Kubo formula

Consider a quantum system described by the (time independent) Hamiltonian $H\_0$. The expectation value of a physical quantity at equilibrium temperature $T$, described by the operator $\backslash hat\{A\}$, can be evaluated as:

:$\backslash left\backslash langle\backslash hat\{A\}\backslash right\backslash rangle\; =\; \{1\; \backslash over\; Z\_0\}\backslash operatorname\{Tr\}\backslash ,\backslash left[\backslash hat\{\backslash rho\_0\}\backslash hat\{A\}\backslash right]\; =\; \{1\; \backslash over\; Z\_0\}\backslash sum\_n\; \backslash left\backslash langle\; n\; \backslash left|\; \backslash hat\{A\}\; \backslash right|\; n\; \backslash right\backslash rangle\; e^\{-\backslash beta\; E\_n\}$,

where $\backslash beta=1/k\_\{\backslash rm\; B\}T$ is the thermodynamic beta, $\backslash hat\{\backslash rho\}\_0$ is density operator, given by

:$\backslash hat\{\backslash rho\_0\}\; =\; e^\{-\backslash beta\; \backslash hat\{H\}\_0\}\; =\; \backslash sum\_n\; |\; n\; \backslash rangle\backslash langle\; n\; |\; e^\{-\backslash beta\; E\_n\}$

and $Z\_0\; =\; \backslash operatorname\{Tr\}\backslash ,\backslash left[\backslash hat\backslash rho\_0\backslash right]$ is the partition function.

Suppose now that just after some time $t\; =\; t\_0$ an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian:

:$\backslash hat\{H\}(t)\; =\; \backslash hat\{H\}\_0\; +\; \backslash hat\{V\}(t)\; \backslash theta\; (t\; -\; t\_0),$

where $\backslash theta\; (t)$ is the Heaviside function (1 for positive times, 0 otherwise) and $\backslash hat\; V(t)$ is hermitian and defined for all t, so that $\backslash hat\; H(t)$ has for positive $t\; -\; t\_0$ again a complete set of real eigenvalues $E\_n(t).$ But these eigenvalues may change with time.

However, one can again find the time evolution of the density matrix $\backslash hat\{\backslash rho\}(t)$ rsp. of the partition function $Z(t)\; =\; \backslash operatorname\{Tr\}\backslash ,\; \backslash left[\backslash hat\backslash rho\; (t)\backslash right],$ to evaluate the expectation value of

:$\backslash left\backslash langle\backslash hat\; A\backslash right\backslash rangle\; =\; \backslash frac\{\backslash operatorname\{Tr\}\backslash ,\backslash left[\backslash hat\; \backslash rho\; (t)\backslash ,\backslash hat\; A\backslash right]\}\{\backslash operatorname\{Tr\}\backslash ,\backslash left[\backslash hat\backslash rho\; (t)\backslash right]\}.$

The time dependence of the states $|n(t)\; \backslash rangle$ is governed by the Schrödinger equation

:$i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\; n(t)\; \backslash rangle\; =\; \backslash hat\{H\}(t)|\; n(t)\; \backslash rangle\; ,$

which thus determines everything, corresponding of course to the Schrödinger picture. But since $\backslash hat\{V\}(t)$ is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, $\backslash left|\backslash hat\; n(t)\; \backslash right\backslash rangle\; ,$ in lowest nontrivial order. The time dependence in this representation is given by $|\; n(t)\; \backslash rangle\; =\; e^\{-i\backslash hat\; H\_0t/\backslash hbar\}\; \backslash left|\; \backslash hat\{n\}(t)\; \backslash right\backslash rangle\; =\; e^\{-i\backslash hat\; H\_0t/\backslash hbar\}\backslash hat\{U\}(t,\; t\_0)\; \backslash left|\; \backslash hat\{n\}(t\_0)\; \backslash right\backslash rangle\; ,$ where by definition for all t and $t\_0$ it is: $\backslash left|\; \backslash hat\{n\}(t\_0)\; \backslash right\backslash rangle\; =\; e^\{i\backslash hat\; H\_0t\_0/\backslash hbar\}\; |\; n(t\_0)\; \backslash rangle$

To linear order in $\backslash hat\{V\}(t)$, we have

:$\backslash hat\{U\}(t,\; t\_0)\; =\; 1\; -\; \backslash frac\{i\}\{\backslash hbar\}\backslash int\_\{t\_0\}^t\; dt\text{'}\; \backslash hat\{V\}\backslash mathord\backslash left(t\text{'}\backslash right)$.

Thus one obtains the expectation value of $\backslash hat\{A\}(t)$ up to linear order in the perturbation:

:$\backslash left\backslash langle\backslash hat\{A\}(t)\backslash right\backslash rangle=\; \backslash left\backslash langle\; \backslash hat\{A\}\; \backslash right\backslash rangle\_0\; -\; \backslash frac\{i\}\{\backslash hbar\}\backslash int\_\{t\_0\}^t\; dt\text{'}\; \{1\; \backslash over\; Z\_0\}\backslash sum\_n\; e^\{-\backslash beta\; E\_n\}\; \backslash left\backslash langle\; n\; (t\_0)\; \backslash left|\; \backslash hat\{A\}(t)\; \backslash hat\{V\}\backslash mathord\backslash left(t\text{'}\backslash right)\; -\; \backslash hat\{V\}\backslash mathord\backslash left(t\text{'}\backslash right)\backslash hat\{A\}(t)\; \backslash right|\; n(t\_0)\; \backslash right\backslash rangle$,

thus{{Cite book |last1=Bruus |first1=Henrik |url=https://books.google.com/books?id=v5vhg1tYLC8C |title=Many-Body Quantum Theory in Condensed Matter Physics: An Introduction |last2=Flensberg |first2=Karsten |last3=Flensberg |first3=ØRsted Laboratory Niels Bohr Institute Karsten |date=2004-09-02 |publisher=OUP Oxford |isbn=978-0-19-856633-5 |language=en}}

{{Equation box 1

|title=Kubo formula (general)

|indent =:

|equation = $\backslash langle\; \backslash hat\; A(t)\; \backslash rangle=\backslash left\backslash langle\; \backslash hat\{A\}\; \backslash right\backslash rangle\_0\; -\; \backslash frac\{i\}\{\backslash hbar\}\backslash int\_\{t\_0\}^t\; dt\text{'}\; \backslash left\backslash langle\; \backslash left[\backslash hat\{A\}(t),\; \backslash hat\{V\}\backslash mathord\backslash left(t\text{'}\backslash right)\backslash right]\backslash right\backslash rangle\_0$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4}}

The brackets $\backslash langle\; \backslash rangle\_0$ mean an equilibrium average with respect to the Hamiltonian $H\_0\; .$ Therefore, although the result is of first order in the perturbation, it involves only the zeroth-order eigenfunctions, which is usually the case in perturbation theory and moves away all complications which otherwise might arise for $t\; >\; t\_0$.

The above expression is true for any kind of operators. (see also Second quantization){{cite book|last=Mahan|first=GD|author-link=Gerald Mahan|title=Many-particle physics|publisher= Springer|location=New York|isbn=0306463385|year=1981}}

• Green–Kubo relations

{{Reflist|2}}

Category:Quantum mechanics