Schrödinger_equation

Quantum mechanics

$\backslash Delta\; x\; \backslash Delta\; p\; \backslash ge\; \backslash frac\{\backslash hbar\}\{2\}$

Uncertainty principle

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For a non-technical introduction to the topic, please see Introduction to quantum mechanics.

In physics, especially quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system varies. According to the Copenhagen interpretation of quantum mechanics, the state vector is used to calculate the probability that a physical system is in a given quantum state. Schrödinger\'s equation is primarily applied to microscopic systems, such as electrons and atoms, but is sometimes applied to macroscopic systems (such as the whole universe). The equation is named after physicist Erwin Schrödinger who proposed the equation in 1926.Schrödinger, Erwin (December 1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules" (PDF). Phys. Rev. 28 (6): 1049 - 1070. 

The Schrödinger equation is commonly written as an operator equation describing how the state vector evolves over time. By specifying the total energy (Hamiltonian) of the quantum system, Schrödinger\'s equation can be solved, the solutions being quantum states.

The Schrödinger equation is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton\'s second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei.

Contents 1 Historical background and development 2 Mathematical forms 2.1 Time-dependent Schrödinger equation 2.2 Time-independent Schrödinger equation 2.3 Bra-ket versions 3 Properties 3.1 Linearity 3.2 Conservation of probability 3.3 Correspondence principle 4 Solutions 5 Free particle Schrödinger equation 6 Relativistic generalisations 7 Applications 8 See also 9 References 9.1 Modern reviews 10 External links

Historical background and development

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Main article: Theoretical and experimental justification for the Schrödinger equation

Although it can\'t be derived from classical arguments, a heuristic derivation of Schrödinger\'s equation follows very naturally from earlier developments:

Heuristic derivation

Assumptions:

• The total energy E of a particle is

$E=\backslash frac\{p^2\}\{2m\}+V$

This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the the kinetic energy, $\backslash frac\{p^2\}\{2m\}$, and the potential energy V. p is the momentum of the particle or mass times velocity. The potential energy is assumed to vary with position, and possibly time as well.

Note that the energy E and momentum p appear in the following two relations:

• Einstein\'s hypothesis of 1905:

$E\; =\; h\; f\backslash ;$

where the frequency f of the quanta of radiation (photons) are related by Planck\'s constant h.

• The de Broglie hypothesis of 1924:

$p=h\; /\; \backslash lambda\backslash ;$

where $\backslash lambda\backslash ,$ is the wavelength of the wave. This hypothesis also requires:

• The association of a wave (with wavefunction $\backslash psi$) with any particle.

Combining the above assumptions yields Schrödinger\'s equation:

Expressed in terms of angular frequency $\backslash omega\; =\; 2\backslash pi\; f\backslash ;$ and wavenumber $k\; =\; 2\backslash pi\; /\; \backslash lambda\backslash ;$, with $\backslash hbar\; =\; h\; /\; 2\; \backslash pi\backslash ;$ we get:

$E=\backslash hbar\; \backslash omega$

and

$\backslash mathbf\{p\}=\backslash hbar\; \backslash mathbf\{k\}\backslash ;$

where we have expressed p and k as vectors.

Schrödinger\'s great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:

$\backslash psi\; \backslash approx\; e^\{i(\backslash mathbf\{k\}\backslash cdot\backslash mathbf\{x\}-\; \backslash omega\; t)\}$

and to realize that since

$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash psi\; =\; -i\backslash omega\; \backslash psi$

then

$E\; \backslash psi\; =\; \backslash hbar\; \backslash omega\; \backslash psi\; =\; i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash psi$

and similarly since:

$\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash psi\; =\; i\; k\_x\; \backslash psi$

then

$p\_x\; \backslash psi\; =\; \backslash hbar\; k\_x\; \backslash psi\; =\; -i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash psi$

and hence:

$p\_x^2\; \backslash psi\; =\; -\backslash hbar^2\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; \backslash psi$

so that, again for a plane wave, he got:

$p^2\; \backslash psi\; =\; (p\_x^2\; +\; p\_y^2\; +\; p\_z^2)\; \backslash psi\; =\; -\backslash hbar^2\backslash left(\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; z^2\}\backslash right)\; \backslash psi\; =\; -\backslash hbar^2\backslash nabla^2\; \backslash psi$

And by inserting these expressions for the energy and momentum into the classical mechanics formula we started with we get Schrödinger\'s famed equation for a single particle in the 3-dimensional case in the presence of a potential V:

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

Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom\'s single negatively charged electron as a wave, $\backslash psi\backslash ;$, moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment,^{[citation needed]} the Bohr model^{[citation needed]} and also the results of Werner Heisenberg\'s matrix mechanics^{[citation needed]} - but without having to introduce Heisenberg\'s concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a series of four papers in 1926.^{[citations needed]}

The Schrödinger equation defines the behaviour of $\backslash psi\backslash ;$, but does not interpret what $\backslash psi\backslash ;$ is. Schrödinger tried unsuccessfully to interpret it as a charge density.^{[citation needed]} In 1926 Max Born, just a few days after Schrödinger\'s fourth and final paper was published, successfully interpreted $\backslash psi\backslash ;$ as a probability amplitude,^{[citation needed]} although Schrödinger was never reconciled to this statistical or probabilistic approach.^{[citation needed]}

Mathematical forms

There are various ways of writing Schrödinger\'s equation, depending on the precise mathematical framework used and whether the wavefunction varies over time.

Time-dependent Schrödinger equation

In operator form, time-dependent Schrödinger equation for a system with total energy $\backslash hat\; H$ is,

$$

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

where $\backslash psi$ is the wavefunction, $\backslash hbar$ is Planck\'s constant and $i$ is the imaginary unit. As with the force occurring in Newton\'s second law, the form of the Hamiltonian is not provided by the Schrödinger equation, but must be independently determined from the physical properties of the system.

As a standard example, a non-relativistic particle with no electric charge and zero spin has a Hamiltonian which is the sum of the kinetic (T) and potential (U) energies :

$\backslash hat\; H\; =\; \backslash left(T\; +\; U\backslash right)\; \backslash \; =\; -\; \backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash nabla^2\; +\; U\backslash left(\backslash mathbf\{r\}\backslash right)$

The Schrödinger equation can then be written explicitly as a partial differential equation

$-\backslash frac\{\backslash hbar^2\}\{2\; m\}\; \backslash left[\backslash frac\{\backslash partial^2\; \backslash psi\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\; \backslash psi\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\; \backslash psi\}\{\backslash partial\; z^2\}\; \backslash right]\; +\; U\; \backslash psi\; =\; i\; \backslash hbar\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; t\}$

where the dependence of $\backslash psi$ on the space and time coordinates has been suppressed for clarity.

Time-independent Schrödinger equation

For many real-world problems the Hamiltonian does not depend on time. Denoting this constant energy by $E$ results in the time-independent Schrödinger equation

$\backslash hat\; H\; \backslash psi\; =\; E\; \backslash psi\; \backslash ,$

Together with Schrödinger\'s time-dependent equation in operator form, this gives,

$i\; \backslash hbar\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; t\}\; =\; E\; \backslash psi$

which can be solved for $\backslash psi$ as

$\backslash psi\backslash left(\backslash mathbf\{r\},\; t\backslash right)\; =\; \backslash phi\backslash left(\backslash mathbf\{r\}\backslash right)\; e^\{-iEt/\backslash hbar\}$

where $\backslash phi\backslash left(\backslash mathbf\{r\}\backslash right)$ is the value of $\backslash psi$ at $t=0$. Also, when such a given solution - representing a stationary state - is substituted into the time-dependent Schrödinger equation, the resulting equation isAn initial condition must be used here, namely, that at time zero the wavefunction must be an eigenstate of $\backslash hat\; H.$ the time-independent Schrödinger equation.

An example of a one-dimensional time-independent Schrödinger equation for a particle of mass m, moving in a potential U(x) is: [1]

$-\backslash frac\{\backslash hbar^2\}\{2\; m\}\; \backslash frac\{d^2\; \backslash phi\; (x)\}\{dx^2\}\; +\; U(x)\; \backslash phi\; (x)\; =\; E\; \backslash phi\; (x).$

The analogous 3-dimensional time-independent equation is, [2]:

$\backslash left[-\backslash frac\{\backslash hbar^2\}\{2\; m\}\; \backslash nabla^2\; +\; U(\backslash mathbf\{r\})\; \backslash right]\; \backslash phi\; (\backslash mathbf\{r\})\; =\; E\; \backslash phi\; (\backslash mathbf\{r\}),$

where $\backslash nabla^2$ is the Laplace operator.

Bra-ket versions

In the mathematical formulation of quantum mechanics, a physical system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a ray (ket) in that space. A state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. It contains all information of the system that is knowable in a quantum mechanical sense. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.

The time-dependent Schrödinger equation can be written using Dirac\'s bra-ket notation as,

$\backslash hat\; H(t)\backslash left|\backslash psi\backslash left(t\backslash right)\backslash right\backslash rangle\; =\; \backslash mathrm\{i\}\backslash hbar\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; \backslash psi\; \backslash left(t\backslash right)\; \backslash right\backslash rangle$

where $|\backslash psi(t)\backslash rangle$ is a ket, $\backslash hbar$ is the reduced Planck\'s constant and $\backslash hat\; H(t)$ is the Hamiltonian (a self-adjoint operator acting on the state space).

The nonzero elements of a Hilbert space are by definition normalizable and it is convenient, although not necessary, to represent a state by an element of the ray which is normalized to unity. This vector is often somewhat loosely referred to as a wavefunction, although in a more rigorous formulation of quantum mechanics a wavefunction is a special case of a state vector. (In fact, a wavefunction is a state in the position representation, see below).

For every time-independent Hamiltonian operator, $\backslash hat\; H$, there exists a set of quantum states, $\backslash left|\backslash psi\_n\backslash right\backslash rang$, known as energy eigenstates, and corresponding real numbers $E\_n$ satisfying the eigenvalue equation,

$\backslash hat\; H\; \backslash left|\backslash psi\_n\backslash right\backslash rang\; =\; E\_n\; \backslash left|\backslash psi\_n\; \backslash right\backslash rang.$

Alternatively, $\backslash psi$ is said to be an eigenstate (eigenket) of $\backslash hat\; H$ with eigenvalue $E$. Such a state possesses a definite total energy, whose value $E\_n$ is the eigenvalue of the Hamiltonian. The corresponding eigenvector $\backslash psi\_n\backslash ,$ is normalizable to unity. This eigenvalue equation is referred to as the time-independent Schrödinger equation. We purposely left out the variable(s) on which the wavefunction $\backslash psi\_n\backslash ,$ depends.

Self-adjoint operators, such as the Hamiltonian, have the property that their eigenvalues are always real numbers, as we would expect, since the energy is a physically observable quantity. Sometimes more than one linearly independent state vector correspond to the same energy $E\_n$. If the maximum number of linearly independent eigenvectors corresponding to $E\_n$ equals k, we say that the energy level $E\_n$ is k-fold degenerate. When k=1 the energy level is called non-degenerate.

On inserting a solution of the time-independent Schrödinger equation into the full Schrödinger equation, we get

$\backslash mathrm\{i\}\; \backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash left|\; \backslash psi\_n\; \backslash left(t\backslash right)\; \backslash right\backslash rangle\; =\; E\_n\; \backslash left|\backslash psi\_n\backslash left(t\backslash right)\backslash right\backslash rang.$

It is relatively easy to solve this equation. One finds that the energy eigenstates (i.e., solutions of the time-independent Schrödinger equation) change as a function of time only trivially, namely, only by a complex phase:

$\backslash left|\; \backslash psi\; \backslash left(t\backslash right)\; \backslash right\backslash rangle\; =\; \backslash mathrm\{e\}^\{-\backslash mathrm\{i\}\; Et\; /\; \backslash hbar\}\; \backslash left|\backslash psi\backslash left(0\backslash right)\backslash right\backslash rang.$

It immediately follows that the probability amplitude,

$\backslash psi(t)^*\backslash psi(t)\; =\; \backslash mathrm\{e\}^\{\backslash mathrm\{i\}\; Et\; /\; \backslash hbar\}\backslash mathrm\{e\}^\{-\backslash mathrm\{i\}\; Et\; /\; \backslash hbar\}$

\psi(0)^*\psi(0) = |\psi(0)|^2, is time-independent. Because of a similar cancellation of phase factors in bra and ket, all average (expectation) values of time-independent observables (physical quantities) computed from $\backslash psi(t)\backslash ,$ are time-independent.

Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors $\backslash left\backslash \{\backslash left|n\backslash right\backslash rang\backslash right\backslash \}$ form a basis for the state space. We introduced here the short-hand notation $|\backslash ,n\backslash ,\backslash rang\; =\; \backslash psi\_n$. Then any state vector that is a solution of the time-dependent Schrödinger equation (with a time-independent $\backslash hat\; H$) $\backslash left|\backslash psi\backslash left(t\backslash right)\backslash right\backslash rang$ can be written as a linear superposition of energy eigenstates:

$\backslash left|\backslash psi\backslash left(t\backslash right)\backslash right\backslash rang\; =\; \backslash sum\_n\; c\_n(t)\; \backslash left|n\backslash right\backslash rang\; \backslash quad,\backslash quad\; \backslash hat\; H\; \backslash left|n\backslash right\backslash rang\; =\; E\_n\; \backslash left|n\backslash right\backslash rang\; \backslash quad,\backslash quad\; \backslash sum\_n\; \backslash left|c\_n\backslash left(t\backslash right)\backslash right|^2\; =\; 1.$

(The last equation enforces the requirement that $\backslash left|\backslash psi\backslash left(t\backslash right)\backslash right\backslash rang$, like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the time-dependent Schrödinger equation in the left-hand side and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain

$\backslash mathrm\{i\}\backslash hbar\; \backslash frac\{\backslash partial\; c\_n\}\{\backslash partial\; t\}\; =\; E\_n\; c\_n\backslash left(t\backslash right).$

Therefore, if we know the decomposition of $\backslash left|\backslash psi\backslash left(t\backslash right)\backslash right\backslash rang$ into the energy basis at time $t\; =\; 0$, its value at any subsequent time is given simply by

$\backslash left|\backslash psi\backslash left(t\backslash right)\backslash right\backslash rang\; =\; \backslash sum\_n\; \backslash mathrm\{e\}^\{-\backslash mathrm\{i\}E\_nt/\backslash hbar\}\; c\_n\backslash left(0\backslash right)\; \backslash left|n\backslash right\backslash rang.$

Note that when some values $c\_n(0)\backslash ,$ are not equal to zero for differing energy values $E\_n\backslash ,$, the left-hand side is not an eigenvector of the energy operator $\backslash hat\; H$. The left-hand is an eigenvector when the only $c\_n(0)\backslash ,$-values not equal to zero belong the same energy, so that $\backslash mathrm\{e\}^\{-\backslash mathrm\{i\}E\_nt/\backslash hbar\}$ can be factored out. In many real-world application this is the case and the state vector $\backslash psi(t)\backslash ,$ (containing time only in its phase factor) is then a solution of the time-independent Schrödinger equation.

Example

Let $|\backslash ,1\backslash ,\backslash rangle$ and $|\backslash ,2\backslash ,\backslash rangle$ be degenerate eigenstates of the time-independent Hamiltonian $\backslash hat\; H\backslash ,$:

$$

\hat H\,|\,1\,\rangle = E |\,1\,\rangle \quad \hbox{and} \quad \hat H\,|\,2\,\rangle = E |\,2\,\rangle. Suppose a solution $\backslash psi(t)\backslash ,$ of the full (time-dependent) Schrödinger equation of $\backslash hat\; H\backslash ,$ has the form at t = 0:

$$

|\,\psi(0)\,\rangle = c_1 |\,1\,\rangle + c_2 |\,2\,\rangle. Hence, because of the discussion above, at t > 0 :

$$

|\,\psi(t)\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 |\,1\,\rangle + \mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 |\,2\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar}

\left( c_1 |\,1\,\rangle +  c_2 |\,2\,\rangle\right) = \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle,

which shows that $\backslash psi(t)\backslash ,$ only depends on time in a trivial way (through its phase), also in the case of degeneracy.

Apply now $\backslash hat\; H\backslash ,$:

$$

\hat H\,|\,\psi(t)\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 E\,|\,1\,\rangle + \mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 E\, |\,2\,\rangle = E\mathrm{e}^{-\mathrm{i}Et/\hbar} \left( c_1 |\,1\,\rangle +

c_2 |\,2\,\rangle\right) 

$$

= E \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle = E\,|\,\psi(t)\,\rangle. Conclusion: The wavefunction $\backslash psi(t)\backslash ,$ with the given initial condition (its form at t = 0), remains a solution of the time-independent Schrödinger equation $\backslash hat\; H\backslash psi(t)\; =\; E\backslash psi(t)$ for all times t > 0.

Properties

Linearity

The Schrödinger equation (in any form) is linear in the wavefunction, meaning that if $\backslash psi(x,\; t)$ and $\backslash phi(x,t)$ are solutions, then so is $a\; \backslash psi\; \backslash pm\; b\; \backslash phi$. This property of the Schrödinger equation has important consequences.

Linearity of the Schrödinger equation

Assumptions:

• The Schrödinger equation:

$\backslash hat\; H(t)\backslash left|\backslash psi(t)\backslash right\backslash rangle\; =\; i\backslash hbar\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; \backslash psi(t)\; \backslash right\backslash rangle$

• $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$ are solutions of the Schrödinger equation.

$\backslash hat\; H(t)\backslash left|a\backslash psi(t)\; \backslash pm\; b\backslash phi(t)\backslash right\backslash rangle$

$=\backslash hat\; H(t)\backslash left|a\backslash psi(t)\backslash right\backslash rangle\; \backslash pm\; \backslash hat\; H(t)\backslash left|b\backslash phi(t)\backslash right\backslash rangle$ (as the Hamiltonian is a linear operator)

$=\; a\; \backslash left(\; \backslash hat\; H(t)\backslash left|\backslash psi(t)\backslash right\backslash rangle\; \backslash right)\; \backslash pm\; b\; \backslash left(\; \backslash hat\; H(t)\backslash left|\backslash phi(t)\backslash right\backslash rangle\; \backslash right)$

$=a\; \backslash left(\; i\backslash hbar\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; \backslash psi(t)\; \backslash right\backslash rangle\; \backslash right)\; \backslash pm\; b\; \backslash left(\; i\backslash hbar\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; \backslash phi(t)\; \backslash right\backslash rangle\; \backslash right)$

$=\; i\; \backslash hbar\; \backslash left(\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; a\backslash psi(t)\; \backslash right\backslash rangle\; \backslash right)\; \backslash pm\; i\; \backslash hbar\; \backslash left(\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; b\backslash phi(t)\; \backslash right\backslash rangle\; \backslash right)$

$=\; i\; \backslash hbar\; \backslash frac\{d\}\{d\; t\}\; \backslash left|\; a\backslash psi(t)\; \backslash pm\; b\backslash phi(t)\; \backslash right\backslash rangle$

Conservation of probability

In order to describe how probability density changes with time, it is acceptable to define probability current or probability flux. The probability flux represents a flowing of probability across space.

For example, consider a Gaussian probability curve centered around $x\_0$ with $x\_0$ moving at speed $v$ to the right. One may say that the probability is flowing towards the right, i.e., there is a probability flux directed to the right.

The probability flux $\backslash mathbf\{j\}$ is defined as:

$\backslash mathbf\{j\}\; =\; \{\backslash hbar\; \backslash over\; m\}\; \backslash cdot\; \{1\; \backslash over\; \{2\; \backslash mathrm\{i\}\}\}\; \backslash left(\; \backslash psi\; ^\{*\}\; \backslash nabla\; \backslash psi\; -\; \backslash psi\; \backslash nabla\; \backslash psi^\{*\}\; \backslash right)\; =\; \{\backslash hbar\; \backslash over\; m\}\; \backslash operatorname\{Im\}\; \backslash left(\; \backslash psi\; ^\{*\}\; \backslash nabla\; \backslash psi\; \backslash right)$

and measured in units of (probability)/(area × time) = r⁻²t⁻¹.

The probability flux satisfies a quantum continuity equation, i.e.:

$\{\; \backslash partial\; \backslash over\; \backslash partial\; t\}\; P\backslash left(x,t\backslash right)\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{j\}\; =\; 0$

where $P\backslash left(x,\; t\backslash right)$ is the probability density and measured in units of (probability)/(volume) = r⁻³. This equation is the mathematical equivalent of the probability conservation law.

It is easy to show that for a plane wave described by the wavefunction,

$\backslash psi\; (x,t)\; =\; \backslash ,\; A\; e^\{\; \backslash mathrm\{i\}\; (k\; x\; -\; \backslash omega\; t\})$

the probability flux is given by

$j\backslash left(x,t\backslash right)\; =\; \backslash left|A\backslash right|^2\; \{k\; \backslash hbar\; \backslash over\; m\}.$

Correspondence principle

Main article: Ehrenfest Theorem

The Schrödinger equation satisfies the correspondence principle.

Solutions

Main article: List of quantum mechanical systems with analytical solutions

Analytical solutions of the time-independent Schrödinger equation can be obtained for a variety of relatively simple conditions. These solutions provide insight into the nature of quantum phenomena and sometimes provide a reasonable approximation of the behavior of more complex systems (e.g., in statistical mechanics, molecular vibrations are often approximated as harmonic oscillators). Several of the more common analytical solutions can be found in the list of quantum mechanical systems with analytical solutions.

For many systems, however, there is no analytic solution to the Schrödinger equation. In these cases, one must resort to approximate solutions. Some of the common techniques are:

• Perturbation theory
• The variational principle underpins many approximate methods (like the popular Hartree-Fock method which is the basis of the post Hartree-Fock methods)
• Quantum Monte Carlo methods
• Density functional theory
• The WKB approximation
• Discrete delta-potential method

Free particle Schrödinger equation

Please help improve this article by expanding this section. See talk page for details. Please remove this message once the section has been expanded.

An important form of the Schrödinger equation results when the potential function for a single particle is zero:

$i\backslash hbar\backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; t\}=-\backslash frac\{\backslash hbar^2\}\{2m\}\backslash nabla^2\backslash psi$

The wave function can then be shown ^[3] to satisfy,

$\backslash psi(x,t)=\; \backslash ,\; Ae^\{i(kx-\; \backslash omega\; t)\}$

Relativistic generalisations

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Main article: Relativistic wave equations

The Schrödinger equation as presented so far in this article does not take into account relativistic effects. Generalisations incorporating ideas from special relativity include the Klein-Gordon equation and the Dirac equation.

Applications

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See also

• Basic quantum mechanics
• Dirac equation
• Klein-Gordon equation
• Pauli equation
• Quantum number
• Schrödinger\'s cat
• Schrödinger field
• Schrödinger picture
• Theoretical and experimental justification for the Schrödinger equation

References

• Erwin Schrödinger, Annalen der Physik, (Leipzig) (1926), [[4] ]

Modern reviews

• David J. Griffiths (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X. 

External links

• Linear Schrödinger Equation at EqWorld: The World of Mathematical Equations.

• Nonlinear Schrödinger Equation at EqWorld: The World of Mathematical Equations.

• The Schrödinger Equation in One Dimension as well as the directory of the book.

• All about 3D Schrödinger Equation

• Mathematical aspects of Schrödinger equation\'s are discussed on the Dispersive PDE Wiki.

• Web-Schrödinger: Interactive solution of the 2D time dependent Schrödinger equation

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