Formalism

Summary: Perturbation theory is the most widely used approximate method. “Time-independent perturbation theory” deals with bound states eg the spectrum of the real hydrogen atom and its response to extermal ﬁelds.

Perturbation theory is applicable when the Hamiltonian

\hat{H}

can be split into two parts, with the ﬁrst part being exactly solvable and the second part being small in comparison. The ﬁrst part is always written

{\hat{H}}^{(0)}

, and we will denote its eigenstates by

| n^{(0)} ⟩

and energies by

E_{n}^{(0)}

(with wave functions

ϕ_{n}^{(0)}

). These we know. The eigenstates and energies of the full Hamiltonian are denoted

| n ⟩

and

E_{n}

, and the aim is to ﬁnd successively better approximations to these. The zeroth-order approximation is simply

| n ⟩ = | n^{(0)} ⟩

and

E_{n} = E_{n}^{(0)}

, which is just another way of saying that the perturbation is small.

Nomenclature for the perturbing Hamiltonian

\hat{H} - {\hat{H}}^{(0)}

varies.

δ V

{\hat{H}}^{(1)}

and

λ {\hat{H}}^{(1)}

are all common. It usually is a perturbing potential but we won’t assume so here, so we won’t use the ﬁrst. The second and third diﬀer in that the third has explicitly identiﬁed a small, dimensionless parameter (eg

α

in EM), so that the residual

{\hat{H}}^{(1)}

isn’t itself small. With the last choice, our expressions for the eigenstates and energies of the full Hamiltonian will be explicitly power series in

λ

, so

E_{n} = E_{n}^{(0)} + λ E_{n}^{(1)} + λ^{2} E_{n}^{(2)} + \dots

etc. With the second choice the small factor is hidden in

{\hat{H}}^{(1)}

, and is implicit in the expansion which then reads

E_{n} = E_{n}^{(0)} + E_{n}^{(1)} + E_{n}^{(2)} + \dots

. In this case one has to remember that anything with a superscript (1) is ﬁrst order in this implicit small factor, or more generally the superscript

(m)

denotes something which is

m

th order. For the derivation of the equations we will retain an explicit

λ

, but thereafter we will set it equal to one to revert to the other formulation. We will take

λ

to be real so that

{\hat{H}}_{1}

is Hermitian.

({\hat{H}}^{(0)} + λ {\hat{H}}^{(1)}) | n ⟩ = E_{n} | n ⟩ .

Then we substitute in

E_{n} = E_{n}^{(0)} + λ E_{n}^{(1)} + λ^{2} E_{n}^{(2)} + \dots

and

| n ⟩ = | n^{(0)} ⟩ + λ | n^{(1)} ⟩ + λ^{2} | n^{(2)} ⟩ + \dots

and expand. Then since

λ

is a free parameter, we have to match terms on each side with the same powers of

λ

, to get

We have to solve these sequentially. The ﬁrst we assume we have already done. The second will yield

E_{n}^{(1)}

and

| n^{(1)} ⟩

. Once we know these, we can use the third equation to yield

E_{n}^{(2)}

and

| n^{(2)} ⟩

, and so on.

In each case, to solve for the energy we take the inner product with

⟨ n^{(0)} |

(ie the same state) whereas for the wave function, we use

⟨ m^{(0)} |

(another state). We use, of course,

⟨ m^{(0)} | {\hat{H}}^{(0)} = E_{m}^{(0)} ⟨ m^{(0)} |

and

⟨ m^{(0)} | n^{(0)} ⟩ = δ_{m n}

E_{n}^{(1)} = ⟨ n^{(0)} | {\hat{H}}^{(1)} | n^{(0)} ⟩ and ⟨ m^{(0)} | n^{(1)} ⟩ = \frac{⟨ m^{(0)} | {\hat{H}}^{(1)} | n^{(0)} ⟩}{E_{n}^{(0)} - E_{m}^{(0)}} \forall m \neq n

The second equation tells us the overlap of

| n^{(1)} ⟩

with all the other

| m^{(0)} ⟩

, but not with

| n^{(0)} ⟩

. This is obviously not constrained, because we can add any amount of

| n^{(0)} ⟩

and the equations will still be satisﬁed. However we need the state to continue to be normalised, and when we expand

⟨ n | n ⟩ = 1

in powers of

λ

we ﬁnd that

⟨ n^{(0)} | n^{(1)} ⟩

is required to be imaginary. Since this is just like a phase rotation of the original state and we can ignore it. Hence

| n^{(1)} ⟩ = \sum_{m \neq n} \frac{⟨ m^{(0)} | {\hat{H}}^{(1)} | n^{(0)} ⟩}{E_{n}^{(0)} - E_{m}^{(0)}} | m^{(0)} ⟩

If the spectrum of

{\hat{H}}^{(0)}

is degenerate, there is a potential problem with this expression because the denominator can be inﬁnite. In that case we have to diagonalise

{\hat{H}}^{(1)}

in the subspace of degenerate states exactly. This is called “degenerate perturbation theory”.

E_{n}^{(2)} = ⟨ n^{(0)} | {\hat{H}}^{(1)} | n^{(1)} ⟩ = \sum_{m \neq n} \frac{{|⟨ m^{(0)} | {\hat{H}}^{(1)} | n^{(0)} ⟩|}^{2}}{E_{n}^{(0)} - E_{m}^{(0)}}

The expression for the second-order shift in the wave function

| n^{(2)} ⟩

can also be found but it is tedious. The main reason we wanted

| n^{(1)} ⟩

was to ﬁnd

E_{n}^{(2)}

anyway, and we’re not planning to ﬁnd

E_{n}^{(3)}

! Note that though the expression for

E_{n}^{(1)}

is generally applicable, those for

| n^{(1)} ⟩

and

E_{n}^{(2)}

would need some modiﬁcation if the Hamiltonian had continuum eigenstates as well as bound states (eg hydrogen atom). Provided the state

| n ⟩

is bound, that is just a matter of integrating rather than summing. This restriction to bound states is why Mandl calls chapter 7 “bound-state perturbation theory”. The perturbation of continuum states (eg scattering states) is usually dealt with separately.

Note that the equations above hold whether we have identiﬁed an explicit small parameter

λ

or not. So from now on we will set

λ

to one, and

E_{n} = E_{n}^{(0)} + E_{n}^{(1)} + E_{n}^{(2)} + \dots

Connection to variational approach:
For the ground state (which is always non-degenerate)

E_{0}^{(0)} + E_{0}^{(1)}

is a variational upper bound on the exact energy

E_{0}

, since it is obtained by using the unperturbed ground state as a trial wavefunction for the full Hamiltonian. It follows that the sum of all higher corrections

E_{0}^{(2)} + \dots

must be negative. We can see indeed that

E_{0}^{(2)}

will always be negative, since for every term in the sum the numerator is positive and the denominator negative.

3.1 Formalism