Schrödinger equation

Template:Quantum mechanics

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 atomic and subatomic 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.

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. To date, the only instance where an exact analytical (purely mathematical) solution to the unconfined Schrödinger equation has been found is in a single-electron system, e.g. the hydrogen atom. Numerical approximations can be found for all other systems, assuming the equation is confined and, if large, that there are boundary conditions that allow computation within the timespan of the universe.

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

The Schrödinger equation is a differential equation that represents the temporal evolution of a wavefunction, where the state vector varies with time, whilst the operator remains static. Although this is opposite in fashion to the Heisenberg picture, the two are in fact alternative mathematical formulations of the same underlying physical quantum-level reality, and both are correct. Whilst the Schrödinger equation separates space from time, and is thus best applied to non-relativistic problems, the Heisenberg picture for solving Quantum Mechanical problems does not distinguish between the two, making it more general in relativistic scope, although less useful in the study of non-relativistic quantum states.

Historical background and development

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:

• 1- The total energy E of a particle is

<math>E=\frac{p^2}{2m}+V</math>

This is the classical mechanics expression for a particle with mass m where the total energy E is the sum of the kinetic energy, <math>\frac{p^2}{2m}</math>, 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:

• 2- Einstein's light quanta hypothesis of 1905:

<math>E = h f\;</math>

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

• 3- The de Broglie hypothesis of 1924:

<math>p=h / \lambda\;</math>

where <math>\lambda\,</math> is the wavelength of the wave. This hypothesis also requires:

• 4- The association of a wave (with wavefunction <math>\psi</math>) with any particle.

Combining the above assumptions yields Schrödinger's equation:

Expressing frequency f in terms of angular frequency <math>\omega = 2\pi f\;</math> and wavelength <math>\lambda\,</math> in terms of wavenumber <math>k = 2\pi / \lambda\;</math>, with <math>\hbar = h / 2 \pi\;</math> we get:

<math>E=\hbar \omega</math>

and

<math>\mathbf{p}=\hbar \mathbf{k}\;</math>

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:

<math>\psi \approx e^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}</math>

and to realize that since

<math> \frac{\partial}{\partial t} \psi = -i\omega \psi </math>

then

<math> E \psi = \hbar \omega \psi = i\hbar\frac{\partial}{\partial t} \psi </math>

and similarly since:

<math> \frac{\partial}{\partial x} \psi = i k_x \psi </math>

then

<math> p_x \psi = \hbar k_x \psi = -i\hbar\frac{\partial}{\partial x} \psi </math>

and hence:

<math> p_x^2 \psi = -\hbar^2\frac{\partial^2}{\partial x^2} \psi </math>

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

<math> p^2 \psi = (p_x^2 + p_y^2 + p_z^2) \psi = -\hbar^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \psi = -\hbar^2\nabla^2 \psi </math>

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:

<math>i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi</math>

Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, <math>\psi\;</math>, moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment for the Lyman, Balmer, Paschen and Brackett series, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - 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 paper in 1926.

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

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 <math>\hat H</math> is,

<math>

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

where <math>\psi</math> is the wavefunction, <math>\hbar</math> is Planck's constant and <math>i</math> 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 (V) energies :

<math>\hat H = \left(T + V\right) \ = - \frac{\hbar^2}{2m} \nabla^2 + V\left(\mathbf{r}\right) </math>

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

<math> -\frac{\hbar^2}{2 m} \left[\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} \right] + V \psi = i \hbar \frac{\partial \psi}{\partial t} </math>

where the dependence of <math>\psi</math> 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 <math>E</math> results in the time-independent Schrödinger equation

<math>\hat H \psi = E \psi \,</math>

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

<math> i \hbar \frac{\partial \psi}{\partial t} = E \psi </math>

which can be solved for <math>\psi</math> as

<math>\psi\left(\mathbf{r}, t\right) = \phi\left(\mathbf{r}\right) e^{-iEt/\hbar}</math>

where <math>\phi\left(\mathbf{r}\right)</math> is the value of <math>\psi</math> at <math>t=0</math>. Also, when such a given solution - representing a stationary state - is substituted into the time-dependent Schrödinger equation, the resulting equation is the time-independent Schrödinger equation.

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

<math> -\frac{\hbar^2}{2 m} \frac{d^2 \phi (x)}{dx^2} + V(x) \phi (x) = E \phi (x). </math>

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

<math> \left[-\frac{\hbar^2}{2 m} \nabla^2 + V(\mathbf{r}) \right] \phi (\mathbf{r}) = E \phi (\mathbf{r}), </math>

where <math>\nabla^2</math> 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,

<math>\hat H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{d}{d t} \left| \psi \left(t\right) \right\rangle</math>

where <math>|\psi(t)\rangle</math> is a ket, <math>\hbar</math> is the reduced Planck's constant and <math>\hat H(t)</math> 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, <math>\hat H</math>, there exists a set of quantum states, <math>\left|\psi_n\right\rang</math>, known as energy eigenstates, and corresponding real numbers <math>E_n</math> satisfying the eigenvalue equation,

<math> \hat H \left|\psi_n\right\rang = E_n \left|\psi_n \right\rang. </math>

Alternatively, <math>\psi</math> is said to be an eigenstate (eigenket) of <math>\hat H</math> with eigenvalue <math>E</math>. Such a state possesses a definite total energy, whose value <math>E_n</math> is the eigenvalue of the Hamiltonian. The corresponding eigenvector <math>\psi_n\,</math> 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 <math>\psi_n\,</math> 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 <math>E_n</math>. If the maximum number of linearly independent eigenvectors corresponding to <math>E_n</math> equals k, we say that the energy level <math>E_n</math> 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

<math>\mathrm{i} \hbar \frac{\partial}{\partial t} \left| \psi_n \left(t\right) \right\rangle = E_n \left|\psi_n\left(t\right)\right\rang. </math>

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:

<math> \left| \psi \left(t\right) \right\rangle = \mathrm{e}^{-\mathrm{i} Et / \hbar} \left|\psi\left(0\right)\right\rang. </math>

It immediately follows that the probability amplitude,

<math>\psi(t)^*\psi(t) = \mathrm{e}^{\mathrm{i} Et / \hbar}\mathrm{e}^{-\mathrm{i} Et / \hbar}

\psi(0)^*\psi(0) = |\psi(0)|^2, </math> 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 <math>\psi(t)\,</math> are time-independent.

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

<math>\left|\psi\left(t\right)\right\rang = \sum_n c_n(t) \left|n\right\rang \quad,\quad \hat H \left|n\right\rang = E_n \left|n\right\rang \quad,\quad \sum_n \left|c_n\left(t\right)\right|^2 = 1.</math>

(The last equation enforces the requirement that <math> \left|\psi\left(t\right)\right\rang </math>, 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

<math>\mathrm{i}\hbar \frac{\partial c_n}{\partial t} = E_n c_n\left(t\right).</math>

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

<math>\left|\psi\left(t\right)\right\rang = \sum_n \mathrm{e}^{-\mathrm{i}E_nt/\hbar} c_n\left(0\right) \left|n\right\rang. </math>

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

Example

Let <math>|\,1\,\rangle</math> and <math>|\,2\,\rangle</math> be degenerate eigenstates of the time-independent Hamiltonian <math>\hat H\,</math>:

<math>

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

<math>

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

<math>

|\,\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,

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

Apply now <math>\hat H\,</math>:

<math>

\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) 

</math>

<math>

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

Properties

Linearity

The Schrödinger equation (in any form) is linear in the wavefunction, meaning that if <math>\psi(x, t)</math> and <math>\phi(x,t)</math> are solutions, then so is <math> a \psi + b \phi</math>, where a and b are any complex numbers. This property of the Schrödinger equation has important consequences.

Linearity of the Schrödinger equation

Assumptions:

• The Schrödinger equation:

<math>\hat H(t)\left|\psi(t)\right\rangle = i\hbar \frac{d}{d t} \left| \psi(t) \right\rangle</math>

• <math>|\psi\rangle</math> and <math>|\phi\rangle</math> are solutions of the Schrödinger equation.

<math>\hat H(t)\left|a\psi(t) + b\phi(t)\right\rangle</math>

<math>=\hat H(t)\left|a\psi(t)\right\rangle + \hat H(t)\left|b\phi(t)\right\rangle</math> (as the Hamiltonian is a linear operator)

<math>= a \left( \hat H(t)\left|\psi(t)\right\rangle \right) + b \left( \hat H(t)\left|\phi(t)\right\rangle \right)</math>

<math>=a \left( i\hbar \frac{d}{d t} \left| \psi(t) \right\rangle \right) + b \left( i\hbar \frac{d}{d t} \left| \phi(t) \right\rangle \right)</math>

<math>= i \hbar \left( \frac{d}{d t} \left| a\psi(t) \right\rangle \right) + i \hbar \left( \frac{d}{d t} \left| b\phi(t) \right\rangle \right)</math>

<math>= i \hbar \frac{d}{d t} \left| a\psi(t) + b\phi(t) \right\rangle</math>

Conservation of probability

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

For example, consider a Gaussian probability curve centered around <math>x_0</math> with <math>x_0</math> moving at speed <math>v</math> 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 <math>\mathbf{j}</math> is defined as:

<math> \mathbf{j} = {\hbar \over m} \cdot {1 \over {2 \mathrm{i}}} \left( \psi ^{*} \nabla \psi - \psi \nabla \psi^{*} \right) = {\hbar \over m} \operatorname{Im} \left( \psi ^{*} \nabla \psi \right) </math>

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

The probability flux satisfies the required continuity equation for a conserved quantity, i.e.:

<math>{ \partial \over \partial t} P\left(x,t\right) + \nabla \cdot \mathbf{j} = 0 </math>

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

A standard calculation shows that for a plane wave described by the wavefunction,

<math> \psi (x,t) = \, A e^{ \mathrm{i} (k x - \omega t)} </math>

the probability flux is given by

<math> j\left(x,t\right) = \left|A\right|^2 {k \hbar \over m}</math>

showing that not only is the probability of finding the particle in a plane wave state the same everywhere at all times, but also that it is moving at constant speed everywhere.

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

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

<math>i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi </math>

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

<math>\psi(x,t)= \, Ae^{i(kx- \omega t)}</math>

i.e. the particle is in a plane wave state.

Relativistic generalisations

Main article: Relativistic wave equations

The Schrödinger equation does not take into account relativistic effects, meaning that the Schrödinger equation is invariant under a Galilean transformation, but not under a Lorentz transformation.

Galilean invariance of the Schrödinger equation

Relativistically valid generalisations incorporating ideas from special relativity include the Klein-Gordon equation and the Dirac equation.

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

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