Cube formula

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The Kubo formula (after Ryōgo Kubo ) is a result of quantum statistics . It gives the linear response function of a measurable quantity ( observable ) in time-dependent perturbation theory at finite temperature as the thermal expectation value of Hermitian operators in the interaction picture .

The numerous applications of the Kubo formula include the computation of magnetic and electrical susceptibilities and abstract generalizations thereof as a result of a time-dependent perturbation of the Hamiltonian of the system.

Details

The cube formula leads to a relationship between

• the quantum statistical expectation of an observable in an undisturbed system with Hamilton operator at a time and ${\displaystyle \langle A\rangle _{0}}$ ${\displaystyle A}$ ${\displaystyle H_{0}}$ ${\displaystyle t_{0}}$
• the expected value of the same observable after introducing a small perturbation of the system in the form of a perturbation operator at a time : ${\displaystyle \langle A(t)\rangle }$ ${\displaystyle V(t)}$ ${\displaystyle t}$

${\displaystyle \langle A(t)\rangle -\langle A\rangle _{0}=-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'\langle [A(t),V(t')]\rangle }$

Designate

• angle brackets the quantum statistical expectation value with the density matrix ${\displaystyle \langle A\rangle =\operatorname {Tr} [\rho A]}$ ${\displaystyle \rho }$
• square brackets denote the commutator ${\displaystyle [A,V]=AV-VA}$
• a subscript  zero the undisturbed system
• i the imaginary unit .

Derivation and formulation

A quantum system has the time-independent Hamilton operator with the energy values ​​assumed to be discrete . The quantum mechanical and thermal expectation value of a physical quantity with the Hermitian operator is then: ${\displaystyle {\hat {H}}_{0}}$ ${\displaystyle E_{n}}$ ${\displaystyle {\hat {A}}}$

${\displaystyle \langle {\hat {A}}\rangle ={1 \over Z_{0}}\operatorname {Tr} [{\hat {\rho }}_{0}{\hat {A}}]={1 \over Z_{0}}\sum _{n}\langle n|{\hat {A}}|n\rangle e^{-\beta E_{n}}}$

${\displaystyle {\hat {\rho }}_{0}=e^{-\beta {\hat {H}}_{0}}=\sum _{n}|n\rangle \langle n|e^{-\beta E_{n}}}$

where is the sum of states and the reciprocal absolute temperature with the Boltzmann constant and the temperature . In the last equals sign was so according to uninterrupted energy eigenstates with developed and exploited its completeness. ${\displaystyle Z_{0}=\operatorname {Tr} [\rho _{0}]}$ ${\displaystyle \beta }$ ${\displaystyle 1/(k_{\text{B}}T)}$ ${\displaystyle k_{\text{B}}}$ ${\displaystyle T}$ ${\displaystyle |n\rangle }$ ${\displaystyle {\hat {H}}_{0}|n\rangle =E_{n}|n\rangle }$

If an external disturbance is switched on at the moment , the system leaves the thermal equilibrium. The perturbation is described by a time-dependent addition to the Hamilton operator: ${\displaystyle t=t_{0}}$

${\displaystyle {\hat {H}}(t)={\hat {H}}_{0}+{\hat {V}}(t)\Theta (t-t_{0}),}$

Here referred to the Heaviside function , which for non-negative values of takes the value one and for all other zero. This takes into account the instantaneous "switch-on process" at the time . is a Hermitian operator defined for all , so that for all has a complete orthonormal system of eigenfunctions and eigenvalues . ${\displaystyle \Theta (t)}$ ${\displaystyle t-t_{0}}$ ${\displaystyle t-t_{0}}$ ${\displaystyle t_{0}}$ ${\displaystyle {\hat {V}}(t)}$ ${\displaystyle t}$ ${\displaystyle {\hat {H}}(t)}$ ${\displaystyle t}$ ${\displaystyle |n(t)\rangle }$ ${\displaystyle E_{n}(t)}$

From the time development of the density matrix ${\displaystyle {\hat {\rho }}(t)}$

${\displaystyle {\hat {\rho }}(t)=\sum _{n}|n(t)\rangle \langle n(t)|e^{-\beta E_{n}(t)}}$

under the assumption that the quantum statistical equilibrium formalism remains valid at any point in time, the thermal expectation value of the operators follows: ${\displaystyle {\hat {A}}}$

${\displaystyle \langle {\hat {A}}\rangle ={\frac {1}{Z(t)}}\operatorname {Tr} [{\hat {\rho }}(t){\hat {A}}]={\frac {1}{Z(t)}}\sum _{n}\langle n(t)|{\hat {A}}|n(t)\rangle e^{-\beta E_{n}(t)},}$

with the partition function . ${\displaystyle Z(t)=\operatorname {Tr} [{\hat {\rho }}(t)]}$

The quantum mechanical Schrödinger picture is still used here, but with time-dependent Hamilton operators. However, it is pointed out at this point that in general both the eigenfunctions and the eigenvalues ​​of the Hamilton operator will also change. The time dependence of follows from the Schrödinger equation. Since “weak” should be, it makes sense to use the lowest order of the time-dependent perturbation theory and to go over to the interaction picture (states ). The result is: ${\displaystyle |n(t)\rangle }$ ${\displaystyle E_{n}(t)}$ ${\displaystyle t}$ ${\displaystyle |n(t)\rangle }$ ${\displaystyle \mathrm {i} \partial _{t}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle .}$ ${\displaystyle {\hat {V}}(t)}$ ${\displaystyle |n\rangle \to |{\hat {n}}\rangle }$

${\displaystyle |n(t)\rangle =:e^{-\mathrm {i} {\hat {H}}_{0}t_{0}}|{\hat {n}}(t)\rangle =e^{-\mathrm {i} {\hat {H}}_{0}t}{\hat {U}}(t,t_{0})|{\hat {n}}(t_{0})\rangle }$, where is by definition . ${\displaystyle |{\hat {n}}(t_{0})\rangle =e^{+\mathrm {i} {\hat {H}}_{0}t_{0}}|n(t_{0})\rangle }$

In linear order in : ${\displaystyle {\hat {V}}(t)}$

${\displaystyle {\hat {U}}(t,t_{0})=1-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'{\hat {V}}(t')}$.

In this way the end result is obtained for in linear order (in this order all the problems mentioned above are also eliminated, because first-order perturbation calculations only need the zeroth-order eigenfunctions): ${\displaystyle {\hat {A}}(t)}$

${\displaystyle {\begin{aligned}\langle {\hat {A}}(t)\rangle _{T}&=\langle {\hat {A}}\rangle _{0,T}-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'{1 \over Z_{0}}\ \sum _{n}e^{-\beta E_{n}}\langle n(t_{0})|{\hat {A}}(t){\hat {V}}(t')-{\hat {V}}(t'){\hat {A}}(t)|n(t_{0})\rangle \\&=\langle {\hat {A}}\rangle _{0,T}-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'\langle [{\hat {A}}(t),{\hat {V}}(t')]\rangle _{0,T}\end{aligned}}}$

Here the expression means a quantum statistical expected value calculated with the Hamilton operator at the temperature , while the expressions above are ordinary quantum mechanical expectation values ​​which do not take the temperature into account. Furthermore, in is meant by the eigenvalues ​​of . ${\displaystyle \langle \dots \rangle _{0,T}}$ ${\displaystyle {\hat {H}}_{0}}$ ${\displaystyle T}$ ${\displaystyle \langle n(t_{0})|\dots |n(t_{0})\rangle ,}$ ${\displaystyle e^{-\beta E_{n}}}$ ${\displaystyle E_{n}}$ ${\displaystyle {\hat {H}}(t_{0})}$

Since the different images are identical at the time, the same applies to the above final result. ${\displaystyle t_{0}}$

Here boson states were considered. For fermionic states there are additional special features. The reduced Planck's constant was set to one. ${\displaystyle \hbar }$

References and footnotes

1. ↑ In the most general case of quantum statistics, it can be replaced by any Hermitian operator whose eigenvalues satisfy the two conditions and . ${\displaystyle {\hat {\rho }}(t)=e^{-\beta {\hat {H}}(t)}}$ ${\displaystyle w_{n}}$ ${\displaystyle \sum w_{n}=1}$ ${\displaystyle \sum w_{n}^{2}<1}$
2. ^ GD Mahan: Many-particle physics . Springer, New York 1981, ISBN 0306463385 .