described by the Hamilton–Jacobi equation of motion.
The Schrödinger equation includes the wavefunction, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.

Non-relativistic quantum mechanics

The quantum mechanics of particles propagating at speeds much less than light is known as non-relativistic quantum mechanics. Following are several forms of Schrödinger's equation in this context for different situations: time independence and dependence, one and three spatial dimensions, and one and N particles. In actuality, the particles constituting the system do not have the numerical labels used in theory. The language of mathematics forces us to label the positions of particles one way or another, otherwise there would be confusion between symbols representing which variables are for which particle.^[23]

Time independent

Diagrammatic summary of the quantities related to the wavefunction, as used in De broglie's hypothesis and development of the Schrödinger equation.^[21]

If the Hamiltonian is not an explicit function of time, the equation is separable into its spatial and temporal parts, i.e. $\Psi(x,t)=\psi(x)\tau(t)$ . Hence the energy operator $\hat{E} = i \hbar \partial / \partial t \,\!$ can then be replaced by the energy eigenvalue E. In abstract form, it is an eigenvalue equation for the Hamiltonian $\hat{H}$ ^[28]

$\hat H \psi = E \psi$

A solution of the time-independent equation is called an energy eigenstate with energy E.^[26]
To find the time dependence of the state, consider starting the time-dependent equation with an initial condition ψ(r). The time derivative at t = 0 is everywhere proportional to the value:

$\left.i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t)\right|_{t=0}= \left.H \Psi(\mathbf{r},t)\right|_{t=0} =E \Psi(\mathbf{r},0) \,$

So initially the whole function just gets rescaled, and it maintains the property that its time derivative is proportional to itself, so for all times t,

$\Psi(\mathbf{r},t)= \tau(t) \psi(\mathbf{r}) \,$

substituting for Ψ:

$i\hbar \frac{\partial \Psi}{\partial t } = E \Psi \rightarrow i\hbar \psi(\mathbf{r})\frac{\partial\tau(t)}{\partial t } = E \tau(t)\psi(\mathbf{r}) \,$

where the ψ(r) cancels, so solving this equation for $\scriptstyle \tau(t)\,\!$ implies the solution of the time-dependent equation with this initial condition is:^[11]

$\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i{E t/\hbar}} = \psi(\mathbf{r}) e^{-i{\omega t}} \,$

This case describes the standing wave solutions of the time-dependent equation, which 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 eigenvalues from this equation form a discrete spectrum of values, so mathematically energy must be quantized. More specifically, the energy eigenstates form a basis – any wavefunction 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, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.
In the case of atoms and molecules, it turns out in spectroscopy that the discrete spectral lines of atoms is evidence that energy is indeed physically quantized in atoms; specifically there are energy levels in atoms, associated with the atomic or molecular orbitals of the electrons (the stationary states, wavefunctions). The spectral lines observed are definite frequencies of light, corresponding to definite energies, by the Planck–Einstein relation and De Broglie relations (above). However, it is not the absolute value of the energy level, but the difference between them, which produces the observed frequencies, due to electronic transitions within the atom emitting/absorbing photons of light.

One-dimensional examples

For a particle in one dimension, the Hamiltonian is:

$\hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x)$

and substituting this into the general Schrödinger equation gives:

$E\Psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\Psi + V\Psi$

This is the only case in which the equation is an ordinary differential equation, rather than a partial differential equation: in higher spatial dimensions and for particles, more spatial derivatives occur for each particle, as shown below. The solutions is always of the form:

$\Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, .$

There is a further restriction — the solution must not grow at infinity, so that it has either a finite L²-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum):^[29]

$\| \psi \|^2 = \int |\psi(x)|^2\, dx.\,$

For N particles in one dimension, the Hamiltonian is:

$\begin{align}\hat{H} &= \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N) \end{align}$

where the position of particle n is x_n. The corresponding Schrödinger equation is:

$E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, .$

with solutions of the form:

$\Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N)$

For non-interacting particles, the potential of the system only influences each particle separately, so the total potential energy is the sum of potential energies for each particle:

$V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .$

and the wavefunction can be written as a product of the wavefunctions for each particle:

$\Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, ,$

In general for interacting particles, the above decompositions are not possible.

Free particle

For no potential, V = 0, so the particle is free and the equation reads:^[30]

$- E \psi = \frac{\hbar^2}{2m}{d^2 \psi \over d x^2}\,$

which has oscillatory solutions for E > 0 (the C_n are arbitrary constants):

$\psi_E(x) = C_1 e^{i\sqrt{2mE/\hbar^2}\,x} + C_2 e^{-i\sqrt{2mE/\hbar^2}\,x}\,$

and exponential solutions for E < 0

$\psi_{-|E|}(x) = C_1 e^{\sqrt{2m|E|/\hbar^2}\,x} + C_2 e^{-\sqrt{2m|E|/\hbar^2}\,x}.\,$

The exponentially growing solutions have an infinite norm, and are not physical. They are not allowed in a finite volume with periodic or fixed boundary conditions.

Constant potential

Animation of a de Broglie wave incident on a barrier.

For a constant potential, V = V₀, the solution is oscillatory for E > V₀ and exponential for E < V₀, corresponding to energies that are allowed or disallowed in classical mechanics. Oscillatory solutions have a classically allowed energy and correspond to actual classical motions, while the exponential solutions have a disallowed energy and describe a small amount of quantum bleeding into the classically disallowed region, due to quantum tunneling. If the potential V₀ grows at infinity, the motion is classically confined to a finite region, which means that in quantum mechanics every solution becomes an exponential far enough away. The condition that the exponential is decreasing restricts the energy levels to a discrete set, called the allowed energies.^[26]

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 wavefunction. (C,D,E,F), but not (G,H), are stationary states (energy eigenstates), which come from solutions to the time-independent Schrödinger Equation.

Main article: Quantum harmonic oscillator

The Schrödinger equation for this situation is

$E\psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi + \frac{1}{2}m\omega^2x^2\psi$

It is a notable quantum system to solve for; since the solutions are exact (but complicated – in terms of Hermite polynomials), and it can describe or at least approximate a wide variety of other systems, including vibrating atoms, molecules,^[31] and atoms or ions in lattices,^[32] and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics.
There is a family of solutions – in the position basis they are

$\psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right)$

where n = 0,1,2..., and the functions H_n are the Hermite polynomials.

Three-dimensional examples

The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the gradient operator.
The Hamiltonian for one particle in three dimensions is:

$\begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \\ & = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \end{align}$

generating the equation:

$E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$

with stationary state solutions of the form:

$\Psi = \psi(\mathbf{r}) e^{-iEt/\hbar}$

where the position of the particle is r = (x, y, z).
For N particles in three dimensions, the Hamiltonian is:

$\begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \end{align}$

where the position of particle n is r_n = (x_n, y_n, z_n), and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle n:

$\nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}$

with corresponding Laplacian operator for particle n:

$\nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}$

The Schrödinger equation is:

$E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi$

with stationary state solutions:

$\Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N)$

Again, for non-interacting particles the potential is the sum of particle potentials and the wavefunction is a product of the particle wavefuntions:

$\Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(\mathbf{r}_n) \, , \quad V(\mathbf{r}_1,\mathbf{r}_2,\cdots \mathbf{r}_N) = \sum_{n=1}^N V(\mathbf{r}_n)$

and this does not apply to interacting particles.
Following are examples where exact solutions are known. See the main articles for further details.

Hydrogen atom

This form of the Schrödinger equation can be applied to the hydrogen atom:^[21]^[24]

$E \psi = -\frac{\hbar^2}{2\mu}\nabla^2\psi - \frac{e^2}{4\pi\epsilon_0 r}\psi$

where e is the electron charge, r is the position of the electron (r = |r| is the magnitude of the position), the potential term is due to the coloumb interaction, wherein ε₀ is the electric constant (permittivity of free space) and

$\mu = \frac{m_em_p}{m_e+m_p}$

is the 2-body reduced mass of the hydrogen nucleus (just a proton) of mass m_p and the electron of mass m_e. 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 principle interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.
The wavefunction for hydrogen is a function of the electron's coordinates, and in fact can be separated into functions of each coordinate.^[33] Usually this is done in spherical polar coordinates:

$\psi(r,\theta,\phi) = R(r)Y_\ell^m(\theta, \phi) = R(r)\Theta(\theta)\Phi(\phi)$

where R are radial functions and $\scriptstyle Y_{\ell}^{m}(\theta, \phi ) \,$ are spherical harmonics of degree ℓ and order m. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximative methods. The family of solutions are:^[34]

$\psi_{n\ell m}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^{\ell} L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^{m}(\theta, \phi )$

where:

$a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2}$ is the Bohr radius,
$L_{n-\ell-1}^{2\ell+1}(\cdots)$ are the generalized Laguerre polynomials of degree n − ℓ − 1.
n, ℓ, m are the principal, azimuthal, and magnetic quantum numbers respectively: which take the values:

$\begin{align} n & = 1,2,3 \cdots \\ \ell & = 0,1,2 \cdots n-1 \\ m & = -\ell\cdots\ell \end{align}$

NB: generalized Laguerre polynomials are defined differently by different authors—see main article on them and the hydrogen atom.

Two-electron atoms or ions

The equation for any two-electron system, such as the neutral helium atom (He, Z = 2), the negative hydrogen ion (H^–, Z = 1), or the positive lithium ion (Li⁺, Z = 3) is:^[22]

$E\psi = -\hbar^2\left[\frac{1}{2\mu}\left(\nabla_1^2 +\nabla_2^2 \right) + \frac{1}{M}\nabla_1\cdot\nabla_2\right] \psi + \frac{e^2}{4\pi\epsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1}+\frac{1}{r_2} \right) \right] \psi$

where r₁ is the position of one electron (r₁ = |r₁| is its magnitude), r₂ is the position of the other electron (r₂ = |r₂| is the magnitude), r₁₂ = |r₁₂| is the magnitude of the separation between them given by

$|\mathbf{r}_{12}| = |\mathbf{r}_2 - \mathbf{r}_1 | \,\!$

μ is again the two-body reduced mass of an electron with respect to the nucleus of mass M, so this time

$\mu = \frac{m_e M}{m_e+M} \,\!$

and Z is the atomic number for the element (not a quantum number).
The cross-term of two laplacians

$\frac{1}{M}\nabla_1\cdot\nabla_2\,\!$

is known as the mass polarization term, which arises due to the motion of atomic nuclei. The wavefunction is a function of the two electron's positions:

$\psi = \psi(\mathbf{r}_1,\mathbf{r}_2).$

There is no closed form solution for this equation.

Time dependent

This is the equation of motion for the quantum state. In the most general form, it is written:^[28]

$i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi.$

For one particle in one dimension:

$\hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)$

generating the equation:

$i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi + V\Psi$

and solutions:

$\Psi = \Psi(x,t)$

For N particles in one dimension, the Hamiltonian is:

$\begin{align}\hat{H} &= \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N,t) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N,t) \end{align}$

where the position of particle n is x_n, generating the equation:

$i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, .$

with solutions:

$\Psi = \Psi(x_1,x_2\cdots x_N,t)$

For one particle in three dimensions, the Hamiltonian is:

$\begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r},t) \\ & = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \\ \end{align}$

generating the equation:

$i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$

with solutions:

$\Psi = \Psi(\mathbf{r},t)$

For N particles in three dimensions, the Hamiltonian is:

$\begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) \end{align}$

generating the equation:

$i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi$

This last equation is in a very high dimension,^[35] so the solutions

$\Psi = \Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t)$

are not easy to visualize.

Solution methods

General techniques:

Perturbation theory
The variational method
Quantum Monte Carlo methods
Density functional theory
The WKB approximation and semi-classical expansion

Methods for special cases:

List of quantum-mechanical systems with analytical solutions
Hartree–Fock method and post Hartree–Fock methods

Properties

The Schrödinger equation has the following properties: some are useful, but there are shortcomings. Ultimately, these properties arise from the Hamiltonian used, and solutions to the equation.

Linearity

Real energy eigenstates

For the time-independent equation, an additional feature of linearity follows: if two wave functions ψ₁ and ψ₂ are solutions to the time-independent equation with the same energy E, then so is any linear combination:

$\hat H (a\psi_1 + b \psi_2 ) = a \hat H \psi_1 + b \hat H \psi_2 = E (a \psi_1 + b\psi_2).$

Two different solutions with the same energy are called degenerate.^[26]
In an arbitrary potential, if a wave function ψ solves the time-independent equation, so does its complex conjugate ψ*. By taking linear combinations, the real and imaginary parts of ψ are each solutions (if there is no degeneracy they can only differ by a factor). Thus, the time-independent eigenvalue problem can be restricted to real-valued wave functions.
In the time-dependent equation, complex conjugate waves move in opposite directions. If Ψ(x, t) is one solution, then so is Ψ(x, –t). The symmetry of complex conjugation is called time-reversal symmetry.

Space and time derivatives

Continuity of the wavefunction and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.

The Schrödinger equation is first order in time and second in space, which describes the time evolution of a quantum state (meaning it determines the future amplitude from the present).
Explicitly for one particle in 3d Cartesian coordinates – the equation is

$i\hbar{\partial \Psi \over \partial t} = - {\hbar^2\over 2m} \left ( {\partial^2 \Psi \over \partial x^2} + {\partial^2 \Psi \over \partial y^2} + {\partial^2 \Psi \over \partial z^2} \right ) + V(x,y,z,t)\Psi.\,\!$

The first time partial derivative implies the initial value (at t = 0) of the wavefunction

$\Psi(x,y,z,0) \,\!$

is an arbitrary constant. Likewise – the second order derivatives with respect to space implies the wavefunction and its first order spatial derivatives

$\begin{align} & \Psi(x_b,y_b,z_b,t) \\ & \frac{\partial}{\partial x}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial y}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial z}\Psi(x_b,y_b,z_b,t) \end{align} \,\!$

are all arbitrary constants at a given set of points, where x_b, y_b, z_b are a set of points describing boundary b (derivatives are evaluated at the boundaries). Typically there are one or two boundaries, such as the step potential and particle in a box respectively.
As the first order derivatives are arbitrary, the wavefunction can be a continuously differentiable function of space, since at any boundary the gradient of the wavefunction can be matched. The prominent case includes waves.
On the contrary, wave equations in physics are usually second order in time, notable are the family of classical wave equations and the quantum Klein–Gordon equation.

Local conservation of probability

Main articles: Probability current and Continuity equation

The Schrödinger equation is consistent with probability conservation – because it can directly derive the continuity equation for probability:^[36]

${ \partial \over \partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0,$

where

$\rho=|\Psi|^2=\Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t)\,\!$

is the probability density (probability per unit volume, * denotes complex conjugate), and

$\mathbf{j} = {1 \over 2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right)\,\!$

is the probability current (flow per unit area).
Hence predictions from the Schrödinger equation do not violate probability conservation. However, the continuity equation is more fundamental and intuitive than the SE itself, and is always true, while SE is not.

Positive energy

If the potential is bounded from below, meaning there is a minimum value of potential energy, the eigenfunctions of the Schrödinger equation have energy which is also bounded from below. This can be seen most easily by using the variational principle, as follows. (See also below).
For any linear operator $\hat{A}$ bounded from below, the eigenvector with the smallest eigenvalue is the vector ψ that minimizes the quantity

$\langle \psi |\hat{A}|\psi \rangle$

over all ψ which are normalized.^[36] In this way, the smallest eigenvalue is expressed through the variational principle. For the Schrödinger Hamiltonian $\hat{H}$ bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of

$\langle \psi|\hat{H}|\psi\rangle = \int \psi^*(\mathbf{r}) \left[ - \frac{\hbar^2}{2m} \nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r})\right] d^3\mathbf{r} = \int \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 + V(\mathbf{r}) |\psi|^2 \right] d^3\mathbf{r} = \langle \hat{H}\rangle$

(using integration by parts). Due to the complex modulus of ψ squared (which is positive definite), the right hand side always greater than the lowest value of V(x). In particular, the ground state energy is positive when V(x) is everywhere positive.
For potentials which are bounded below and are not infinite over a region, there is a ground state which minimizes the integral above. This lowest energy wavefunction is real and positive definite – meaning the wavefunction can increase and decrease, but is positive for all positions. It physically cannot be negative: if it were, smoothing out the bends at the sign change (to minimize the wavefunction) rapidly reduces the gradient contribution to the integral and hence the kinetic energy, while the potential energy changes linearly and less quickly. The kinetic and potential energy are both changing at different rates, so the total energy is not constant, which can't happen (conservation). The solutions are consistent with Schrödinger equation if this wavefunction is positive definite.
The lack of sign changes also shows that the ground state is nondegenerate, since if there were two ground states with common energy E, not proportional to each other, there would be a linear combination of the two that would also be a ground state resulting in a zero solution.

Analytic continuation to diffusion

The above properties (positive definiteness of energy) allow the analytic continuation of the Schrödinger equation to be identified as a stochastic process. This can be interpreted as the Huygens–Fresnel principle applied to De Broglie waves; the spreading wavefronts are diffusive probability amplitudes.^[36]
For a particle in a random walk (again for which V = 0), the continuation is to let:^[37]

$\tau = it \,, \!$

substituting into the time-dependent Schrödinger equation gives:

${\partial \over \partial \tau} \Psi(\mathbf{r},\tau) = \frac{\hbar}{2m} \nabla ^2 \Psi(\mathbf{r},\tau),$

which has the same form as the diffusion equation, with diffusion coefficient ħ/2m.

Galilean and Lorentz transformations

Non-relativistic

Main article: Galilean invariance

The solutions to the Schrödinger equation are not Galilean invariant, so the equation itself is not either, as outlined below. Changing inertial reference frames requires a transformation of the wavefunction analogous to requiring gauge invariance. This transformation introduces a phase factor that is normally ignored as non-physical, but has application in some problems.^[38]
Galilean transformations (or "boosts") look at the system from the point of view of an observer moving with a steady velocity –v. A boost must change the physical properties of a wavepacket in the same way as in classical mechanics:^[36]

$\mathbf{r}'= \mathbf{r} + \mathbf{v}t \quad \mathbf{p}'= \mathbf{p} + m\mathbf{v}. \,$

For a free-particle solution (plane wave), Galilean invariance requires the phase (complex exponent) to remain in the same form:

$\Psi = Ae^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar} = Ae^{i(\mathbf{p}'\cdot\mathbf{r}' - E' t)/\hbar} \,\!$

but instead the phase terms transform according to

$\begin{align} (\mathbf{p}\cdot\mathbf{r} - E t)/\hbar & \rightarrow \left[(\mathbf{p}' - m\mathbf{v})\cdot(\mathbf{r}' - \mathbf{v}t) - (\mathbf{p}'-m\mathbf{v})\cdot(\mathbf{p}'-m\mathbf{v})t/2m \right]/\hbar \\ & = \left[(\mathbf{p}' \cdot\mathbf{r}' + E' t) + (\mathbf{p} \cdot \mathbf{r} - Et) \right]/\hbar \\ \end{align}$

(energy is kinetic energy, since V = 0) therefore the plane wave transforms by

$\begin{align} \Psi = Ae^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar} & \rightarrow & \Psi' & = Ae^{i(\mathbf{p}'\cdot\mathbf{r}' - E' t)/\hbar} e^{i(\mathbf{p} \cdot \mathbf{r} - Et)/\hbar} \\ & & & = \Psi e^{i(\mathbf{p}'\cdot\mathbf{r}' - E' t)/\hbar} \\ & & & \neq Ae^{i(\mathbf{p}'\cdot\mathbf{r}' - E' t)/\hbar}. \\ \end{align}\,\!$

So by contradiction, the extra phase factor implies the Schrödinger equation is not Galilean invariant: the plane wave solutions do not remain in the same form under Galilean transformations. A linear combination of plane waves, with different values of p and E, will also transform in the same way due to linearity. In general the transformation of any solution to the free-particle Schrödinger equation, Ψ(r, t) results in other "boosted" solutions:

$\Psi'(\mathbf{r}',t) = \Psi(\mathbf{r},t) e^{ i (\mathbf{p}' \cdot \mathbf{r}' - E't)/\hbar} = \Psi(\mathbf{r}' - \mathbf{v}t,t) e^{ i (\mathbf{p}' \cdot \mathbf{r}' - E't)/\hbar}. \,$

Relativistic

Main article: Lorentz invariance

The Lorentz transformations are (slightly) more complicated than the Galilean ones, so the solutions to the Schrödinger equation are certainly not Lorentz invariant either, in turn not consistent with special relativity. Also, as shown above in the plausibility argument – the Schrödinger equation was constructed from classical energy conservation rather than the relativistic energy–momentum relation

$E^2 = (pc)^2 + (m_0c^2)^2 \, .$

This relativistic equation is Lorentz invariant. The classical equation is not – it is the low-velocity limit of the relativistic equation (velocities much less than the speed of light). This further shows that the Schrödinger equation itself, not just the solutions, is not Lorentz invariant.
Secondly, the equation requires the particles to be the same type, and the number of particles in the system to be constant, since their masses are constants in the equation (kinetic energy terms).^[39] This alone means the Schrödinger equation is not compatible with relativity – even the simple equation

$\displaystyle E = mc^2$

allows (in high-energy processes) particles of matter to completely transform into energy by particle–antiparticle annihilation, and enough energy can re-create other particle–antiparticle pairs. So the number of particles and types of particles is not necessarily fixed. For all other intrinsic properties of the particles which may enter the potential function, including mass (such as the harmonic oscillator) and charge (such as electrons in atoms), which will also be constants in the equation, the same problem follows.

Relativistic quantum mechanics

Quantum mechanics applied together with special relativity is relativistic quantum mechanics. The general form of the Schrödinger equation is still applicable, but the Hamiltonian operators are much less obvious, and more complicated.
One wishes to build relativistic wave equations from the relativistic energy–momentum relation. The Klein–Gordon equation and the Dirac equation are two such equations. The KG equation was the first such equation to be obtained, even before the non-relativistic one, and applies to massive spinless particles. The Dirac equation arose from taking the "square root" of the KG equation by factorizing the entire relativistic wave operator into a product of two operators – one of these is the operator for the entire Dirac equation.
The Dirac equation introduces spin matrices, in the particular case of the Dirac equation they are the gamma matrices for spin-1/2 particles, and the solutions are 4-component spinor fields with two components corresponding to the particle and the other two for the antiparticle. In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of position and momentum operators, 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.

Quantum field theory

The general equation is also valid and used in quantum field theory, both in relativistic and non-relativistic situations. However, the solution ψ is no longer interpreted as a "wave", but more like a "field".

Friday, 30 August 2013

Schrödinger equation

Equation

Time-dependent equation

Time-independent equation

Implications

Total, kinetic, and potential energy

Quantization

Measurement and uncertainty

Quantum tunneling

Particles as waves

Interpretation of the wave function

Historical background and development

The wave equation for particles

Assumptions

Solution to equation

Solutions to equation

Wave and particle motion

Non-relativistic quantum mechanics

Time independent

One-dimensional examples

Free particle

Constant potential

Harmonic oscillator

Three-dimensional examples

Hydrogen atom

Two-electron atoms or ions

Time dependent

Solution methods

Properties

Linearity

Real energy eigenstates

Space and time derivatives

Local conservation of probability

Positive energy

Analytic continuation to diffusion

Galilean and Lorentz transformations

Non-relativistic

Relativistic

Relativistic quantum mechanics

Quantum field theory

See also

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