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![{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c) |
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The Kubo formula, named for Ryogo Kubo who first presented the formula in 1957,[1][2] is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.
Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well.
Consider a quantum system described by the (time independent) Hamiltonian
. The expectation value of a physical quantity at equilibrium temperature
, described by the operator
, can be evaluated as:
,
where
is the thermodynamic beta,
is density operator, given by
![{\displaystyle {\hat {\rho _{0}}}=e^{-\beta {\hat {H}}_{0}}=\sum _{n}|n\rangle \langle n|e^{-\beta E_{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/388d7d4a94081f75dea2247ffc7b979bef47eea1)
and
is the partition function.
Suppose now that just above some time
an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian:
![{\displaystyle {\hat {H}}(t)={\hat {H}}_{0}+{\hat {V}}(t)\theta (t-t_{0}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3597c7b056100a19d95977b8fc529299ad7dff17)
where
is the Heaviside function (1 for positive times, 0 otherwise) and
is hermitian and defined for all t, so that
has for positive
again a complete set of real eigenvalues
But these eigenvalues may change with time.
However, one can again find the time evolution of the density matrix
rsp. of the partition function
to evaluate the expectation value of
![{\displaystyle \left\langle {\hat {A}}\right\rangle ={\frac {\operatorname {Tr} \,\left[{\hat {\rho }}(t)\,{\hat {A}}\right]}{\operatorname {Tr} \,\left[{\hat {\rho }}(t)\right]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8174b5e1ecaeff80db790fd7cdbb0e3836771f99)
The time dependence of the states
is governed by the Schrödinger equation
![{\displaystyle i\hbar {\frac {\partial }{\partial t}}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77065d69793b457279b0fb893aa94a3a5c692bc4)
which thus determines everything, corresponding of course to the Schrödinger picture. But since
is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation,
in lowest nontrivial order. The time dependence in this representation is given by
where by definition for all t and
it is:
To linear order in
, we have
.
Thus one obtains the expectation value of
up to linear order in the perturbation:
,
thus[3]
Kubo formula (general)
The brackets
mean an equilibrium average with respect to the Hamiltonian
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
.
The above expression is true for any kind of operators. (see also Second quantization)[4]
See also
References
- ^ Kubo, Ryogo (1957). "Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems". J. Phys. Soc. Jpn. 12 (6): 570–586. doi:10.1143/JPSJ.12.570.
- ^ Kubo, Ryogo; Yokota, Mario; Nakajima, Sadao (1957). "Statistical-Mechanical Theory of Irreversible Processes. II. Response to Thermal Disturbance". J. Phys. Soc. Jpn. 12 (11): 1203–1211. doi:10.1143/JPSJ.12.1203.
- ^ Bruus, Henrik; Flensberg, Karsten; Flensberg, ØRsted Laboratory Niels Bohr Institute Karsten (2004-09-02). Many-Body Quantum Theory in Condensed Matter Physics: An Introduction. OUP Oxford. ISBN 978-0-19-856633-5.
- ^ Mahan, GD (1981). Many-particle physics. New York: Springer. ISBN 0306463385.