In quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a 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. In all these formulations, they 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; free, bound, or localized. It is not a simple algebraic equation, but (in general) a linear partial differential equation. The differential equation encases the wavefunction of the system, also called the quantum state or state vector.
In the standard interpretation of quantum mechanics, the wavefunction 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 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)
Failed to parse (Missing texvc executable; please see math/README to configure.): i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi
|
where Ψ is the wave function of the quantum system, i is the imaginary unit, ħ is the reduced Planck constant, and Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{H}
is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.
A
wave function which 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):
-
| Time-dependent Schrödinger equation (single non-relativistic particle)
Failed to parse (Missing texvc executable; please see math/README to configure.): i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \frac{-\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t) + V(\mathbf{r},t) \Psi(\mathbf{r},t)
|
where m is the particle's mass, V is its potential energy, ∇2 is the Laplacian, and Ψ is the wavefunction (more precisely, in this context, it is called the "position-space wavefunction"). In words, this equation could be described as "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 wave equation, hence the name "Schrödinger wave 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 wavefunction, which contains information about the system.
[edit] Time-independent equation
Each of these three rows is a wavefunction which satisfies the time-dependent Schrödinger equation for a
harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wavefunction. Right: The
probability distribution of finding the particle with this wavefunction 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 wavefunctions 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)
Failed to parse (Missing texvc executable; please see math/README to configure.): E\Psi=\hat H \Psi
|
In words, the equation states:
-
- When the Hamiltonian operator acts on the wavefunction Ψ, the result might be proportional to the same wavefunction 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)
Failed to parse (Missing texvc executable; please see math/README to configure.): E \Psi(\mathbf{r}) = \frac{-\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r}) + V(\mathbf{r}) \Psi(\mathbf{r})
|
with definitions as above.
The Schrödinger equation, and its solutions, introduced a breakthrough in thinking about physics. His equation was the first of its type, and solutions led to very unusual and unexpected consequences for the time.
The overall form of the equation is not unusual or unexpected. The terms of the nonrelativistic Schrödinger equation can be interpreted as:
-
- (Total energy) = (kinetic energy) + (potential energy)
In this respect, it is just the same as in classical physics. For example, a frictionless roller coaster has constant total energy; therefore it travels slower (low kinetic energy) when it is high off the ground (high gravitational potential energy) and vice versa.
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.
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 a famous example of the 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 wavefunction of a particle. However, even if the wavefunction is known exactly, the result of a specific measurement on the wavefunction is uncertain.
In classical physics, when a ball is rolled slowly towards 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.
Time development of wave packet, as described by the solution to Schrödinger’s equation for a 1-d step potential system, shown in slices of position-time coordinates (the third axis the probability amplitude
Ψ). The particle is shown as the blue circles, the
opacity corresponds to the
probability density of the particle at the location shown. The step potential is the dotted line. The probability of transmission is greater than reflection, 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 fuzzyness of a particle's position illustrated, which is not definite in quantum mechanics.
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 particles can exhibit behavior usually attributed to waves.
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). (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 different states 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.
Propagation of
de Broglie waves in 1d - real part of the
complex amplitude is blue, imaginary part is green. The probability (shown as the colour
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 decreases, so the decreases again, and vice versa - the result is an alternating amplitude: a wave.
The Schrödinger equation provides a way to calculate the possible wavefunctions of a system and how they dynamically change in time. However, the Schrödinger equation does not directly say what, exactly, the wavefunction is. Interpretations of quantum mechanics address questions such as what the relation is between the wavefunction, 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.
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.
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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 quantisation of angular momentum:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): n \lambda = 2 \pi r.\,
This approach essentially confined the electron wave in one dimension, along a circular orbit.
Following up on these 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.[8] A modern version of his reasoning is reproduced below. The equation he found is:[9]
- Failed to parse (Missing texvc executable; please see math/README to configure.): i\hbar \frac{\partial}{\partial t}\Psi(\bold{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\bold{r},\,t) + V(\bold{r})\Psi(\bold{r},\,t).
However, by that time, Arnold Sommerfeld had refined the Bohr model with relativistic corrections.[10][11] 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):
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.[12]
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.[13] 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[14]
|
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.[15]
The Schrödinger equation details the behaviour 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.[16] 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.[17] 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.[18]
The Schrödinger equation was developed principally from the De Broglie hypothesis, a wave equation that would describe particles,[19] and can be constructed in the following way.[20] For a more rigorous mathematical derivation of Schrödinger's equation, see also.[21]
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): E = \frac{p^2}{2m}+V(x,t)=H.
For three dimensions, the position vector r and momentum vector p must be used:
- Failed to parse (Missing texvc executable; please see math/README to configure.): E = \frac{\bold{p}\cdot\bold{p}}{2m}+V(\bold{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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): E=\sum_{n=1}^N \frac{\bold{p}_n\cdot\bold{p}_n}{2m_n} + V(\bold{r}_1,\bold{r}_2\cdots\bold{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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): p = \frac{h}{\lambda} = \hbar k\;,
in three dimensions:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 some function of the state of the system - the wavefunction (denoted Ψ). The wavefunction 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 wavefunction can be used to predict how the particles will behave under the influence of the specified potential and with each other.
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 wavefunction is the displacement of matter, and electromagnetic waves from Maxwell's equations, where the wavefunctions 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 wavefunction is a description of the system.[22] 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[23]
|
Wave-particle duality follows from the Schrödinger equation, as presented below.
The Planck-Einstein and De Broglie relations
- Failed to parse (Missing texvc executable; please see math/README to configure.): E=\hbar\omega, \quad \bold{p}=\hbar\bold{k}
illuminate the deep connections between space with momentum, and energy with time. This is more apparent using natural units, setting ħ = 1 makes these equations into identities:
- Failed to parse (Missing texvc executable; please see math/README to configure.): E=\omega, \quad \bold{p}=\bold{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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = Ae^{i(\bold{k}\cdot\bold{r}-\omega t)} = Ae^{i(\bold{p}\cdot\bold{r}-Et)/\hbar}
and to realize that the first order partial derivatives with respect to space
- Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla\Psi = \dfrac{i}{\hbar}\bold{p}Ae^{i(\bold{p}\cdot\bold{r}-Et)/\hbar} = \dfrac{i}{\hbar}\bold{p}\Psi
and time
- Failed to parse (Missing texvc executable; please see math/README to configure.): \dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} Ae^{i(\bold{p}\cdot\bold{r}-Et)/\hbar} = -\dfrac{i E}{\hbar} \Psi
imply the derivatives
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} -i\hbar\nabla\Psi = \bold{p}\Psi & \rightarrow & -\dfrac{\hbar^2}{2m}\nabla^2\Psi = \dfrac{1}{2m}\bold{p}\cdot\bold{p}\Psi \\ \dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} \Psi & \rightarrow & i\hbar\dfrac{\partial \Psi}{\partial t} = E \Psi \\ \end{matrix}
Multiplying the energy equation by Ψ
- Failed to parse (Missing texvc executable; please see math/README to configure.): E= \dfrac{\bold{p}\cdot\bold{p}}{2m}+V \rightarrow E\Psi= \dfrac{\bold{p}\cdot\bold{p}}{2m}\Psi+V\Psi
immediately lead Schrödinger to his equation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): i\hbar\dfrac{\partial \Psi}{\partial t}= -\dfrac{\hbar^2}{2m}\nabla^2\Psi +V\Psi
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,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{E}= i\hbar\dfrac{\partial}{\partial t}
and the momentum operator, corresponding to the spatial derivatives (the gradient),
- Failed to parse (Missing texvc executable; please see math/README to configure.): \bold{\hat{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. Substitution of these operators in the energy equation and multiplication by Ψ returns the same wave equation. 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.
Given the derivatives, wave-particle duality can be assessed as follows. The kinetic energy T is related to the square of momentum p, and hence the magnitude of the second spatial derivatives, so the kinetic energy is also proportional to the magnitude of the curvature of the wave. As the particle's momentum increases, the kinetic energy increases more rapidly, but the since the wavenumber k increases the wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \lambda
decreases, mathematically
- Failed to parse (Missing texvc executable; please see math/README to configure.): \bold{p}\cdot\bold{p} \propto \bold{k}\cdot\bold{k} \propto T \propto |\nabla^2\Psi| \propto \dfrac{1}{\lambda^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.[19]
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi(\bold{r},t) = \sum_{n=1}^\infty A_n e^{i(\bold{k}_n\cdot\bold{r}-\omega_n t)} \,\!
and for continuous k the sum becomes an integral, the Fourier transform of a momentum space wavefunction:[24]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi(\bold{r},t) = \frac{1}{(\sqrt{2\pi})^3}\int\Phi(\bold{k})e^{i(\bold{k}\cdot\bold{r}-\omega t)}d^3\bold{k} \,\!
where d3k = dkxdkydkz 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.
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 (not just plane waves) at position r with wavevector k will move along the trajectory determined by classical mechanics in the limit that the wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \lambda \,\!
is small, i.e. for a large k and therefore large p in comparison to Planck's reduced constant ħ. Equivalently, in the limit ħ becomes zero, the equations of classical mechanics are restored from quantum mechanics.[25]
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 px (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
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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)
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 qi 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).[25]
Substituting
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
Following are several forms of Schrödinger's equation 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.[21]
Diagrammatic summary of the quantities related to the wavefunction, as used in De broglie's hypothesis and development of the Schrödinger equation.
[19]
If the Hamiltonian is not an explicit function of time, the equation is separable into its spatial and temporal parts. Hence the energy operator Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{H}
[26]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \hat H \psi = E \psi
A solution of the time-independent equation is called an energy eigenstate with energy E.[24]
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left.i\hbar \frac{\partial}{\partial t} \Psi(\bold{r},t)\right|_{t=0}= \left.H \Psi(\bold{r},t)\right|_{t=0} =E \Psi(\bold{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,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi(\bold{r},t)= \tau(t) \psi(\bold{r}) \,
substituting for Ψ:
- Failed to parse (Missing texvc executable; please see math/README to configure.): i\hbar \frac{\partial \Psi}{\partial t } = E \Psi \rightarrow i\hbar \psi(\bold{r})\frac{\partial\tau(t)}{\partial t } = E \tau(t)\psi(\bold{r}) \,
where the ψ(r) cancels, so solving this equation for Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \tau(t)\,\!
implies the solution of the time-dependent equation with this initial condition is:[9]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi(\bold{r},t) = \psi(\bold{r}) e^{-i{E t/\hbar}} = \psi(\bold{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 Eigen states 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 Plank-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.
Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.
|
One particle |
N particles |
| One dimension |
Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \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 xn.
|
| Failed to parse (Missing texvc executable; please see math/README to configure.): E\Psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\Psi + V\Psi |
Failed to parse (Missing texvc executable; please see math/README to configure.): E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, . |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \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):[27]
Failed to parse (Missing texvc executable; please see math/README to configure.): \| \psi \|^2 = \int |\psi(x)|^2\, dx.\,
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N)
for non-interacting particles
Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , \quad V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .
|
| Three dimensions |
Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align}\hat{H} & = \frac{\bold{\hat{p}}\cdot\bold{\hat{p}}}{2m} + V(\bold{r}) \\ & = -\frac{\hbar^2}{2m}\nabla^2 + V(\bold{r}) \\ \end{align}
where the position of the particle is r = (x, y, z).
|
Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\bold{\hat{p}}_n\cdot\bold{\hat{p}}_n}{2m_n} + V(\bold{r}_1,\bold{r}_2,\cdots\bold{r}_N) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\bold{r}_1,\bold{r}_2,\cdots\bold{r}_N) \end{align}
where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is
Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla_n^2=\frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}
|
| Failed to parse (Missing texvc executable; please see math/README to configure.): E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi |
Failed to parse (Missing texvc executable; please see math/README to configure.): E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = \psi(\bold{r}) e^{-iEt/\hbar} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = e^{-iEt/\hbar}\psi(\bold{r}_1,\bold{r}_2\cdots \bold{r}_N)
for non-interacting particles
Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(\bold{r}_n) \, , \quad V(\bold{r}_1,\bold{r}_2,\cdots \bold{r}_N) = \sum_{n=1}^N V(\bold{r}_n)
|
Following are examples where exact solutions are known. See the main articles for further details.
For no potential, V = 0, so the particle is free and the equation reads:[28]
- Failed to parse (Missing texvc executable; please see math/README to configure.): - E \psi = \frac{\hbar^2}{2m}{d^2 \psi \over d x^2}\,
which has oscillatory solutions for E > 0 (the Cn are arbitrary constants):
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
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.[24]
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.
The Schrödinger equation for this situation is
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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,[29] and atoms or ions in lattices,[30] 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Hn are the Hermite polynomials.
This form of the Schrödinger equation can be applied to the Hydrogen atom:[19][22]
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mu = \frac{m_em_p}{m_e+m_p}
is the 2-body reduced mass of the Hydrogen nucleus (just a proton) of mass mp and the electron of mass me. 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.[31] Usually this is done in spherical polar coordinates:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(r,\theta,\phi) = R(r)Y_\ell^m(\theta, \phi) = R(r)\Theta(\theta)\Phi(\phi)
where R are radial functions and Failed to parse (Missing texvc executable; please see math/README to configure.): \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:[32]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi_{n\ell m}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]^3} } 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2}
is the Bohr radius,
- Failed to parse (Missing texvc executable; please see math/README to configure.): L_{n-\ell-1}^{2\ell+1}(\cdots)
are the generalized Laguerre polynomials of degree n − ℓ − 1.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
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:[20]
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): |\bold{r}_{12}| = |\bold{r}_2 - \bold{r}_1 | \,\!
μ is again the two-body reduced mass of an electron with respect to the nucleus of mass M, so this time
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi = \psi(\bold{r}_1,\bold{r}_2)
There is no closed form solution for this equation.
This is the equation of motion for the quantum state. In the most general form, it is written:[26]
- Failed to parse (Missing texvc executable; please see math/README to configure.): i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.
|
One particle |
N particles |
| One dimension |
Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \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 xn.
|
| Failed to parse (Missing texvc executable; please see math/README to configure.): i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi + V\Psi |
Failed to parse (Missing texvc executable; please see math/README to configure.): 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 \, . |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = \Psi(x,t) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = \Psi(x_1,x_2\cdots x_N,t) |
| Three dimensions |
Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align}\hat{H} & = \frac{\bold{\hat{p}}\cdot\bold{\hat{p}}}{2m} + V(\bold{r},t) \\ & = -\frac{\hbar^2}{2m}\nabla^2 + V(\bold{r},t) \\ \end{align} |
Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\bold{\hat{p}}_n\cdot\bold{\hat{p}}_n}{2m_n} + V(\bold{r}_1,\bold{r}_2,\cdots\bold{r}_N,t) \\ & = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2 + V(\bold{r}_1,\bold{r}_2,\cdots\bold{r}_N,t) \end{align} |
| Failed to parse (Missing texvc executable; please see math/README to configure.): i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi |
Failed to parse (Missing texvc executable; please see math/README to configure.): 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,[33] so the solutions are not easy to visualize.
|
| Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = \Psi(\bold{r},t) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = \Psi(\bold{r}_1,\bold{r}_2,\cdots\bold{r}_N,t) |
|
|
Methods for special cases:
|
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.
In the development above, the Schrödinger equation was made to be linear for generality, though this has other implications. If two wave functions ψ1 and ψ2 are solutions, then so is any linear combination of the two:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \psi = a\psi_1 + b \psi_2
where a and b are any complex numbers (the sum can be extended for any number of wavefunctions). This property allows superpositions of quantum states to be solutions of the Schrödinger equation. In particular a given solution can be multiplied by any complex number, this allows one to solve for a wave function without normalizing it first.
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.[24]
In an arbitrary potential, there is one degeneracy: 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. 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.
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
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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 xb, yb, zb 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 equation and the quantum Klein–Gordon equation.
The Schrödinger equation is consistent with probability conservation - because it can directly derive the continuity equation for probability:[34]
- Failed to parse (Missing texvc executable; please see math/README to configure.): { \partial \over \partial t} \rho\left(\bold{r},t\right) + \nabla \cdot \mathbf{j} = 0,
where
- Failed to parse (Missing texvc executable; please see math/README to configure.): \rho=|\Psi|^2=\Psi^*(\bold{r},t)\Psi(\bold{r},t)\,\!
is the probability density (probability per unit volume, * denotes complex conjugate), and
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{j} = {i\hbar \over 2m} \left( \Psi\nabla \Psi^* - \Psi^*\nabla \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.
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 Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{A}
bounded from below, the eigenvector with the smallest eigenvalue is the vector ψ that minimizes the quantity
- Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \psi |\hat{A}|\psi \rangle
over all ψ which are normalized.[34] In this way, the smallest eigenvalue is expressed through the variational principle. For the Schrödinger Hamiltonian Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{H}
bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of
- Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \psi|\hat{H}|\psi\rangle = \int \psi^*(\bold{r}) \left[ - \frac{\hbar^2}{2m} \nabla^2\psi(\bold{r}) + V(\bold{r})\psi(\bold{r})\right] d^3\bold{r} = \int \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 + V(\bold{r}) |\psi|^2 \right] d^3\bold{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.
The above properties (positive definiteness of energy) allow the analytic continuation of the Schrödinger equation to be identified as a stochastic process, which can be represented by a path integral. This can be interpreted as the Huygens–Fresnel principle applied to De Broglie waves, the spreading wavefronts are diffusive probability amplitudes.[34]
For a particle which is random walking (again for which V = 0), the continuation is to let:[35]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = it \,\!
so substituting into the time-dependent Schrödinger equation gives:
- Failed to parse (Missing texvc executable; please see math/README to configure.): {\partial \over \partial \tau} \Psi(\bold{r},\tau) = \frac{\hbar}{2m} \nabla ^2 \Psi(\bold{r},\tau)
which has the same form as the Diffusion equation, with diffusion coefficient ħ/2m.
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.[36]
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:[34]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \bold{r}'= \bold{r} + \bold{v}t \quad \bold{p}'= \bold{p} + m\bold{v} \,
For a free-particle solution (plane wave), Galilean invariance requires the phase (complex exponent) to remain in the same form:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi = Ae^{i(\bold{p}\cdot\bold{r} - E t)/\hbar} = Ae^{i(\bold{p}'\cdot\bold{r}' - E' t)/\hbar} \,\!
but instead the phase terms transform according to
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} (\bold{p}\cdot\bold{r} - E t)/\hbar & \rightarrow \left[(\bold{p}' - m\bold{v})\cdot(\bold{r}' - \bold{v}t) - (\bold{p}'-m\bold{v})\cdot(\bold{p}'-m\bold{v})t/2m \right]\hbar \\ & = \left[(\bold{p}' \cdot\bold{r}' + E' t) + (\bold{p} \cdot \bold{r} - Et) \right]/\hbar \\ \end{align}
(energy is kinetic energy, since V = 0) therefore the plane wave transforms by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \Psi = Ae^{i(\bold{p}\cdot\bold{r} - E t)/\hbar} & \rightarrow & \Psi' & = Ae^{i(\bold{p}'\cdot\bold{r}' - E' t)/\hbar} e^{i(\bold{p} \cdot \bold{r} - Et)/\hbar} \\ & & & = \Psi e^{i(\bold{p}'\cdot\bold{r}' - E' t)/\hbar} \\ & & & \neq Ae^{i(\bold{p}'\cdot\bold{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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Psi'(\bold{r}',t) = \Psi(\bold{r},t) e^{ i (\bold{p}' \cdot \bold{r}' - E't)/\hbar} = \Psi(\bold{r}' - \bold{v}t,t) e^{ i (\bold{p}' \cdot \bold{r}' - E't)/\hbar} \,
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 mass–energy relation
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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).[37] This alone means the Schrödinger equation is not compatible with relativity - even the simple equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
In order to extend Schrödinger's formalism to include relativity, the physical picture must be transformed. The Klein–Gordon equation and the Dirac equation are built from the relativistic mass–energy relation; so as a result these equations are relativistically invariant, and replace the Schrödinger equation in relativistic quantum mechanics. In attempt to extend the scope of these equations further, other relativistic wave equations have developed. By no means is the Schrödinger equation obsolete: it is still in use for both teaching and research - particularly in physical and quantum chemistry to understand the properties of atoms and molecules, but understood to be an approximation to real behaviour of them, for speeds much less than light.
- ^ Schrödinger, E. (1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules". Physical Review 28 (6): 1049–1070. Bibcode 1926PhRv...28.1049S. DOI:10.1103/PhysRev.28.1049. Archived from the original on 2008-12-17. http://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf.
- ^ Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Kluwer Academic/Plenum Publishers. p. 143. ISBN 978-0-306-44790-7.
- ^ Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman
- ^ O Donati G F Missiroli G Pozzi May 1973 An Experiment on Electron Interference American Journal of Physics 41 639–644
- ^ Brian Greene, The Elegant Universe, p. 110
- ^ Feynman Lectures on Physics (Vol. 3), R. Feynman, R.B. Leighton, M. Sands, Addison-Wesley, 1965, ISBN 0-201-02118-8
- ^ de Broglie, L. (1925). "Recherches sur la théorie des quanta [On the Theory of Quanta]". Annales de Physique 10 (3): 22–128. http://tel.archives-ouvertes.fr/docs/00/04/70/78/PDF/tel-00006807.pdf. Translated version.
- ^ Schrodinger, E. (1984). Collected papers. Friedrich Vieweg und Sohn. ISBN 3-7001-0573-8. See introduction to first 1926 paper.
- ^ a b Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) ISBN 0-89573-752-3
- ^ Sommerfeld, A. (1919). Atombau und Spektrallinien. Braunschweig: Friedrich Vieweg und Sohn. ISBN 3-87144-484-7.
- ^ For an English source, see Haar, T.. The Old Quantum Theory.
- ^ Rhodes, R. (1986). Making of the Atomic Bomb. Touchstone. ISBN 0-671-44133-7.
- ^ Schrödinger, E. (1926). "Quantisierung als Eigenwertproblem; von Erwin Schrödinger". Annalen der Physik, (Leipzig): 361–377. http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination.
- ^ Erwin Schrödinger, "The Present situation in Quantum Mechanics," p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in in Proceedings of the American Philosophical Society, 124, 323-38. It later appeared as Section I.11 of Part I of Quantum Theory and Measurement by J.A. Wheeler and W.H. Zurek, eds., Princeton University Press, New Jersey 1983).
- ^ Einstein, A.; et. al.. Letters on Wave Mechanics: Schrodinger-Planck-Einstein-Lorentz.
- ^ Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. p. 219. ISBN 0-521-43767-9.
- ^ Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. p. 220. ISBN 0-521-43767-9.
- ^ It is clear that even in his last year of life, as shown in a letter to Max Born, that Schrödinger never accepted the Copenhagen interpretation (cf. p. 220). Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. p. 479. ISBN 0-521-43767-9.
- ^ a b c d Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1
- ^ a b Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2
- ^ a b Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
- ^ a b Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0
- ^ The New Quantum Universe, T.Hey, P.Walters, Cambridge University Press, 2009, ISBN 978-0-521-56457-1
- ^ a b c d Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9
- ^ a b Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
- ^ a b Shankar, R. (1994). Principles of Quantum Mechanics. Kluwer Academic/Plenum Publishers. pp. 143ff. ISBN 978-0-306-44790-7.
- ^ Feynman, R.P.; Leighton, R.B.; Sand, M. (1964). "Operators". The Feynman Lectures on Physics. 3. Addison-Wesley. pp. 20–7. ISBN 0-201-02115-3.
- ^ Shankar, R. (1994). Principles of Quantum Mechanics. Kluwer Academic/Plenum Publishers. pp. 151ff. ISBN 978-0-306-44790-7.
- ^ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7
- ^ Solid State Physics (2nd Edition), J.R. Hook, H.E. Hall, Manchester Physics Series, John Wiley & Sons, 2010, ISBN 978-0-471-92804-1
- ^ Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7
- ^ David Griffiths (2008). Introduction to elementary particles. Wiley-VCH. pp. 162–. ISBN 978-3-527-40601-2. http://books.google.com/books?id=w9Dz56myXm8C&pg=PA162. Retrieved 27 June 2011.
- ^ Shankar, R. (1994). Principles of Quantum Mechanics. Kluwer Academic/Plenum Publishers. p. 141. ISBN 978-0-306-44790-7.
- ^ a b c d Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
- ^ http://www.stt.msu.edu/~mcubed/Relativistic.pdf
- ^ van Oosten, A. B. (2006). "Covariance of the Schrödinger equation under low velocity boost". Apeiro 13 (2): 449–454.
- ^ Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
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