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 (e.g. its energy
levels and eigenstates) 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. In effect, it is describing
a complicated unsolved system using a simple,
solved system.
== Approximate Hamiltonians ==
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
system.
== Applying perturbation theory ==
Perturbation theory is applicable if the problem
at hand cannot be solved exactly, but can
be formulated by adding a "small" term to
the mathematical description of the exactly
solvable problem.
For example, by adding a perturbative electric
potential to the quantum mechanical model
of the hydrogen atom, tiny shifts in the spectral
lines of hydrogen caused by the presence of
an electric field (the Stark effect) can be
calculated. This is only approximate because
the sum of a Coulomb potential with a linear
potential is unstable (has no true bound states)
although the tunneling time (decay rate) 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 (being asymptotic series). There
exist ways to convert them into convergent
series, which can be evaluated for large-expansion
parameters, most efficiently by the variational
method.
In the theory of quantum electrodynamics (QED),
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.
=== Limitations ===
==== Large perturbations ====
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
(the expansion parameter) becomes too large.
==== Non-adiabatic states ====
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 (i.e. non-interacting) 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.
==== Difficult computations ====
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
(see next section). In time-independent perturbation
theory the perturbation Hamiltonian is static
(i.e., possesses no time dependence). 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:
H
0
|
n
(
0
)
⟩
=
E
n
(
0
)
|
n
(
0
)
⟩
,
n
=
1
,
2
,
3
,
⋯
{\displaystyle H_{0}\left|n^{(0)}\right\rangle
=E_{n}^{(0)}\left|n^{(0)}\right\rangle ,\qquad
n=1,2,3,\cdots }
For simplicity, we have assumed that the energies
are discrete. The (0) 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. (Thus,
V is formally a Hermitian operator.) Let λ
be a dimensionless parameter that can take
on values ranging continuously from 0 (no
perturbation) to 1 (the full perturbation).
The perturbed Hamiltonian is
H
=
H
0
+
λ
V
{\displaystyle H=H_{0}+\lambda V}
The energy levels and eigenstates of the perturbed
Hamiltonian are again given by the Schrödinger
equation:
(
H
0
+
λ
V
)
|
n
⟩
=
E
n
|
n
⟩
.
{\displaystyle \left(H_{0}+\lambda V\right)|n\rangle
=E_{n}|n\rangle .}
Our goal is to express En and
|
n
⟩
{\displaystyle |n\rangle }
in terms of the energy levels and eigenstates
of the old Hamiltonian. If the perturbation
is sufficiently weak, we can write them as
a (Maclaurin) power series in λ:
E
n
=
E
n
(
0
)
+
λ
E
n
(
1
)
+
λ
2
E
n
(
2
)
+
⋯
|
n
⟩
=
|
n
(
0
)
⟩
+
λ
|
n
(
1
)
⟩
+
λ
2
|
n
(
2
)
⟩
+
⋯
{\displaystyle {\begin{aligned}E_{n}&=E_{n}^{(0)}+\lambda
E_{n}^{(1)}+\lambda ^{2}E_{n}^{(2)}+\cdots
\\|n\rangle &=\left|n^{(0)}\right\rangle +\lambda
\left|n^{(1)}\right\rangle +\lambda ^{2}\left|n^{(2)}\right\rangle
+\cdots \end{aligned}}}
where
E
n
(
k
)
=
1
k
!
d
k
E
n
d
λ
k
|
λ
=
0
|
n
(
k
)
⟩
=
1
k
!
d
k
|
n
⟩
d
λ
k
|
λ
=
0
{\displaystyle {\begin{aligned}E_{n}^{(k)}&={\frac
{1}{k!}}{\frac {d^{k}E_{n}}{d\lambda ^{k}}}{\bigg
|}_{\lambda =0}\\\left|n^{(k)}\right\rangle
&={\frac {1}{k!}}{\frac {d^{k}|n\rangle }{d\lambda
^{k}}}{\bigg |}_{\lambda =0}\end{aligned}}}
When k = 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
(
H
0
+
λ
V
)
(
|
n
(
0
)
⟩
+
λ
|
n
(
1
)
⟩
+
⋯
)
=
(
E
n
(
0
)
+
λ
E
n
(
1
)
+
⋯
)
(
|
n
(
0
)
⟩
+
λ
|
n
(
1
)
⟩
+
⋯
)
{\displaystyle \left(H_{0}+\lambda V\right)\left(\left|n^{(0)}\right\rangle
+\lambda \left|n^{(1)}\right\rangle +\cdots
\right)=\left(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\cdots
\right)\left(\left|n^{(0)}\right\rangle +\lambda
\left|n^{(1)}\right\rangle +\cdots \right)}
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
H
0
|
n
(
1
)
⟩
+
V
|
n
(
0
)
⟩
=
E
n
(
0
)
|
n
(
1
)
⟩
+
E
n
(
1
)
|
n
(
0
)
⟩
{\displaystyle H_{0}\left|n^{(1)}\right\rangle
+V\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(1)}\right\rangle
+E_{n}^{(1)}\left|n^{(0)}\right\rangle }
Operating through by
⟨
n
(
0
)
|
{\displaystyle \langle n^{(0)}|}
, the first term on the left-hand side cancels
the first term on the right-hand side. (Recall,
the unperturbed Hamiltonian is Hermitian).
This leads to the first-order energy shift:
E
n
(
1
)
=
⟨
n
(
0
)
|
V
|
n
(
0
)
⟩
{\displaystyle E_{n}^{(1)}=\left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle
}
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
|
n
(
0
)
⟩
{\displaystyle |n^{(0)}\rangle }
, which is a valid quantum state though no
longer an energy eigenstate. The perturbation
causes the average energy of this state to
increase by
⟨
n
(
0
)
|
V
|
n
(
0
)
⟩
{\displaystyle \langle n^{(0)}|V|n^{(0)}\rangle
}
. However, the true energy shift is slightly
different, because the perturbed eigenstate
is not exactly the same as
|
n
(
0
)
⟩
{\displaystyle |n^{(0)}\rangle }
. 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
⟨
n
(
0
)
|
n
(
0
)
⟩
=
1
,
{\displaystyle \left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle
=1,}
but perturbation theory assumes we also have
⟨
n
|
n
⟩
=
1
{\displaystyle \langle n|n\rangle =1}
. It follows that at first order in λ, we
must have
(
⟨
n
(
0
)
|
+
λ
⟨
n
(
1
)
|
)
(
|
n
(
0
)
⟩
+
λ
|
n
(
1
)
⟩
)
=
1
{\displaystyle \left(\left\langle n^{(0)}\right|+\lambda
\left\langle n^{(1)}\right|\right)\left(\left|n^{(0)}\right\rangle
+\lambda \left|n^{(1)}\right\rangle \right)=1}
⟨
n
(
0
)
|
n
(
0
)
⟩
+
λ
⟨
n
(
0
)
|
n
(
1
)
⟩
+
λ
⟨
n
(
1
)
|
n
(
0
)
⟩
+
λ
2
⟨
n
(
1
)
|
n
(
1
)
⟩
=
1
{\displaystyle \left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle
+\lambda \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle
+\lambda \left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle
+{\cancel {\lambda ^{2}\left\langle n^{(1)}\right|\left.n^{(1)}\right\rangle
}}=1}
⟨
n
(
0
)
|
n
(
1
)
⟩
+
⟨
n
(
1
)
|
n
(
0
)
⟩
=
0.
{\displaystyle \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle
+\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle
=0.}
Since the overall phase is not determined
in quantum mechanics, without loss of generality,
we may assume
⟨
n
(
0
)
|
n
(
1
)
⟩
{\displaystyle \langle n^{(0)}|n^{(1)}\rangle
}
is purely real. Therefore,
⟨
n
(
0
)
|
n
(
1
)
⟩
=
−
⟨
n
(
1
)
|
n
(
0
)
⟩
,
{\displaystyle \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle
=-\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle
,}
and we deduce
⟨
n
(
0
)
|
n
(
1
)
⟩
=
0.
{\displaystyle \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle
=0.}
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,
V
|
n
(
0
)
⟩
=
(
∑
k
≠
n
|
k
(
0
)
⟩
⟨
k
(
0
)
|
)
V
|
n
(
0
)
⟩
+
(
|
n
(
0
)
⟩
⟨
n
(
0
)
|
)
V
|
n
(
0
)
⟩
=
∑
k
≠
n
|
k
(
0
)
⟩
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
+
E
n
(
1
)
|
n
(
0
)
⟩
,
{\displaystyle {\begin{aligned}V\left|n^{(0)}\right\rangle
&=\left(\sum _{k\neq n}\left|k^{(0)}\right\rangle
\left\langle k^{(0)}\right|\right)V\left|n^{(0)}\right\rangle
+\left(\left|n^{(0)}\right\rangle \left\langle
n^{(0)}\right|\right)V\left|n^{(0)}\right\rangle
\\&=\sum _{k\neq n}\left|k^{(0)}\right\rangle
\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
+E_{n}^{(1)}\left|n^{(0)}\right\rangle ,\end{aligned}}}
where the
|
k
(
0
)
⟩
{\displaystyle |k^{(0)}\rangle }
are in the orthogonal complement of
|
n
(
0
)
⟩
{\displaystyle |n^{(0)}\rangle }
. The first-order equation may thus be expressed
as
(
E
n
(
0
)
−
H
0
)
|
n
(
1
)
⟩
=
∑
k
≠
n
|
k
(
0
)
⟩
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
{\displaystyle \left(E_{n}^{(0)}-H_{0}\right)\left|n^{(1)}\right\rangle
=\sum _{k\neq n}\left|k^{(0)}\right\rangle
\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
}
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
|
n
(
0
)
⟩
{\displaystyle |n^{(0)}\rangle }
with the energy
E
n
(
0
)
{\displaystyle E_{n}^{(0)}}
. After renaming the summation dummy index
above as
k
′
{\displaystyle k'}
, we can pick any
k
≠
n
{\displaystyle k\neq n}
, and multiply through by
⟨
k
(
0
)
|
{\displaystyle \langle k^{(0)}|}
giving
(
E
n
(
0
)
−
E
k
(
0
)
)
⟨
k
(
0
)
|
n
(
1
)
⟩
=
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
{\displaystyle \left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left\langle
k^{(0)}\right.\left|n^{(1)}\right\rangle =\left\langle
k^{(0)}\right|V\left|n^{(0)}\right\rangle
}
We see that the above
⟨
k
(
0
)
|
n
(
1
)
⟩
{\displaystyle \langle k^{(0)}|n^{(1)}\rangle
}
also gives us the component of the first-order
correction along
|
k
(
0
)
⟩
{\displaystyle |k^{(0)}\rangle }
.
Thus in total we get,
|
n
(
1
)
⟩
=
∑
k
≠
n
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
E
n
(
0
)
−
E
k
(
0
)
|
k
(
0
)
⟩
{\displaystyle \left|n^{(1)}\right\rangle
=\sum _{k\neq n}{\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
}{E_{n}^{(0)}-E_{k}^{(0)}}}\left|k^{(0)}\right\rangle
}
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
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
{\displaystyle \langle k^{(0)}|V|n^{(0)}\rangle
}
, 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
2
⟨
n
(
0
)
|
n
(
2
)
⟩
+
⟨
n
(
1
)
|
n
(
1
)
⟩
=
0.
{\displaystyle 2\left\langle n^{(0)}\right|\left.n^{(2)}\right\rangle
+\left\langle n^{(1)}\right|\left.n^{(1)}\right\rangle
=0.}
Up to second order, the expressions for the
energies and (normalized) eigenstates are:
E
n
(
λ
)
=
E
n
(
0
)
+
λ
⟨
n
(
0
)
|
V
|
n
(
0
)
⟩
+
λ
2
∑
k
≠
n
|
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
|
2
E
n
(
0
)
−
E
k
(
0
)
+
O
(
λ
3
)
{\displaystyle E_{n}(\lambda )=E_{n}^{(0)}+\lambda
\left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle
+\lambda ^{2}\sum _{k\neq n}{\frac {\left|\left\langle
k^{(0)}\right|V\left|n^{(0)}\right\rangle
\right|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}}+O(\lambda
^{3})}
|
n
(
λ
)
⟩
=
|
n
(
0
)
⟩
+
λ
∑
k
≠
n
|
k
(
0
)
⟩
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
E
n
(
0
)
−
E
k
(
0
)
+
λ
2
∑
k
≠
n
∑
ℓ
≠
n
|
k
(
0
)
⟩
⟨
k
(
0
)
|
V
|
ℓ
(
0
)
⟩
⟨
ℓ
(
0
)
|
V
|
n
(
0
)
⟩
(
E
n
(
0
)
−
E
k
(
0
)
)
(
E
n
(
0
)
−
E
ℓ
(
0
)
)
−
λ
2
∑
k
≠
n
|
k
(
0
)
⟩
⟨
n
(
0
)
|
V
|
n
(
0
)
⟩
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
(
E
n
(
0
)
−
E
k
(
0
)
)
2
−
1
2
λ
2
|
n
(
0
)
⟩
∑
k
≠
n
⟨
n
(
0
)
|
V
|
k
(
0
)
⟩
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
(
E
n
(
0
)
−
E
k
(
0
)
)
2
+
O
(
λ
3
)
.
{\displaystyle {\begin{aligned}|n(\lambda
)\rangle =\left|n^{(0)}\right\rangle &+\lambda
\sum _{k\neq n}\left|k^{(0)}\right\rangle
{\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
}{E_{n}^{(0)}-E_{k}^{(0)}}}+\lambda ^{2}\sum
_{k\neq n}\sum _{\ell \neq n}\left|k^{(0)}\right\rangle
{\frac {\left\langle k^{(0)}\right|V\left|\ell
^{(0)}\right\rangle \left\langle \ell ^{(0)}\right|V\left|n^{(0)}\right\rangle
}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left(E_{n}^{(0)}-E_{\ell
}^{(0)}\right)}}\\&-\lambda ^{2}\sum _{k\neq
n}\left|k^{(0)}\right\rangle {\frac {\left\langle
n^{(0)}\right|V\left|n^{(0)}\right\rangle
\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}}-{\frac
{1}{2}}\lambda ^{2}\left|n^{(0)}\right\rangle
\sum _{k\neq n}{\frac {\left\langle n^{(0)}\right|V\left|k^{(0)}\right\rangle
\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle
}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}}+O(\lambda
^{3}).\end{aligned}}}
Extending the process further, the third-order
energy correction can be shown to be
E
n
(
3
)
=
∑
k
≠
n
∑
m
≠
n
⟨
n
(
0
)
|
V
|
m
(
0
)
⟩
⟨
m
(
0
)
|
V
|
k
(
0
)
⟩
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
(
E
n
(
0
)
−
E
m
(
0
)
)
(
E
n
(
0
)
−
E
k
(
0
)
)
−
⟨
n
(
0
)
|
V
|
n
(
0
)
⟩
∑
m
≠
n
|
⟨
n
(
0
)
|
V
|
m
(
0
)
⟩
|
2
(
E
n
(
0
)
−
E
m
(
0
)
)
2
.
{\displaystyle E_{n}^{(3)}=\sum _{k\neq n}\sum
_{m\neq n}{\frac {\langle n^{(0)}|V|m^{(0)}\rangle
\langle m^{(0)}|V|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle
}{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)\left(E_{n}^{(0)}-E_{k}^{(0)}\right)}}-\langle
n^{(0)}|V|n^{(0)}\rangle \sum _{m\neq n}{\frac
{|\langle n^{(0)}|V|m^{(0)}\rangle |^{2}}{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)^{2}}}.}
=== 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 various eigenstates
for a given energy will perturb with different
energies, or may well possess no continuous
family of perturbations at all.
This is manifested in the calculation of the
perturbed eigenstate via the fact that the
operator
E
n
(
0
)
−
H
0
{\displaystyle E_{n}^{(0)}-H_{0}}
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
of H are non-zero, so complete mixing of at
least some of these states is assured. Typically,
the eigenvalues will split, and the eigenspaces
will become simple (one-dimensional), or at
least of smaller dimension than D.
The successful perturbations will not be "small"
relative to a poorly chosen basis of D. Instead,
we consider the perturbation "small" if the
new eigenstate is close to the subspace D.
The new Hamiltonian must be diagonalized in
D, or a slight variation of D, so to speak.
These perturbed eigenstates in D are now the
basis for the perturbation expansion,
|
n
⟩
=
∑
k
∈
D
α
n
k
|
k
(
0
)
⟩
+
λ
|
n
(
1
)
⟩
.
{\displaystyle |n\rangle =\sum _{k\in D}\alpha
_{nk}|k^{(0)}\rangle +\lambda |n^{(1)}\rangle
.}
For the first-order perturbation, we need
solve the perturbed Hamiltonian restricted
to the degenerate subspace D,
V
|
k
(
0
)
⟩
=
ϵ
k
|
k
(
0
)
⟩
+
small
∀
|
k
(
0
)
⟩
∈
D
,
{\displaystyle V|k^{(0)}\rangle =\epsilon
_{k}|k^{(0)}\rangle +{\text{small}}\qquad
\forall |k^{(0)}\rangle \in D,}
simultaneously for all the degenerate eigenstates,
where
ϵ
k
{\displaystyle \epsilon _{k}}
are first-order corrections to the degenerate
energy levels, and "small" is a vector of
O
(
λ
)
{\displaystyle O(\lambda )}
orthogonal to D. This amounts to diagonalizing
the matrix
⟨
k
(
0
)
|
V
|
l
(
0
)
⟩
=
V
k
l
∀
|
k
(
0
)
⟩
,
|
l
(
0
)
⟩
∈
D
.
{\displaystyle \langle k^{(0)}|V|l^{(0)}\rangle
=V_{kl}\qquad \forall \;|k^{(0)}\rangle ,|l^{(0)}\rangle
\in D.}
This procedure is approximate, since we neglected
states outside the D subspace ("small"). The
splitting of degenerate energies
ϵ
k
{\displaystyle \epsilon _{k}}
is generally observed. Although the splitting
may be small,
O
(
λ
)
{\displaystyle O(\lambda )}
, 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
outside D can be found in the same way as
for the non-degenerate case,
(
E
n
(
0
)
−
H
0
)
|
n
(
1
)
⟩
=
∑
k
∉
D
(
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
)
|
k
(
0
)
⟩
.
{\displaystyle \left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle
=\sum _{k\not \in D}\left(\langle k^{(0)}|V|n^{(0)}\rangle
\right)|k^{(0)}\rangle .}
The operator on the left-hand side is not
singular when applied to eigenstates outside
D, so we can write
|
n
(
1
)
⟩
=
∑
k
∉
D
⟨
k
(
0
)
|
V
|
n
(
0
)
⟩
E
n
(
0
)
−
E
k
(
0
)
|
k
(
0
)
⟩
,
{\displaystyle |n^{(1)}\rangle =\sum _{k\not
\in D}{\frac {\langle k^{(0)}|V|n^{(0)}\rangle
}{E_{n}^{(0)}-E_{k}^{(0)}}}|k^{(0)}\rangle
,}
but the effect on the degenerate states is
of
O
(
λ
)
{\displaystyle O(\lambda )}
.
Near-degenerate states should also be treated
similarly, when the original Hamiltonian splits
aren't 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
x
μ
=
(
x
1
,
x
2
,
⋯
)
{\displaystyle x^{\mu }=(x^{1},x^{2},\cdots
)}
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 calculates 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
(
x
1
,
x
2
,
⋯
)
{\displaystyle (x^{1},x^{2},\cdots )}
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
|
n
(
x
μ
)
⟩
{\displaystyle |n(x^{\mu })\rangle }
be the n-th eigenenergy and eigenstate of
H(x μ) respectively. In the language of
differential geometry, the states
|
n
(
x
μ
)
⟩
{\displaystyle |n(x^{\mu })\rangle }
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
E
n
(
x
0
μ
)
{\displaystyle E_{n}(x_{0}^{\mu })}
and
|
n
(
x
0
μ
)
⟩
{\displaystyle |n(x_{0}^{\mu })\rangle }
at an unperturbed reference point
x
0
μ
{\displaystyle x_{0}^{\mu }}
, how to estimate the En(x μ) and
|
n
(
x
μ
)
⟩
{\displaystyle |n(x^{\mu })\rangle }
at x μ close to that reference point.
Without loss of generality, the coordinate
system can be shifted, such that the reference
point
x
0
μ
=
0
{\displaystyle x_{0}^{\mu }=0}
is set to be the origin. The following linearly
parameterized Hamiltonian is frequently used
H
(
x
μ
)
=
H
(
0
)
+
x
μ
F
μ
.
{\displaystyle H(x^{\mu })=H(0)+x^{\mu }F_{\mu
}.}
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 (like Taylor expansion)
of the parameters:
E
n
(
x
μ
)
=
E
n
+
x
μ
∂
μ
E
n
+
1
2
!
x
μ
x
ν
∂
μ
∂
ν
E
n
+
⋯
|
n
(
x
μ
)
⟩
=
|
n
⟩
+
x
μ
|
∂
μ
n
⟩
+
1
2
!
x
μ
x
ν
|
∂
μ
∂
ν
n
⟩
+
⋯
{\displaystyle {\begin{aligned}E_{n}(x^{\mu
})&=E_{n}+x^{\mu }\partial _{\mu }E_{n}+{\frac
{1}{2!}}x^{\mu }x^{\nu }\partial _{\mu }\partial
_{\nu }E_{n}+\cdots \\\left|n(x^{\mu })\right\rangle
&=|n\rangle +x^{\mu }|\partial _{\mu }n\rangle
+{\frac {1}{2!}}x^{\mu }x^{\nu }|\partial
_{\mu }\partial _{\nu }n\rangle +\cdots \end{aligned}}}
Here ∂μ denotes the derivative with respect
to x μ. When applying to the state
|
∂
μ
n
⟩
{\displaystyle |\partial _{\mu }n\rangle }
, 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
|
n
⟩
≡
|
n
(
0
)
⟩
{\displaystyle |n\rangle \equiv |n(0)\rangle
}
. 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.
==== Hellmann–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,
∂
μ
E
n
=
⟨
n
|
∂
μ
H
|
n
⟩
{\displaystyle \partial _{\mu }E_{n}=\langle
n|\partial _{\mu }H|n\rangle }
The second Hellmann–Feynman theorem gives
the derivative of the state (resolved by the
complete basis with m ≠ n),
⟨
m
|
∂
μ
n
⟩
=
⟨
m
|
∂
μ
H
|
n
⟩
E
n
−
E
m
,
⟨
∂
μ
m
|
n
⟩
=
⟨
m
|
∂
μ
H
|
n
⟩
E
m
−
E
n
.
{\displaystyle \langle m|\partial _{\mu }n\rangle
={\frac {\langle m|\partial _{\mu }H|n\rangle
}{E_{n}-E_{m}}},\qquad \langle \partial _{\mu
}m|n\rangle ={\frac {\langle m|\partial _{\mu
}H|n\rangle }{E_{m}-E_{n}}}.}
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
H
|
n
⟩
=
E
n
|
n
⟩
,
{\displaystyle H|n\rangle =E_{n}|n\rangle
,}
which reads
∂
μ
H
|
n
⟩
+
H
|
∂
μ
n
⟩
=
∂
μ
E
n
|
n
⟩
+
E
n
|
∂
μ
n
⟩
.
{\displaystyle \partial _{\mu }H|n\rangle
+H|\partial _{\mu }n\rangle =\partial _{\mu
}E_{n}|n\rangle +E_{n}|\partial _{\mu }n\rangle
.}
Then overlap with the state
⟨
m
|
{\displaystyle \langle m|}
from left and make use of the Schrödinger
equation
⟨
m
|
H
=
⟨
m
|
E
m
{\displaystyle \langle m|H=\langle m|E_{m}}
again,
⟨
m
|
∂
μ
H
|
n
⟩
+
E
m
⟨
m
|
∂
μ
n
⟩
=
∂
μ
E
n
⟨
m
|
n
⟩
+
E
n
⟨
m
|
∂
μ
n
⟩
.
{\displaystyle \langle m|\partial _{\mu }H|n\rangle
+E_{m}\langle m|\partial _{\mu }n\rangle =\partial
_{\mu }E_{n}\langle m|n\rangle +E_{n}\langle
m|\partial _{\mu }n\rangle .}
Given that the eigenstates of the Hamiltonian
always form an orthonormal basis
⟨
m
|
n
⟩
=
δ
m
n
{\displaystyle \langle m|n\rangle =\delta
_{mn}}
, 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
E
n
(
x
μ
)
=
⟨
n
|
H
|
n
⟩
+
⟨
n
|
∂
μ
H
|
n
⟩
x
μ
+
ℜ
∑
m
≠
n
⟨
n
|
∂
ν
H
|
m
⟩
⟨
m
|
∂
μ
H
|
n
⟩
E
n
−
E
m
x
μ
x
ν
+
⋯
,
{\displaystyle E_{n}(x^{\mu })=\langle n|H|n\rangle
+\langle n|\partial _{\mu }H|n\rangle x^{\mu
}+\Re \sum _{m\neq n}{\frac {\langle n|\partial
_{\nu }H|m\rangle \langle m|\partial _{\mu
}H|n\rangle }{E_{n}-E_{m}}}x^{\mu }x^{\nu
}+\cdots ,}
where
ℜ
{\displaystyle \Re }
denotes the real part function.
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
⟨
n
|
∂
ν
H
|
n
⟩
{\displaystyle \langle n|\partial _{\nu }H|n\rangle
}
, which reads
∂
μ
∂
ν
E
n
=
⟨
∂
μ
n
|
∂
ν
H
|
n
⟩
+
⟨
n
|
∂
μ
∂
ν
H
|
n
⟩
+
⟨
n
|
∂
ν
H
|
∂
μ
n
⟩
.
{\displaystyle \partial _{\mu }\partial _{\nu
}E_{n}=\langle \partial _{\mu }n|\partial
_{\nu }H|n\rangle +\langle n|\partial _{\mu
}\partial _{\nu }H|n\rangle +\langle n|\partial
_{\nu }H|\partial _{\mu }n\rangle .}
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,
∂
μ
∂
ν
E
n
=
∑
m
(
⟨
∂
μ
n
|
m
⟩
⟨
m
|
∂
ν
H
|
n
⟩
+
⟨
n
|
∂
ν
H
|
m
⟩
⟨
m
|
∂
μ
n
⟩
)
,
{\displaystyle \partial _{\mu }\partial _{\nu
}E_{n}=\sum _{m}\left(\langle \partial _{\mu
}n|m\rangle \langle m|\partial _{\nu }H|n\rangle
+\langle n|\partial _{\nu }H|m\rangle \langle
m|\partial _{\mu }n\rangle \right),}
then all parts can be calculated using the
Hellmann–Feynman theorems. In terms of Lie
derivatives,
⟨
∂
μ
n
|
n
⟩
=
⟨
n
|
∂
μ
n
⟩
=
0
{\displaystyle \langle \partial _{\mu }n|n\rangle
=\langle n|\partial _{\mu }n\rangle =0}
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
|
n
(
x
μ
)
⟩
=
|
n
⟩
+
∑
m
≠
n
⟨
m
|
∂
μ
H
|
n
⟩
E
n
−
E
m
|
m
⟩
x
μ
+
(
∑
m
≠
n
∑
l
≠
n
⟨
m
|
∂
μ
H
|
l
⟩
⟨
l
|
∂
ν
H
|
n
⟩
(
E
n
−
E
m
)
(
E
n
−
E
l
)
|
m
⟩
−
∑
m
≠
n
⟨
m
|
∂
μ
H
|
n
⟩
⟨
n
|
∂
ν
H
|
n
⟩
(
E
n
−
E
m
)
2
|
m
⟩
−
1
2
∑
m
≠
n
⟨
n
|
∂
μ
H
|
m
⟩
⟨
m
|
∂
ν
H
|
n
⟩
(
E
n
−
E
m
)
2
|
m
⟩
)
x
μ
x
ν
+
⋯
.
{\displaystyle {\begin{aligned}\left|n\left(x^{\mu
}\right)\right\rangle =|n\rangle &+\sum _{m\neq
n}{\frac {\langle m|\partial _{\mu }H|n\rangle
}{E_{n}-E_{m}}}|m\rangle x^{\mu }\\&+\left(\sum
_{m\neq n}\sum _{l\neq n}{\frac {\langle m|\partial
_{\mu }H|l\rangle \langle l|\partial _{\nu
}H|n\rangle }{(E_{n}-E_{m})(E_{n}-E_{l})}}|m\rangle
-\sum _{m\neq n}{\frac {\langle m|\partial
_{\mu }H|n\rangle \langle n|\partial _{\nu
}H|n\rangle }{(E_{n}-E_{m})^{2}}}|m\rangle
-{\frac {1}{2}}\sum _{m\neq n}{\frac {\langle
n|\partial _{\mu }H|m\rangle \langle m|\partial
_{\nu }H|n\rangle }{(E_{n}-E_{m})^{2}}}|m\rangle
\right)x^{\mu }x^{\nu }+\cdots .\end{aligned}}}
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
H
L
{\displaystyle {\mathcal {H}}_{L}}
or in the high-energy subspace
H
H
{\displaystyle {\mathcal {H}}_{H}}
, such that there is no matrix element in
H(0) connecting the low- and the high-energy
subspaces, i.e.
⟨
m
|
H
(
0
)
|
l
⟩
=
0
{\displaystyle \langle m|H(0)|l\rangle =0}
if
m
∈
H
L
,
l
∈
H
H
{\displaystyle m\in {\mathcal {H}}_{L},l\in
{\mathcal {H}}_{H}}
. 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
H
m
n
eff
(
x
μ
)
=
⟨
m
|
H
|
n
⟩
+
δ
n
m
⟨
m
|
∂
μ
H
|
n
⟩
x
μ
+
1
2
!
∑
l
∈
H
H
(
⟨
m
|
∂
μ
H
|
l
⟩
⟨
l
|
∂
ν
H
|
n
⟩
E
m
−
E
l
+
⟨
m
|
∂
ν
H
|
l
⟩
⟨
l
|
∂
μ
H
|
n
⟩
E
n
−
E
l
)
x
μ
x
ν
+
⋯
.
{\displaystyle H_{mn}^{\text{eff}}\left(x^{\mu
}\right)=\langle m|H|n\rangle +\delta _{nm}\langle
m|\partial _{\mu }H|n\rangle x^{\mu }+{\frac
{1}{2!}}\sum _{l\in {\mathcal {H}}_{H}}\left({\frac
{\langle m|\partial _{\mu }H|l\rangle \langle
l|\partial _{\nu }H|n\rangle }{E_{m}-E_{l}}}+{\frac
{\langle m|\partial _{\nu }H|l\rangle \langle
l|\partial _{\mu }H|n\rangle }{E_{n}-E_{l}}}\right)x^{\mu
}x^{\nu }+\cdots .}
Here
m
,
n
∈
H
L
{\displaystyle m,n\in {\mathcal {H}}_{L}}
are restricted in the low energy subspace.
The above result can be derived by power series
expansion of
⟨
m
|
H
(
x
μ
)
|
n
⟩
{\displaystyle \langle m|H(x^{\mu })|n\rangle
}
.
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 (perturbation
theory) 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 (eigenvectors)
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
(i.e. an oscillating electric potential),
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
(see line broadening) 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
|
n
⟩
{\displaystyle {|n\rangle }}
for the unperturbed system. (We drop the (0)
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
(of the Hamiltonian)
|
j
⟩
{\displaystyle |j\rangle }
at time t = 0, its state at subsequent times
varies only by a phase (in the Schrödinger
picture, where state vectors evolve in time
and operators are constant),
|
j
(
t
)
⟩
=
e
−
i
E
j
t
/
ℏ
|
j
⟩
.
{\displaystyle |j(t)\rangle =e^{-iE_{j}t/\hbar
}|j\rangle ~.}
Now, introduce a time-dependent perturbing
Hamiltonian V(t). The Hamiltonian of the perturbed
system is
H
=
H
0
+
V
(
t
)
.
{\displaystyle H=H_{0}+V(t)~.}
Let
|
ψ
(
t
)
⟩
{\displaystyle |\psi (t)\rangle }
denote the quantum state of the perturbed
system at time t. It obeys the time-dependent
Schrödinger equation,
H
|
ψ
(
t
)
⟩
=
i
ℏ
∂
∂
t
|
ψ
(
t
)
⟩
.
{\displaystyle H|\psi (t)\rangle =i\hbar {\frac
{\partial }{\partial t}}|\psi (t)\rangle ~.}
The quantum state at each instant can be expressed
as a linear combination of the complete eigenbasis
of
|
n
⟩
{\displaystyle |n\rangle }
:
where the cn(t)s are to be determined complex
functions of t which we will refer to as amplitudes
(strictly speaking, they are the amplitudes
in the Dirac picture).
We have explicitly extracted the exponential
phase factors
exp
⁡
(
−
i
E
n
t
/
ℏ
)
{\displaystyle \exp(-iE_{n}t/\hbar )}
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
|
j
⟩
{\displaystyle |j\rangle }
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
|
c
n
(
t
)
|
2
=
|
⟨
n
|
ψ
(
t
)
⟩
|
2
.
{\displaystyle \left|c_{n}(t)\right|^{2}=\left|\langle
n|\psi (t)\rangle \right|^{2}~.}
Plugging into the Schrödinger equation and
using the fact that ∂/∂t acts by a product
rule, one obtains
∑
n
(
i
ℏ
∂
c
n
∂
t
−
c
n
(
t
)
V
(
t
)
)
e
−
i
E
n
t
/
ℏ
|
n
⟩
=
0
.
{\displaystyle \sum _{n}\left(i\hbar {\frac
{\partial c_{n}}{\partial t}}-c_{n}(t)V(t)\right)e^{-iE_{n}t/\hbar
}|n\rangle =0~.}
By resolving the identity in front of V and
multiplying through by the bra
⟨
n
|
{\displaystyle \langle n|}
on the left, this can be reduced to a set
of coupled differential equations for the
amplitudes,
∂
c
n
∂
t
=
−
i
ℏ
∑
k
⟨
n
|
V
(
t
)
|
k
⟩
c
k
(
t
)
e
−
i
(
E
k
−
E
n
)
t
/
ℏ
.
{\displaystyle {\frac {\partial c_{n}}{\partial
t}}={\frac {-i}{\hbar }}\sum _{k}\langle n|V(t)|k\rangle
\,c_{k}(t)\,e^{-i(E_{k}-E_{n})t/\hbar }~.}
where we have used equation (1) to evaluate
the sum on n in the second term, then used
the fact that
⟨
k
|
Ψ
(
t
)
⟩
=
c
k
(
t
)
e
−
i
E
k
t
/
ℏ
{\displaystyle \langle k|\Psi (t)\rangle =c_{k}(t)e^{-iE_{k}t/\hbar
}}
.
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. ( E.g., such oscillations are
useful for managing radiative transitions
in a laser.)
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 (i.e.,
non-perturbative) solution. This is easily
done when there are only two energy levels
(n = 1, 2), 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,
c
n
(
t
)
=
c
n
(
0
)
+
−
i
ℏ
∑
k
∫
0
t
d
t
′
⟨
n
|
V
(
t
′
)
|
k
⟩
c
k
(
t
′
)
e
−
i
(
E
k
−
E
n
)
t
′
/
ℏ
.
{\displaystyle c_{n}(t)=c_{n}(0)+{\frac {-i}{\hbar
}}\sum _{k}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle
\,c_{k}(t')\,e^{-i(E_{k}-E_{n})t'/\hbar }~.}
Repeatedly substituting this expression for
cn back into right hand side, yields an iterative
solution,
c
n
(
t
)
=
c
n
(
0
)
+
c
n
(
1
)
+
c
n
(
2
)
+
⋯
{\displaystyle c_{n}(t)=c_{n}^{(0)}+c_{n}^{(1)}+c_{n}^{(2)}+\cdots
}
where, for example, the first-order term is
c
n
(
1
)
(
t
)
=
−
i
ℏ
∑
k
∫
0
t
d
t
′
⟨
n
|
V
(
t
′
)
|
k
⟩
c
k
(
0
)
e
−
i
(
E
k
−
E
n
)
t
′
/
ℏ
.
{\displaystyle c_{n}^{(1)}(t)={\frac {-i}{\hbar
}}\sum _{k}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle
\,c_{k}(0)\,e^{-i(E_{k}-E_{n})t'/\hbar }~.}
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
H
(
t
)
|
ψ
(
t
)
⟩
=
i
ℏ
∂
|
ψ
(
t
)
⟩
∂
t
{\displaystyle H(t)|\psi (t)\rangle =i\hbar
{\frac {\partial |\psi (t)\rangle }{\partial
t}}}
has the formal solution
|
ψ
(
t
)
⟩
=
T
exp
⁡
[
−
i
ℏ
∫
t
0
t
d
t
′
H
(
t
′
)
]
|
ψ
(
t
0
)
⟩
,
{\displaystyle |\psi (t)\rangle =T\exp {\left[-{\frac
{i}{\hbar }}\int _{t_{0}}^{t}dt'H(t')\right]}|\psi
(t_{0})\rangle ~,}
where T is the time ordering operator,
T
A
(
t
1
)
A
(
t
2
)
=
{
A
(
t
1
)
A
(
t
2
)
t
1
>
t
2
A
(
t
2
)
A
(
t
1
)
t
2
>
t
1
.
{\displaystyle TA(t_{1})A(t_{2})={\begin{cases}A(t_{1})A(t_{2})&t_{1}>t_{2}\\A(t_{2})A(t_{1})&t_{2}>t_{1}\end{cases}}~.}
Thus, the exponential represents the following
Dyson series,
|
ψ
(
t
)
⟩
=
[
1
−
i
ℏ
∫
t
0
t
d
t
1
H
(
t
1
)
−
1
ℏ
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
H
(
t
1
)
H
(
t
2
)
+
…
]
|
ψ
(
t
0
)
⟩
.
{\displaystyle |\psi (t)\rangle =\left[1-{\frac
{i}{\hbar }}\int _{t_{0}}^{t}dt_{1}H(t_{1})-{\frac
{1}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int
_{t_{0}}^{t_{1}}dt_{2}H(t_{1})H(t_{2})+\ldots
\right]|\psi (t_{0})\rangle ~.}
Note that in the second term, the 1/2! factor
exactly cancels the double contribution due
to the time-ordering operator, etc.
Consider the following perturbation problem
[
H
0
+
λ
V
(
t
)
]
|
ψ
(
t
)
⟩
=
i
ℏ
∂
|
ψ
(
t
)
⟩
∂
t
,
{\displaystyle [H_{0}+\lambda V(t)]|\psi (t)\rangle
=i\hbar {\frac {\partial |\psi (t)\rangle
}{\partial t}}~,}
assuming that the parameter λ is small and
that the problem
H
0
|
n
⟩
=
E
n
|
n
⟩
{\displaystyle H_{0}|n\rangle =E_{n}|n\rangle
}
has been solved.
Perform the following unitary transformation
to the interaction picture (or Dirac picture),
|
ψ
(
t
)
⟩
=
e
−
i
ℏ
H
0
(
t
−
t
0
)
|
ψ
I
(
t
)
⟩
.
{\displaystyle |\psi (t)\rangle =e^{-{\frac
{i}{\hbar }}H_{0}(t-t_{0})}|\psi _{I}(t)\rangle
~.}
Consequently, the Schrödinger equation simplifies
to
λ
e
i
ℏ
H
0
(
t
−
t
0
)
V
(
t
)
e
−
i
ℏ
H
0
(
t
−
t
0
)
|
ψ
I
(
t
)
⟩
=
i
ℏ
∂
|
ψ
I
(
t
)
⟩
∂
t
,
{\displaystyle \lambda e^{{\frac {i}{\hbar
}}H_{0}(t-t_{0})}V(t)e^{-{\frac {i}{\hbar
}}H_{0}(t-t_{0})}|\psi _{I}(t)\rangle =i\hbar
{\frac {\partial |\psi _{I}(t)\rangle }{\partial
t}}~,}
so it is solved through the above Dyson series,
|
ψ
I
(
t
)
⟩
=
[
1
−
i
λ
ℏ
∫
t
0
t
d
t
1
e
i
ℏ
H
0
(
t
1
−
t
0
)
V
(
t
1
)
e
−
i
ℏ
H
0
(
t
1
−
t
0
)
−
λ
2
ℏ
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
e
i
ℏ
H
0
(
t
1
−
t
0
)
V
(
t
1
)
e
−
i
ℏ
H
0
(
t
1
−
t
0
)
e
i
ℏ
H
0
(
t
2
−
t
0
)
V
(
t
2
)
e
−
i
ℏ
H
0
(
t
2
−
t
0
)
+
…
]
|
ψ
(
t
0
)
⟩
,
{\displaystyle |\psi _{I}(t)\rangle =\left[1-{\frac
{i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}e^{{\frac
{i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac
{i}{\hbar }}H_{0}(t_{1}-t_{0})}-{\frac {\lambda
^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int
_{t_{0}}^{t_{1}}dt_{2}e^{{\frac {i}{\hbar
}}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar
}}H_{0}(t_{1}-t_{0})}e^{{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-{\frac
{i}{\hbar }}H_{0}(t_{2}-t_{0})}+\ldots \right]|\psi
(t_{0})\rangle ~,}
as a perturbation series with small λ.
Using the solution of the unperturbed problem
H
0
|
n
⟩
=
E
n
|
n
⟩
{\displaystyle H_{0}|n\rangle =E_{n}|n\rangle
}
and
∑
n
|
n
⟩
⟨
n
|
=
1
{\displaystyle \sum _{n}|n\rangle \langle
n|=1}
(for the sake of simplicity assume a pure
discrete spectrum), yields, to first order,
|
ψ
I
(
t
)
⟩
=
[
1
−
i
λ
ℏ
∑
m
∑
n
∫
t
0
t
d
t
1
⟨
m
|
V
(
t
1
)
|
n
⟩
e
−
i
ℏ
(
E
n
−
E
m
)
(
t
1
−
t
0
)
|
m
⟩
⟨
n
|
+
…
]
|
ψ
(
t
0
)
⟩
.
{\displaystyle |\psi _{I}(t)\rangle =\left[1-{\frac
{i\lambda }{\hbar }}\sum _{m}\sum _{n}\int
_{t_{0}}^{t}dt_{1}\langle m|V(t_{1})|n\rangle
e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle
\langle n|+\ldots \right]|\psi (t_{0})\rangle
~.}
Thus, the system, initially in the unperturbed
state
|
α
⟩
=
|
ψ
(
t
0
)
⟩
{\displaystyle |\alpha \rangle =|\psi (t_{0})\rangle
}
, by dint of the perturbation can go into
the state
|
β
⟩
{\displaystyle |\beta \rangle }
. The corresponding transition probability
amplitude to first order is
A
α
β
=
−
i
λ
ℏ
∫
t
0
t
d
t
1
⟨
β
|
V
(
t
1
)
|
α
⟩
e
−
i
ℏ
(
E
α
−
E
β
)
(
t
1
−
t
0
)
,
{\displaystyle A_{\alpha \beta }=-{\frac {i\lambda
}{\hbar }}\int _{t_{0}}^{t}dt_{1}\langle \beta
|V(t_{1})|\alpha \rangle e^{-{\frac {i}{\hbar
}}(E_{\alpha }-E_{\beta })(t_{1}-t_{0})}~,}
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
U
(
t
)
=
1
−
i
λ
ℏ
∫
t
0
t
d
t
1
e
i
ℏ
H
0
(
t
1
−
t
0
)
V
(
t
1
)
e
−
i
ℏ
H
0
(
t
1
−
t
0
)
−
λ
2
ℏ
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
e
i
ℏ
H
0
(
t
1
−
t
0
)
V
(
t
1
)
e
−
i
ℏ
H
0
(
t
1
−
t
0
)
e
i
ℏ
H
0
(
t
2
−
t
0
)
V
(
t
2
)
e
−
i
ℏ
H
0
(
t
2
−
t
0
)
+
⋯
{\displaystyle U(t)=1-{\frac {i\lambda }{\hbar
}}\int _{t_{0}}^{t}dt_{1}e^{{\frac {i}{\hbar
}}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar
}}H_{0}(t_{1}-t_{0})}-{\frac {\lambda ^{2}}{\hbar
^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}e^{{\frac
{i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac
{i}{\hbar }}H_{0}(t_{1}-t_{0})}e^{{\frac {i}{\hbar
}}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-{\frac {i}{\hbar
}}H_{0}(t_{2}-t_{0})}+\cdots }
and take the perturbation V to be time-independent.
Using the identity resolution
∑
n
|
n
⟩
⟨
n
|
=
1
{\displaystyle \sum _{n}|n\rangle \langle
n|=1}
with
H
0
|
n
⟩
=
E
n
|
n
⟩
{\displaystyle H_{0}|n\rangle =E_{n}|n\rangle
}
for a pure discrete spectrum, write
U
(
t
)
=
1
−
{
i
λ
ℏ
∫
t
0
t
d
t
1
∑
m
∑
n
⟨
m
|
V
|
n
⟩
e
−
i
ℏ
(
E
n
−
E
m
)
(
t
1
−
t
0
)
|
m
⟩
⟨
n
|
}
−
{
λ
2
ℏ
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
∑
m
∑
n
∑
q
e
−
i
ℏ
(
E
n
−
E
m
)
(
t
1
−
t
0
)
⟨
m
|
V
|
n
⟩
⟨
n
|
V
|
q
⟩
e
−
i
ℏ
(
E
q
−
E
n
)
(
t
2
−
t
0
)
|
m
⟩
⟨
q
|
}
+
⋯
{\displaystyle {\begin{aligned}U(t)=1&-\left\{{\frac
{i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}\sum
_{m}\sum _{n}\langle m|V|n\rangle e^{-{\frac
{i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle
\langle n|\right\}\\&-\left\{{\frac {\lambda
^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int
_{t_{0}}^{t_{1}}dt_{2}\sum _{m}\sum _{n}\sum
_{q}e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}\langle
m|V|n\rangle \langle n|V|q\rangle e^{-{\frac
{i}{\hbar }}(E_{q}-E_{n})(t_{2}-t_{0})}|m\rangle
\langle q|\right\}+\cdots \end{aligned}}}
It is evident that, at second order, one must
sum on all the intermediate states. Assume
t
0
=
0
{\displaystyle t_{0}=0}
and the asymptotic limit of larger times.
This means that, at each contribution of the
perturbation series, one has to add a multiplicative
factor
e
−
ϵ
t
{\displaystyle e^{-\epsilon t}}
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
U
(
t
)
=
1
−
i
λ
ℏ
∑
n
⟨
n
|
V
|
n
⟩
t
−
i
λ
2
ℏ
∑
m
≠
n
⟨
n
|
V
|
m
⟩
⟨
m
|
V
|
n
⟩
E
n
−
E
m
t
−
1
2
λ
2
ℏ
2
∑
m
,
n
⟨
n
|
V
|
m
⟩
⟨
m
|
V
|
n
⟩
t
2
+
…
+
λ
∑
m
≠
n
⟨
m
|
V
|
n
⟩
E
n
−
E
m
|
m
⟩
⟨
n
|
+
λ
2
∑
m
≠
n
∑
q
≠
n
∑
n
⟨
m
|
V
|
n
⟩
⟨
n
|
V
|
q
⟩
(
E
n
−
E
m
)
(
E
q
−
E
n
)
|
m
⟩
⟨
q
|
+
…
{\displaystyle {\begin{aligned}U(t)=1&-{\frac
{i\lambda }{\hbar }}\sum _{n}\langle n|V|n\rangle
t-{\frac {i\lambda ^{2}}{\hbar }}\sum _{m\neq
n}{\frac {\langle n|V|m\rangle \langle m|V|n\rangle
}{E_{n}-E_{m}}}t-{\frac {1}{2}}{\frac {\lambda
^{2}}{\hbar ^{2}}}\sum _{m,n}\langle n|V|m\rangle
\langle m|V|n\rangle t^{2}+\ldots \\&+\lambda
\sum _{m\neq n}{\frac {\langle m|V|n\rangle
}{E_{n}-E_{m}}}|m\rangle \langle n|+\lambda
^{2}\sum _{m\neq n}\sum _{q\neq n}\sum _{n}{\frac
{\langle m|V|n\rangle \langle n|V|q\rangle
}{(E_{n}-E_{m})(E_{q}-E_{n})}}|m\rangle \langle
q|+\ldots \end{aligned}}}
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 (
|
n
(
λ
)
⟩
=
U
(
0
;
λ
)
|
n
⟩
)
{\displaystyle |n(\lambda )\rangle =U(0;\lambda
)|n\rangle )}
.)
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
H
(
t
)
|
ψ
(
t
)
⟩
=
i
ℏ
∂
|
ψ
(
t
)
⟩
∂
t
{\displaystyle H(t)|\psi (t)\rangle =i\hbar
{\frac {\partial |\psi (t)\rangle }{\partial
t}}}
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
[
H
0
+
λ
V
(
t
)
]
|
ψ
(
t
)
⟩
=
i
ℏ
∂
|
ψ
(
t
)
⟩
∂
t
{\displaystyle [H_{0}+\lambda V(t)]|\psi (t)\rangle
=i\hbar {\frac {\partial |\psi (t)\rangle
}{\partial t}}}
being λ→ ∞. Our aim is to find a solution
in the form
|
ψ
⟩
=
|
ψ
0
⟩
+
1
λ
|
ψ
1
⟩
+
1
λ
2
|
ψ
2
⟩
+
…
{\displaystyle |\psi \rangle =|\psi _{0}\rangle
+{\frac {1}{\lambda }}|\psi _{1}\rangle +{\frac
{1}{\lambda ^{2}}}|\psi _{2}\rangle +\ldots
}
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
τ
=
λ
t
{\displaystyle \tau =\lambda t}
producing the following meaningful equations
V
(
t
)
|
ψ
0
⟩
=
i
ℏ
∂
|
ψ
0
⟩
∂
τ
{\displaystyle V(t)|\psi _{0}\rangle =i\hbar
{\frac {\partial |\psi _{0}\rangle }{\partial
\tau }}}
V
(
t
)
|
ψ
1
⟩
+
H
0
|
ψ
0
⟩
=
i
ℏ
∂
|
ψ
1
⟩
∂
τ
{\displaystyle V(t)|\psi _{1}\rangle +H_{0}|\psi
_{0}\rangle =i\hbar {\frac {\partial |\psi
_{1}\rangle }{\partial \tau }}}
⋮
{\displaystyle \vdots }
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
V
(
t
)
{\displaystyle V(t)}
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
|
ψ
(
t
)
⟩
=
e
−
i
ℏ
λ
V
(
t
−
t
0
)
|
ψ
F
(
t
)
⟩
{\displaystyle |\psi (t)\rangle =e^{-{\frac
{i}{\hbar }}\lambda V(t-t_{0})}|\psi _{F}(t)\rangle
}
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
e
i
ℏ
λ
V
(
t
−
t
0
)
H
0
e
−
i
ℏ
λ
V
(
t
−
t
0
)
|
ψ
F
(
t
)
⟩
=
i
ℏ
∂
|
ψ
F
(
t
)
⟩
∂
t
{\displaystyle e^{{\frac {i}{\hbar }}\lambda
V(t-t_{0})}H_{0}e^{-{\frac {i}{\hbar }}\lambda
V(t-t_{0})}|\psi _{F}(t)\rangle =i\hbar {\frac
{\partial |\psi _{F}(t)\rangle }{\partial
t}}}
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
|
ψ
F
(
t
)
⟩
=
[
1
−
i
ℏ
∫
t
0
t
d
t
1
e
i
ℏ
λ
V
(
t
1
−
t
0
)
H
0
e
−
i
ℏ
λ
V
(
t
1
−
t
0
)
−
1
ℏ
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
e
i
ℏ
λ
V
(
t
1
−
t
0
)
H
0
e
−
i
ℏ
λ
V
(
t
1
−
t
0
)
e
i
ℏ
λ
V
(
t
2
−
t
0
)
H
0
e
−
i
ℏ
λ
V
(
t
2
−
t
0
)
+
…
]
|
ψ
(
t
0
)
⟩
.
{\displaystyle |\psi _{F}(t)\rangle =\left[1-{\frac
{i}{\hbar }}\int _{t_{0}}^{t}dt_{1}e^{{\frac
{i}{\hbar }}\lambda V(t_{1}-t_{0})}H_{0}e^{-{\frac
{i}{\hbar }}\lambda V(t_{1}-t_{0})}-{\frac
{1}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int
_{t_{0}}^{t_{1}}dt_{2}e^{{\frac {i}{\hbar
}}\lambda V(t_{1}-t_{0})}H_{0}e^{-{\frac {i}{\hbar
}}\lambda V(t_{1}-t_{0})}e^{{\frac {i}{\hbar
}}\lambda V(t_{2}-t_{0})}H_{0}e^{-{\frac {i}{\hbar
}}\lambda V(t_{2}-t_{0})}+\ldots \right]|\psi
(t_{0})\rangle .}
After the rescaling in time
τ
=
λ
t
{\displaystyle \tau =\lambda t}
we can see that this is indeed a series in
1
/
λ
{\displaystyle 1/\lambda }
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
H
0
=
p
2
/
2
m
{\displaystyle H_{0}=p^{2}/2m}
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
H
=
−
ℏ
2
2
m
∂
2
∂
x
2
+
m
ω
2
x
2
2
+
λ
x
4
{\displaystyle H=-{\frac {\hbar ^{2}}{2m}}{\frac
{\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega
^{2}x^{2}}{2}}+\lambda x^{4}}
The ground state of the harmonic oscillator
is
ψ
0
=
(
α
π
)
1
4
e
−
α
x
2
/
2
{\displaystyle \psi _{0}=\left({\frac {\alpha
}{\pi }}\right)^{\frac {1}{4}}e^{-\alpha x^{2}/2}}
(
α
=
m
ω
/
ℏ
{\displaystyle \alpha =m\omega /\hbar }
) and the energy of unperturbed ground state
is
E
0
(
0
)
=
1
2
ℏ
ω
.
{\displaystyle E_{0}^{(0)}={\tfrac {1}{2}}\hbar
\omega .\,}
Using the first order correction formula we
get
E
0
(
1
)
=
λ
(
α
π
)
1
2
∫
e
−
α
x
2
/
2
x
4
e
−
α
x
2
/
2
d
x
=
λ
(
α
π
)
1
2
∂
2
∂
α
2
∫
e
−
α
x
2
d
x
{\displaystyle E_{0}^{(1)}=\lambda \left({\frac
{\alpha }{\pi }}\right)^{\frac {1}{2}}\int
e^{-\alpha x^{2}/2}x^{4}e^{-\alpha x^{2}/2}dx=\lambda
\left({\frac {\alpha }{\pi }}\right)^{\frac
{1}{2}}{\frac {\partial ^{2}}{\partial \alpha
^{2}}}\int e^{-\alpha x^{2}}dx}
or
E
0
(
1
)
=
λ
(
α
π
)
1
2
∂
2
∂
α
2
(
π
α
)
1
2
=
λ
3
4
1
α
2
=
3
4
ℏ
2
λ
m
2
ω
2
{\displaystyle E_{0}^{(1)}=\lambda \left({\frac
{\alpha }{\pi }}\right)^{\frac {1}{2}}{\frac
{\partial ^{2}}{\partial \alpha ^{2}}}\left({\frac
{\pi }{\alpha }}\right)^{\frac {1}{2}}=\lambda
{\frac {3}{4}}{\frac {1}{\alpha ^{2}}}={\frac
{3}{4}}{\frac {\hbar ^{2}\lambda }{m^{2}\omega
^{2}}}}
=== Example of first and second order perturbation
theory – quantum pendulum ===
Consider the quantum mathematical pendulum
with the Hamiltonian
H
=
−
ℏ
2
2
m
a
2
∂
2
∂
ϕ
2
−
λ
cos
⁡
ϕ
{\displaystyle H=-{\frac {\hbar ^{2}}{2ma^{2}}}{\frac
{\partial ^{2}}{\partial \phi ^{2}}}-\lambda
\cos \phi }
with the potential energy
−
λ
cos
⁡
ϕ
{\displaystyle -\lambda \cos \phi }
taken as the perturbation i.e.
V
=
−
cos
⁡
ϕ
{\displaystyle V=-\cos \phi }
The unperturbed normalized quantum wave functions
are those of the rigid rotor and are given
by
ψ
n
(
ϕ
)
=
e
i
n
ϕ
2
π
{\displaystyle \psi _{n}(\phi )={\frac {e^{in\phi
}}{\sqrt {2\pi }}}}
and the energies
E
n
(
0
)
=
ℏ
2
n
2
2
m
a
2
{\displaystyle E_{n}^{(0)}={\frac {\hbar ^{2}n^{2}}{2ma^{2}}}}
The first order energy correction to the rotor
due to the potential energy is
E
n
(
1
)
=
−
1
2
π
∫
e
−
i
n
ϕ
cos
⁡
ϕ
e
i
n
ϕ
=
−
1
2
π
∫
cos
⁡
ϕ
=
0
{\displaystyle E_{n}^{(1)}=-{\frac {1}{2\pi
}}\int e^{-in\phi }\cos \phi e^{in\phi }=-{\frac
{1}{2\pi }}\int \cos \phi =0}
Using the formula for the second order correction
one gets
E
n
(
2
)
=
m
a
2
2
π
2
ℏ
2
∑
k
|
∫
e
−
i
k
ϕ
cos
⁡
ϕ
e
i
n
ϕ
|
2
n
2
−
k
2
{\displaystyle E_{n}^{(2)}={\frac {ma^{2}}{2\pi
^{2}\hbar ^{2}}}\sum _{k}{\frac {\left|\int
e^{-ik\phi }\cos \phi e^{in\phi }\right|^{2}}{n^{2}-k^{2}}}}
or
E
n
(
2
)
=
m
a
2
2
ℏ
2
∑
k
|
(
δ
n
,
1
−
k
+
δ
n
,
−
1
−
k
)
|
2
n
2
−
k
2
{\displaystyle E_{n}^{(2)}={\frac {ma^{2}}{2\hbar
^{2}}}\sum _{k}{\frac {\left|\left(\delta
_{n,1-k}+\delta _{n,-1-k}\right)\right|^{2}}{n^{2}-k^{2}}}}
or
E
n
(
2
)
=
m
a
2
2
ℏ
2
(
1
2
n
−
1
+
1
−
2
n
−
1
)
=
m
a
2
ℏ
2
1
4
n
2
−
1
{\displaystyle E_{n}^{(2)}={\frac {ma^{2}}{2\hbar
^{2}}}\left({\frac {1}{2n-1}}+{\frac {1}{-2n-1}}\right)={\frac
{ma^{2}}{\hbar ^{2}}}{\frac {1}{4n^{2}-1
