Variation method

The variational method is in quantum mechanics , a method of approximation to an upper bound for the eigenvalues of a quantum mechanical observables with discrete spectrum to find. A generalization of the method leads to the min-max principle .

Procedure

Basic state

The method is based on the fact that the eigenvalue of the ground state is a lower bound for the expected value of the measurement of the observable: If the degeneracy of an eigenvalue , then any state can be described as ${\displaystyle g_{i}}$ ${\displaystyle i}$

${\displaystyle |\psi \rangle =\sum _{i}\sum _{j=1}^{g_{i}}c_{i,j}|\psi _{i,j}\rangle }$

write, which form a complete orthonormal system. The following then applies to the expected value of the state when measuring an observable with eigenvalues ${\displaystyle |\psi _{i,j}\rangle }$ ${\displaystyle H}$ ${\displaystyle E_{i}}$

${\displaystyle \langle \psi |H|\psi \rangle =\sum _{i}\sum _{j=1}^{g_{i}}E_{i}|c_{i,j}|^{2}\geq E_{0}\sum _{i}\sum _{j=1}^{g_{i}}|c_{i,j}|^{2}=E_{0}\langle \psi |\psi \rangle }$.

An upper bound can therefore be found for if one calculates the expectation value for a family of states and looks for the infimum: ${\displaystyle E_{0}}$ ${\displaystyle |\psi _{\alpha }\rangle }$

${\displaystyle E_{0}\leq \inf _{\alpha }{\frac {\langle \psi _{\alpha }|H|\psi _{\alpha }\rangle }{\langle \psi _{\alpha }|\psi _{\alpha }\rangle }}}$.

Excited states

If the eigenfunction is a (not degenerate) ground state with an eigenvalue , one can write for any state ${\displaystyle |\psi _{0}\rangle }$ ${\displaystyle E_{0}}$ ${\displaystyle |\psi \rangle }$

${\displaystyle H|\psi \rangle =c_{0}E_{0}|\psi _{0}\rangle +\varepsilon |\varphi \rangle }$,

where . If one breaks down into eigenstates as above, one obtains under the secondary condition ${\displaystyle |\varphi \rangle \perp |\psi _{0}\rangle }$ ${\displaystyle |\varphi \rangle }$ ${\displaystyle \langle \varphi |\psi _{0}\rangle =0}$

${\displaystyle E_{1}\leq \inf _{\alpha }{\frac {\langle \varphi _{\alpha }|H|\varphi _{\alpha }\rangle }{\langle \varphi _{\alpha }|\varphi _{\alpha }\rangle }}}$,

because the total is missing. ${\displaystyle i=0}$

The search for further eigenstates is carried out analogously, with orthogonality to several subspaces that span the lower eigenvalues ​​then being minimized.

literature

• Alberto Galindo, Pedro Pascual: Quantum Mechanics II. Chapter 10.9; Springer, 1991
• Torsten Fließbach: Quantum Mechanics, Textbook on Theoretical Physics III. Chapter 44; Spectrum, 2008