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{{Quantum book backlink|Mathematical structure and systems}}
In most physical systems, the Hamiltonian contains interactions that make exact solutions intractable. Approximation methods provide systematic procedures to estimate eigenvalues, eigenfunctions, and dynamical behavior with accuracy. These methods are used in atomic physics, molecular physics, condensed matter physics, and quantum field theory.<ref name="cohen">Cohen-Tannoudji, C., Diu, B., & Laloë, F. (1977). ''Quantum Mechanics''. Wiley.</ref>

[[File:Quantum Approximation Methods.jpg|thumb|450px|Quantum Approximation Methods]]
= Quantum Approximation Methods =
Quantum approximation methods are a group of analytical and semi-analytical techniques used to find approximate solutions to quantum mechanical problems that cannot be solved exactly. Since exact solutions of the Schrödinger equation exist only for a limited number of systems, approximation methods are vital tools in both theoretical and applied quantum mechanics.[<ref name="griffiths">Griffiths, D. J. (2018). ''Introduction to Quantum Mechanics''. Cambridge University Press.</ref><ref name="sakurai">Sakurai, J. J., & Napolitano, J. (2017). ''Modern Quantum Mechanics''. Cambridge University Press.</ref>

== Time-Independent Perturbation Theory ==

Time-independent perturbation theory applies when the Hamiltonian can be written as:

<math>H = H_0 + \lambda V</math>

where <math>H_0</math> is a solvable Hamiltonian and <math>V</math> is a small perturbation. The method yields corrections to energy levels and eigenstates as power series in the perturbation parameter.<ref name="griffiths" />

For non-degenerate systems, the first-order energy correction is:

<math>E_n^{(1)} = \langle n^{(0)} | V | n^{(0)} \rangle</math>

Degenerate perturbation theory is required when energy levels of <math>H_0</math> are degenerate.

== Time-Dependent Perturbation Theory ==

Time-dependent perturbation theory describes transitions between quantum states due to a time-dependent interaction. It is essential for understanding processes such as absorption and emission of radiation.<ref name="sakurai" />

A central result is Fermi’s Golden Rule:

<math>\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 \rho(E_f)</math>

where <math>\rho(E_f)</math> is the density of final states.

== Variational Method ==

The variational method provides an upper bound on the ground state energy of a system. Given a trial wavefunction <math>\psi_t</math>, the energy expectation value:

<math>E[\psi_t] = \frac{\langle \psi_t | H | \psi_t \rangle}{\langle \psi_t | \psi_t \rangle}</math>

satisfies:

<math>E[\psi_t] \geq E_0</math>

where <math>E_0</math> is the true ground state energy.<ref name="griffiths" />

This method is widely used in atomic and molecular calculations.

== WKB Approximation ==

The Wentzel–Kramers–Brillouin (WKB) approximation is a semiclassical method applicable when the potential varies slowly compared to the particle’s wavelength.<ref name="cohen" />

The approximate solution to the Schrödinger equation is:

<math>\psi(x) \approx \frac{1}{\sqrt{p(x)}} \exp\left( \pm \frac{i}{\hbar} \int p(x)\,dx \right)</math>

where <math>p(x) = \sqrt{2m(E - V(x))}</math>.

It is particularly useful for tunneling problems and bound states in slowly varying potentials.

== Adiabatic Approximation ==

The adiabatic approximation applies when the Hamiltonian changes slowly in time. A system initially in an eigenstate of the Hamiltonian remains in the corresponding instantaneous eigenstate, up to a phase factor.<ref name="sakurai" />

This leads to the concept of the Berry phase, a geometric phase acquired during cyclic evolution.

== Born Approximation ==

The Born approximation is used in scattering theory when the interaction potential is weak. It provides an approximate expression for the scattering amplitude:

<math>f(\mathbf{k}', \mathbf{k}) \propto \int e^{-i\mathbf{k}' \cdot \mathbf{r}} V(\mathbf{r}) e^{i\mathbf{k} \cdot \mathbf{r}} d^3r</math>

It is valid when the incident wave is only weakly distorted by the potential.<ref name="griffiths" />

== Applications ==

Approximation methods are fundamental in:

* Atomic structure calculations
* Molecular bonding and spectroscopy
* Solid-state physics and band structure
* Quantum optics and laser physics
* Nuclear and particle physics

They provide practical tools for connecting theoretical models with experimental observations.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Schrödinger equation]]
* [[Perturbation theory (quantum mechanics)]]
* [[Scattering theory]]

== References ==
{{reflist|3}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Physics]]

{{Sourceattribution|Quantum Approximation Methods|1}}

Physics:Quantum Approximation Methods - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template