In quantum mechanics, perturbation
theory is a set of approximation schemes
directly related to mathematical
perturbation for describing a
complicated quantum system in terms of a
simpler one. The idea is to start with a
simple system for which a mathematical
solution is known, and add an additional
"perturbing" Hamiltonian representing a
weak disturbance to the system. If the
disturbance is not too large, the
various physical quantities associated
with the perturbed system can be
expressed as "corrections" to those of
the simple system. These corrections,
being small compared to the size of the
quantities themselves, can be calculated
using approximate methods such as
asymptotic series. The complicated
system can therefore be studied based on
knowledge of the simpler one.
Applications of perturbation theory
Perturbation theory is an important tool
for describing real quantum systems, as
it turns out to be very difficult to
find exact solutions to the Schrödinger
equation for Hamiltonians of even
moderate complexity. The Hamiltonians to
which we know exact solutions, such as
the hydrogen atom, the quantum harmonic
oscillator and the particle in a box,
are too idealized to adequately describe
most systems. Using perturbation theory,
we can use the known solutions of these
simple Hamiltonians to generate
solutions for a range of more
complicated systems.
For example, by adding a perturbative
electric potential to the quantum
mechanical model of the hydrogen atom,
we can calculate the tiny shifts in the
spectral lines of hydrogen caused by the
presence of an electric field. This is
only approximate because the sum of a
Coulomb potential with a linear
potential is unstable although the
tunneling time is very long. This
instability shows up as a broadening of
the energy spectrum lines, which
perturbation theory fails to reproduce
entirely.
The expressions produced by perturbation
theory are not exact, but they can lead
to accurate results as long as the
expansion parameter, say α, is very
small. Typically, the results are
expressed in terms of finite power
series in α that seem to converge to the
exact values when summed to higher
order. After a certain order n ~ 1/α
however, the results become increasingly
worse since the series are usually
divergent. There exist ways to convert
them into convergent series, which can
be evaluated for large-expansion
parameters, most efficiently by
Variational method.
In the theory of quantum
electrodynamics, in which the
electron–photon interaction is treated
perturbatively, the calculation of the
electron's magnetic moment has been
found to agree with experiment to eleven
decimal places. In QED and other quantum
field theories, special calculation
techniques known as Feynman diagrams are
used to systematically sum the power
series terms.
Under some circumstances, perturbation
theory is an invalid approach to take.
This happens when the system we wish to
describe cannot be described by a small
perturbation imposed on some simple
system. In quantum chromodynamics, for
instance, the interaction of quarks with
the gluon field cannot be treated
perturbatively at low energies because
the coupling constant becomes too large.
Perturbation theory also fails to
describe states that are not generated
adiabatically from the "free model",
including bound states and various
collective phenomena such as solitons.
Imagine, for example, that we have a
system of free particles, to which an
attractive interaction is introduced.
Depending on the form of the
interaction, this may create an entirely
new set of eigenstates corresponding to
groups of particles bound to one
another. An example of this phenomenon
may be found in conventional
superconductivity, in which the
phonon-mediated attraction between
conduction electrons leads to the
formation of correlated electron pairs
known as Cooper pairs. When faced with
such systems, one usually turns to other
approximation schemes, such as the
variational method and the WKB
approximation. This is because there is
no analogue of a bound particle in the
unperturbed model and the energy of a
soliton typically goes as the inverse of
the expansion parameter. However, if we
"integrate" over the solitonic
phenomena, the nonperturbative
corrections in this case will be tiny;
of the order of exp(−1/g) or exp(−1/g2)
in the perturbation parameter g.
Perturbation theory can only detect
solutions "close" to the unperturbed
solution, even if there are other
solutions for which the perturbative
expansion is not valid.
The problem of non-perturbative systems
has been somewhat alleviated by the
advent of modern computers. It has
become practical to obtain numerical
non-perturbative solutions for certain
problems, using methods such as density
functional theory. These advances have
been of particular benefit to the field
of quantum chemistry. Computers have
also been used to carry out perturbation
theory calculations to extraordinarily
high levels of precision, which has
proven important in particle physics for
generating theoretical results that can
be compared with experiment.
Time-independent perturbation theory
Time-independent perturbation theory is
one of two categories of perturbation
theory, the other being time-dependent
perturbation. In time-independent
perturbation theory the perturbation
Hamiltonian is static. Time-independent
perturbation theory was presented by
Erwin Schrödinger in a 1926 paper,
shortly after he produced his theories
in wave mechanics. In this paper
Schrödinger referred to earlier work of
Lord Rayleigh, who investigated harmonic
vibrations of a string perturbed by
small inhomogeneities. This is why this
perturbation theory is often referred to
as Rayleigh–Schrödinger perturbation
theory.
= First order corrections=
We begin with an unperturbed Hamiltonian
H0, which is also assumed to have no
time dependence. It has known energy
levels and eigenstates, arising from the
time-independent Schrödinger equation:
For simplicity, we have assumed that the
energies are discrete. The superscripts
denote that these quantities are
associated with the unperturbed system.
Note the use of bra–ket notation.
We now introduce a perturbation to the
Hamiltonian. Let V be a Hamiltonian
representing a weak physical
disturbance, such as a potential energy
produced by an external field. Let λ be
a dimensionless parameter that can take
on values ranging continuously from 0 to
1. The perturbed Hamiltonian is
The energy levels and eigenstates of the
perturbed Hamiltonian are again given by
the Schrödinger equation:
Our goal is to express En and  in terms
of the energy levels and eigenstates of
the old Hamiltonian. If the perturbation
is sufficiently weak, we can write them
as power series in λ:
where
When λ = 0, these reduce to the
unperturbed values, which are the first
term in each series. Since the
perturbation is weak, the energy levels
and eigenstates should not deviate too
much from their unperturbed values, and
the terms should rapidly become smaller
as we go to higher order.
Substituting the power series expansion
into the Schrödinger equation, we obtain
Expanding this equation and comparing
coefficients of each power of λ results
in an infinite series of simultaneous
equations. The zeroth-order equation is
simply the Schrödinger equation for the
unperturbed system. The first-order
equation is
Operating through by . The first term on
the left-hand side cancels with the
first term on the right-hand side.. This
leads to the first-order energy shift:
This is simply the expectation value of
the perturbation Hamiltonian while the
system is in the unperturbed state. This
result can be interpreted in the
following way: suppose the perturbation
is applied, but we keep the system in
the quantum state , which is a valid
quantum state though no longer an energy
eigenstate. The perturbation causes the
average energy of this state to increase
by . However, the true energy shift is
slightly different, because the
perturbed eigenstate is not exactly the
same as . These further shifts are given
by the second and higher order
corrections to the energy.
Before we compute the corrections to the
energy eigenstate, we need to address
the issue of normalization. We may
suppose
but perturbation theory assumes we also
have . It follows that at first order in
λ, we must have
Since the overall phase is not
determined in quantum mechanics, without
loss of generality, we may assume  is
purely real. Therefore,
and we deduce
To obtain the first-order correction to
the energy eigenstate, we insert our
expression for the first-order energy
correction back into the result shown
above of equating the first-order
coefficients of λ. We then make use of
the resolution of the identity,
where the  are in the orthogonal
complement of . The result is
For the moment, suppose that the
zeroth-order energy level is not
degenerate, i.e. there is no eigenstate
of H0 in the orthogonal complement of 
with the energy . We multiply through by
, which gives
and hence the component of the
first-order correction along  since by
assumption . In total we get
The first-order change in the n-th
energy eigenket has a contribution from
each of the energy eigenstates k ≠ n.
Each term is proportional to the matrix
element , which is a measure of how much
the perturbation mixes eigenstate n with
eigenstate k; it is also inversely
proportional to the energy difference
between eigenstates k and n, which means
that the perturbation deforms the
eigenstate to a greater extent if there
are more eigenstates at nearby energies.
We see also that the expression is
singular if any of these states have the
same energy as state n, which is why we
assumed that there is no degeneracy.
= Second-order and higher corrections=
We can find the higher-order deviations
by a similar procedure, though the
calculations become quite tedious with
our current formulation. Our
normalization prescription gives that
Up to second order, the expressions for
the energies and eigenstates are:
Extending the process further, the
third-order energy correction can be
shown to be 
= Effects of degeneracy=
Suppose that two or more energy
eigenstates are degenerate. The
first-order energy shift is not well
defined, since there is no unique way to
choose a basis of eigenstates for the
unperturbed system. The calculation of
the change in the eigenstate is
problematic as well, because the
operator
does not have a well-defined inverse.
Let D denote the subspace spanned by
these degenerate eigenstates. No matter
how small the perturbation is, in the
degenerate subspace D the energy
differences between the eigenstates H0
are zero, so complete mixing of at least
some of these states is assured. Thus
the perturbation can not be considered
small in the D subspace and in that
subspace the new Hamiltonian must be
diagonalized first. These correct
perturbed eigenstates in D are now the
basis for the perturbation expansion:
where only eigenstates outside of the D
subspace are considered to be small. For
the first-order perturbation we need to
solve the perturbed Hamiltonian
restricted to the degenerate subspace D
simultaneously for all the degenerate
eigenstates, where  are first-order
corrections to the degenerate energy
levels. This is equivalent to
diagonalizing the matrix
This procedure is approximate, since we
neglected states outside the D subspace.
The splitting of degenerate energies  is
generally observed. Although the
splitting may be small compared to the
range of energies found in the system,
it is crucial in understanding certain
details, such as spectral lines in
Electron Spin Resonance experiments.
Higher-order corrections due to other
eigenstates can be found in the same way
as for the non-degenerate case
The operator on the left hand side is
not singular when applied to eigenstates
outside D, so we can write
but the effect on the degenerate states
is minuscule, proportional to the square
of the first-order correction .
Near-degenerate states should also be
treated in the above manner, since the
original Hamiltonian won't be larger
than the perturbation in the
near-degenerate subspace. An application
is found in the nearly free electron
model, where near-degeneracy treated
properly gives rise to an energy gap
even for small perturbations. Other
eigenstates will only shift the absolute
energy of all near-degenerate states
simultaneously.
= Generalization to multi-parameter
case=
The generalization of the
time-independent perturbation theory to
the case where there are multiple small
parameters  in place of λ can be
formulated more systematically using the
language of differential geometry, which
basically defines the derivatives of the
quantum states and calculate the
perturbative corrections by taking
derivatives iteratively at the
unperturbed point.
Hamiltonian and force operator
From the differential geometric point of
view, a parameterized Hamiltonian is
considered as a function defined on the
parameter manifold that maps each
particular set of parameters  to an
Hermitian operator H(x μ) that acts on
the Hilbert space. The parameters here
can be external field, interaction
strength, or driving parameters in the
quantum phase transition. Let En(x μ)
and  be the n-th eigenenergy and
eigenstate of H(x μ) respectively. In
the language of differential geometry,
the states  form a vector bundle over
the parameter manifold, on which
derivatives of these states can be
defined. The perturbation theory is to
answer the following question: given 
and  at an unperturbed reference point ,
how to estimate the En(x μ) and  at x μ
close to that reference point.
Without loss of generality, the
coordinate system can be shifted, such
that the reference point  is set to be
the origin. The following linearly
parameterized Hamiltonian is frequently
used
If the parameters x μ are considered as
generalized coordinates, then Fμ should
be identified as the generalized force
operators related to those coordinates.
Different indices μ label the different
forces along different directions in the
parameter manifold. For example, if x μ
denotes the external magnetic field in
the μ-direction, then Fμ should be the
magnetization in the same direction.
Perturbation theory as power series
expansion
The validity of the perturbation theory
lies on the adiabatic assumption, which
assumes the eigenenergies and
eigenstates of the Hamiltonian are
smooth functions of parameters such that
their values in the vicinity region can
be calculated in power series of the
parameters:
Here ∂μ denotes the derivative with
respect to x μ. When applying to the
state , it should be understood as the
covariant derivative if the vector
bundle is equipped with non-vanishing
connection. All the terms on the
right-hand-side of the series are
evaluated at x μ = 0, e.g. En ≡ En(0)
and . This convention will be adopted
throughout this subsection, that all
functions without the parameter
dependence explicitly stated are assumed
to be evaluated at the origin. The power
series may converge slowly or even not
converge when the energy levels are
close to each other. The adiabatic
assumption breaks down when there is
energy level degeneracy, and hence the
perturbation theory is not applicable in
that case.
Hellman–Feynman theorems
The above power series expansion can be
readily evaluated if there is a
systematic approach to calculate the
derivates to any order. Using the chain
rule, the derivatives can be broken down
to the single derivative on either the
energy or the state. The
Hellmann–Feynman theorems are used to
calculate these single derivatives. The
first Hellmann–Feynman theorem gives the
derivative of the energy,
The second Hellmann–Feynman theorem
gives the derivative of the state,
For the linearly parameterized
Hamiltonian, ∂μH simply stands for the
generalized force operator Fμ.
The theorems can be simply derived by
applying the differential operator ∂μ to
both sides of the Schrödinger equation 
which reads
Then overlap with the state  from left
and make use of the Schrödinger equation
again,
Given that the eigenstates of the
Hamiltonian always form an orthonormal
basis , the cases of m = n and m ≠ n can
be discussed separately. The first case
will lead to the first theorem and the
second case to the second theorem, which
can be shown immediately by rearranging
the terms. With the differential rules
given by the Hellmann–Feynman theorems,
the perturbative correction to the
energies and states can be calculated
systematically.
Correction of energy and state
To the second order, the energy
correction reads
The first order derivative ∂μEn is given
by the first Hellmann–Feynman theorem
directly. To obtain the second order
derivative ∂μ∂νEn, simply applying the
differential operator ∂μ to the result
of the first order derivative , which
reads
Note that for linearly parameterized
Hamiltonian, there is no second
derivative ∂μ∂νH = 0 on the operator
level. Resolve the derivative of state
by inserting the complete set of basis,
then all parts can be calculated using
the Hellmann–Feynman theorems. In terms
of Lie derivatives,  according to the
definition of the connection for the
vector bundle. Therefore the case m = n
can be excluded from the summation,
which avoids the singularity of the
energy denominator. The same procedure
can be carried on for higher order
derivatives, from which higher order
corrections are obtained.
The same computational scheme is
applicable for the correction of states.
The result to the second order is as
follows
Both energy derivatives and state
derivatives will be involved in
deduction. Whenever a state derivative
is encountered, resolve it by inserting
the complete set of basis, then the
Hellmann-Feynman theorem is applicable.
Because differentiation can be
calculated systematically, the series
expansion approach to the perturbative
corrections can be coded on computers
with symbolic processing software like
Mathematica.
Effective Hamiltonian
Let H(0) be the Hamiltonian completely
restricted either in the low-energy
subspace  or in the high-energy subspace
, such that there is no matrix element
in H(0) connecting the low- and the
high-energy subspaces, i.e.  if . Let Fμ
= ∂μH be the coupling terms connecting
the subspaces. Then when the high energy
degrees of freedoms are integrated out,
the effective Hamiltonian in the low
energy subspace reads
Here  are restricted in the low energy
subspace. The above result can be
derived by power series expansion of .
In a formal way it is possible to define
an effective Hamiltonian that gives
exactly the low-lying energy states and
wavefunctions. In practice, some kind of
approximation is generally required.
Time-dependent perturbation theory
= Method of variation of constants=
Time-dependent perturbation theory,
developed by Paul Dirac, studies the
effect of a time-dependent perturbation
V(t) applied to a time-independent
Hamiltonian H0.
Since the perturbed Hamiltonian is
time-dependent, so are its energy levels
and eigenstates. Thus, the goals of
time-dependent perturbation theory are
slightly different from time-independent
perturbation theory. One is interested
in the following quantities:
The time-dependent expectation value of
some observable A, for a given initial
state.
The time-dependent amplitudes of those
quantum states that are energy eigenkets
in the unperturbed system.
The first quantity is important because
it gives rise to the classical result of
an A measurement performed on a
macroscopic number of copies of the
perturbed system. For example, we could
take A to be the displacement in the
x-direction of the electron in a
hydrogen atom, in which case the
expected value, when multiplied by an
appropriate coefficient, gives the
time-dependent dielectric polarization
of a hydrogen gas. With an appropriate
choice of perturbation, this allows one
to calculate the AC permittivity of the
gas.
The second quantity looks at the
time-dependent probability of occupation
for each eigenstate. This is
particularly useful in laser physics,
where one is interested in the
populations of different atomic states
in a gas when a time-dependent electric
field is applied. These probabilities
are also useful for calculating the
"quantum broadening" of spectral lines
and particle decay in particle physics
and nuclear physics.
We will briefly examine the method
behind Dirac's formulation of
time-dependent perturbation theory.
Choose an energy basis  for the
unperturbed system. superscripts for the
eigenstates, because it is not useful to
speak of energy levels and eigenstates
for the perturbed system.)
If the unperturbed system is in
eigenstate  at time t = 0, its state at
subsequent times varies only by a phase,
Now, introduce a time-dependent
perturbing Hamiltonian V(t). The
Hamiltonian of the perturbed system is
Let  denote the quantum state of the
perturbed system at time t. It obeys the
time-dependent Schrödinger equation,
The quantum state at each instant can be
expressed as a linear combination of the
complete eigenbasis of :
where the cn(t)s are to be determined
complex functions of t which we will
refer to as amplitudes.
We have explicitly extracted the
exponential phase factors  on the right
hand side. This is only a matter of
convention, and may be done without loss
of generality. The reason we go to this
trouble is that when the system starts
in the state  and no perturbation is
present, the amplitudes have the
convenient property that, for all t,
cj(t) = 1 and cn(t) = 0 if n ≠ j.
The square of the absolute amplitude
cn(t) is the probability that the system
is in state n at time t, since
Plugging into the Schrödinger equation
and using the fact that ∂/∂t acts by a
chain rule, one obtains
By resolving the identity in front of V,
this can be reduced to a set of partial
differential equations for the
amplitudes,
The matrix elements of V play a similar
role as in time-independent perturbation
theory, being proportional to the rate
at which amplitudes are shifted between
states. Note, however, that the
direction of the shift is modified by
the exponential phase factor. Over times
much longer than the energy difference
Ek − En, the phase winds around 0
several times. If the time-dependence of
V is sufficiently slow, this may cause
the state amplitudes to oscillate.
Up to this point, we have made no
approximations, so this set of
differential equations is exact. By
supplying appropriate initial values
cn(t), we could in principle find an
exact solution. This is easily done when
there are only two energy levels, and
this solution is useful for modelling
systems like the ammonia molecule.
However, exact solutions are difficult
to find when there are many energy
levels, and one instead looks for
perturbative solutions. These may be
obtained by expressing the equations in
an integral form,
Repeatedly substituting this expression
for cn back into right hand side, yields
an iterative solution,
where, for example, the first-order term
is
Several further results follow from
this, such as Fermi's golden rule, which
relates the rate of transitions between
quantum states to the density of states
at particular energies; or the Dyson
series, obtained by applying the
iterative method to the time evolution
operator, which is one of the starting
points for the method of Feynman
diagrams.
= Method of Dyson series=
Time-dependent perturbations can be
reorganized through the technique of the
Dyson series. The Schrödinger equation
has the formal solution
where T is the time ordering operator,
Thus, the exponential represents the
following Dyson series,
Consider the following perturbation
problem
assuming that the parameter λ is small
and that the problem  has been solved.
Perform the following unitary
transformation to the interaction
picture,
Consequently, the Schrödinger equation
simplifies to
so it is solved through the above Dyson
series,
as a perturbation series with small λ.
Using the solution of the unperturbed
problem  and , yields, to first order,
Thus, the system, initially in the
unperturbed state , by dint of the
perturbation can go into the state . The
corresponding transition probability
amplitude to first order is
as detailed in the previous
section——while the corresponding
transition probability to a continuum is
furnished by Fermi's golden rule.
As an aside, note that time-independent
perturbation theory is also organized
inside this time-dependent perturbation
theory Dyson series. To see this, write
the unitary evolution operator, obtained
from the above Dyson series, as
and take the perturbation V to be
time-independent.
Using the identity resolution
with  for a pure discrete spectrum,
write
It is evident that, at second order, one
must sum on all the intermediate states.
Assume  and the asymptotic limit of
larger times. This means that, at each
contribution of the perturbation series,
one has to add a multiplicative factor 
in the integrands for ε arbitrarily
small. Thus the limit t → ∞ gives back
the final state of the system by
eliminating all oscillating terms, but
keeping the secular ones. The integrals
are thus computable, and, separating the
diagonal terms from the others yields
where the time secular series yields the
eigenvalues of the perturbed problem
specified above, recursively; whereas
the remaining time-constant part yields
the corrections to the stationary
eigenfunctions also given above
The unitary evolution operator is
applicable to arbitrary eigenstates of
the unperturbed problem and, in this
case, yields a secular series that holds
at small times.
Strong perturbation theory
In a similar way as for small
perturbations, it is possible to develop
a strong perturbation theory. Let us
consider as usual the Schrödinger
equation
and we consider the question if a dual
Dyson series exists that applies in the
limit of a perturbation increasingly
large. This question can be answered in
an affirmative way  and the series is
the well-known adiabatic series. This
approach is quite general and can be
shown in the following way. Let us
consider the perturbation problem
being λ→ ∞. Our aim is to find a
solution in the form
but a direct substitution into the above
equation fails to produce useful
results. This situation can be adjusted
making a rescaling of the time variable
as  producing the following meaningful
equations
that can be solved once we know the
solution of the leading order equation.
But we know that in this case we can use
the adiabatic approximation. When  does
not depend on time one gets the
Wigner-Kirkwood series that is often
used in statistical mechanics. Indeed,
in this case we introduce the unitary
transformation
that defines a free picture as we are
trying to eliminate the interaction
term. Now, in dual way with respect to
the small perturbations, we have to
solve the Schrödinger equation
and we see that the expansion parameter
λ appears only into the exponential and
so, the corresponding Dyson series, a
dual Dyson series, is meaningful at
large λs and is
After the rescaling in time  we can see
that this is indeed a series in 
justifying in this way the name of dual
Dyson series. The reason is that we have
obtained this series simply
interchanging H0 and V and we can go
from one to another applying this
exchange. This is called duality
principle in perturbation theory. The
choice  yields, as already said, a
Wigner-Kirkwood series that is a
gradient expansion. The Wigner-Kirkwood
series is a semiclassical series with
eigenvalues given exactly as for WKB
approximation.
Examples
= Example of first order perturbation
theory – ground state energy of the
quartic oscillator=
Let us consider the quantum harmonic
oscillator with the quartic potential
perturbation and the Hamiltonian
The ground state of the harmonic
oscillator is
() and the energy of unperturbed ground
state is
Using the first order correction formula
we get
or
= Example of first and second order
perturbation theory – quantum pendulum=
Consider the quantum mathematical
pendulum with the Hamiltonian
with the potential energy  taken as the
perturbation i.e.
The unperturbed normalized quantum wave
functions are those of the rigid rotor
and are given by
and the energies
The first order energy correction to the
rotor due to the potential energy is
Using the formula for the second order
correction one gets
or
or
See also
Fermi's Golden Rule
References
