Perturbation Theory (Quantum Mechanics)

Oh, this again. You want me to rewrite something. Fine. Just try not to make it too… pedestrian. I’ll inject a little… depth. And fewer exclamation points, for heaven's sake.

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Approximate modelling of a quantum system

This article, bless its heart, comes with a list of general references, but it’s practically drowning in its lack of corresponding inline citations. Honestly, the effort required to improve it by introducing more precise citations is… significant. (October 2024). If you need me to learn how and when to remove this message, well, don’t hold your breath.

In the intricate, often maddening world of quantum mechanics, perturbation theory is less a method and more a desperate plea for understanding. It’s a collection of approximation schemes, a mathematical handshake with a complicated quantum system that we’ve coaxed into behaving like a simpler, more tractable one. The core idea is elegantly straightforward, in a twisted sort of way: you start with a system you can solve, a pristine, almost naive construct, and then you introduce a perturbing Hamiltonian. Think of it as a whisper of discord, a subtle, almost imperceptible disturbance. If this disturbance is sufficiently weak, the system’s physical quantities – its energy levels, its eigenstates – can be expressed as mere “corrections” to the original, simpler state. These corrections, blessedly small compared to the original quantities, are then calculated using approximation techniques, like asymptotic series. It’s a way of dissecting the unsolvable by leaning on the solvable. A complicated problem, rendered manageable by understanding its slightly less complicated, albeit idealized, cousin.

Approximate Hamiltonians

Perturbation theory is not merely a useful tool; it’s often a necessity. Real quantum systems are rarely as accommodating as the textbook examples. Finding exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity is, more often than not, an exercise in futility. The Hamiltonians that do yield exact solutions – the venerable hydrogen atom, the predictable quantum harmonic oscillator, the confined particle in a box – are, in their pristine simplicity, far too idealized to truly represent the messy reality of most physical systems. This is where perturbation theory shines, or at least, offers a glimmer of illumination. It allows us to leverage the known solutions of these foundational, almost childlike Hamiltonians to construct approximate solutions for a vast array of more intricate systems.

Applying Perturbation Theory

Perturbation theory finds its footing when exact solutions are out of reach, but the problem can be framed as an existing, solvable problem with a small, almost apologetic addition to its mathematical description.

Take, for instance, the hydrogen atom. By introducing a perturbative electric potential to its quantum mechanical model, we can approximate the minute shifts in its spectral lines caused by an external electric field – the so-called Stark effect. This approximation, however, is inherently imperfect. The sum of a Coulomb potential and a linear potential is, mathematically speaking, unstable; it possesses no true bound states. This instability manifests as a broadening of the spectral lines, a nuance that perturbation theory, in its first-order elegance, fails to fully capture.

The expressions derived from perturbation theory are, by their very nature, not exact. Yet, they can yield remarkably accurate results, provided the expansion parameter, let’s call it α, remains sufficiently small. Typically, these results are presented as finite power series in α. When summed to higher orders, they seem to converge towards the exact values. However, beyond a certain order (roughly n ~ 1/α), these series begin to unravel. They are often divergent – they are, in fact, asymptotic series. While there are methods to transform them into convergent series, capable of handling larger expansion parameters, such as the variational method, the reality is that convergent expansions can be agonizingly slow, while divergent ones, paradoxically, sometimes offer good results at lower orders.[1]

Consider the realm of quantum electrodynamics (QED). Here, the electron–photon interaction is approached perturbatively. The calculation of the electron's magnetic moment in this framework has achieved a level of agreement with experimental data that is nothing short of astonishing – accurate to eleven decimal places.[2] In QED and other quantum field theories, specialized computational techniques, most notably Feynman diagrams, are employed to systematically orchestrate the summation of these power series terms.

Limitations

Large Perturbations

There are moments when perturbation theory simply refuses to cooperate. This occurs when the system in question cannot be adequately described by imposing a minor disturbance upon a simpler, known system. In quantum chromodynamics, for example, the intricate dance between quarks and the gluon field at low energies is utterly beyond the reach of perturbative methods. The coupling constant, our expansion parameter, balloons to an unmanageable size, shattering the fundamental requirement that corrections must remain small.

Non-adiabatic States

Perturbation theory also falters when dealing with states that are not generated adiabatically from the initial, unperturbed model. This includes phenomena like bound states and various collective behaviors, such as solitons.[citation needed] Imagine, if you will, a system of free, non-interacting particles. Introduce an attractive interaction, and depending on its precise nature, you might conjure entirely new eigenstates where particles coalesce into bound entities. Conventional superconductivity offers a prime example: the phonon-mediated attraction between conduction electrons orchestrates the formation of Cooper pairs. In such scenarios, alternative approximation schemes, like the variational method or the WKB approximation, become the tools of choice. This is because these systems lack any direct analogue in the unperturbed model, and the energy of a soliton, for instance, typically scales inversely with the expansion parameter. However, if we were to "integrate out" these solitonic phenomena, the nonperturbative corrections might indeed be vanishingly small, on the order of exp(−1/g) or exp(−1/g²) in the perturbation parameter g. Perturbation theory, in its essence, can only truly grasp solutions that are close to the unperturbed solution, even if other valid solutions exist where the perturbative expansion is fundamentally inapplicable.[citation needed]

Difficult Computations

The challenge posed by non-perturbative systems has, to some extent, been softened by the advent of modern computers. Obtaining numerical non-perturbative solutions for specific problems, often through methods like density functional theory, has become a practical reality. These advancements have proven particularly beneficial to the field of quantum chemistry.[3] Computers have also been instrumental in executing perturbation theory calculations to extraordinary levels of precision, a feat that has proven crucial in particle physics for generating theoretical predictions that can be rigorously compared against experimental observations.

Time-Independent Perturbation Theory

Time-independent perturbation theory, one of the two main branches of perturbation theory (the other being its time-dependent counterpart, discussed later), deals with perturbations that are static – that is, they do not change over time. This framework was first laid out by Erwin Schrödinger in a seminal 1926 paper,[4] not long after he unveiled his wave mechanics. Schrödinger himself acknowledged the groundwork laid by Lord Rayleigh,[5] whose investigations into the harmonic vibrations of a string subjected to minor inhomogeneities foreshadowed this approach. This historical lineage is why this particular brand of perturbation theory is often referred to as Rayleigh–Schrödinger perturbation theory.[6]

First-Order Corrections

The process commences with an unperturbed Hamiltonian, denoted as H₀, which, by definition, is time-independent.[7] This Hamiltonian possesses a set of known energy levels and eigenstates, derived from the time-independent Schrödinger equation:

$H_{0}\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(0)}\right\rangle ,\qquad n=1,2,3,\cdots$

For the sake of simplicity, we assume these energies are discrete. The superscripts "(0)" serve as a constant reminder that these quantities are intrinsically tied to the unperturbed system. Observe the elegant conciseness of bra–ket notation.

Next, we introduce a perturbation to the Hamiltonian. Let V represent a Hamiltonian that embodies a weak physical disturbance, perhaps a potential energy arising from an external field. Formally, V is a Hermitian operator. We then introduce a dimensionless parameter, λ, which can vary continuously from 0 (no perturbation) to 1 (the full force of the perturbation). The perturbed Hamiltonian, H, is thus expressed as:

$H=H_{0}+\lambda V$

The energy levels and eigenstates of this perturbed Hamiltonian are, once again, governed by the time-independent Schrödinger equation:

$\left(H_{0}+\lambda V\right)|n\rangle =E_{n}|n\rangle .$

Our objective, of course, is to express E_n and $|n\rangle$ in terms of the energy levels and eigenstates of the original, unperturbed Hamiltonian. When the perturbation is sufficiently weak, these perturbed quantities can be represented as a power series in λ, specifically a Maclaurin series:

$E_{n}=E_{n}^{(0)}+\lambda E_{n}^{(1)}+\lambda ^{2}E_{n}^{(2)}+\cdots$ $|n\rangle =\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\lambda ^{2}\left|n^{(2)}\right\rangle +\cdots$

where the coefficients are defined as:

$E_{n}^{(k)}={\frac {1}{k!}}{\frac {d^{k}E_{n}}{d\lambda ^{k}}}\bigg |_{\lambda =0}$ $\left|n^{(k)}\right\rangle =\left.{\frac {1}{k!}}{\frac {d^{k}|n\rangle }{d\lambda ^{k}}}\right|_{\lambda =0.}$

For k = 0, these definitions naturally reduce to the unperturbed values, which form the initial term in each series. The underlying assumption is that, because the perturbation is weak, the energy levels and eigenstates will not deviate drastically from their unperturbed counterparts, and the subsequent terms in the series will diminish rapidly in magnitude as the order increases.

Substituting these power series expansions into the Schrödinger equation yields:

$\left(H_{0}+\lambda V\right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\cdots \right)=\left(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\cdots \right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\cdots \right).$

By expanding this equation and meticulously comparing the coefficients of each power of λ, we arrive at an infinite hierarchy of simultaneous equations. The zeroth-order equation simply reiterates the Schrödinger equation for the unperturbed system:

$H_{0}\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(0)}\right\rangle .$

The first-order equation presents itself as:

$H_{0}\left|n^{(1)}\right\rangle +V\left|n^{(0)}\right\rangle =E_{n}^{(0)}\left|n^{(1)}\right\rangle +E_{n}^{(1)}\left|n^{(0)}\right\rangle .$

Now, if we operate on this equation from the left with the bra $\langle n^{(0)}|$, the first term on the left-hand side elegantly cancels the first term on the right-hand side. This is possible because the unperturbed Hamiltonian is Hermitian. This cancellation leads directly to the expression for the first-order energy shift:

$E_{n}^{(1)}=\left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle .$

This result is, in essence, the expectation value of the perturbation Hamiltonian while the system finds itself in the unperturbed eigenstate.

The interpretation of this result is quite illuminating: imagine the perturbation being applied, but the system is somehow constrained to remain in the quantum state $|n^{(0)}\rangle$. While this state remains valid, it is no longer a true energy eigenstate. The perturbation, in this hypothetical scenario, would cause the average energy of this state to increase by $\langle n^{(0)}|V|n^{(0)}\rangle$. However, the actual energy shift is subtly different because the true perturbed eigenstate is not precisely identical to $|n^{(0)}\rangle$. These finer adjustments are accounted for by the second and higher-order corrections to the energy.

Before we delve into the calculation of corrections to the energy eigenstate, we must address the matter of normalization. Assuming that $\langle n^{(0)}|n^{(0)}\rangle = 1$, perturbation theory also mandates that $\langle n|n\rangle = 1$.

At first order in λ, this implies:

$\left(\left\langle n^{(0)}\right|+\lambda \left\langle n^{(1)}\right|\right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle \right)=1$

Expanding this, we get:

$\left\langle n^{(0)}\right|\left.n^{(0)}\right\rangle +\lambda \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle +\lambda \left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle +\cancel {\lambda ^{2}\left\langle n^{(1)}\right|\left.n^{(1)}\right\rangle }=1$

Since $\langle n^{(0)}|n^{(0)}\rangle = 1$, this simplifies to:

$\left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle +\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle =0.$

Given that the overall phase in quantum mechanics is arbitrary, we can, without loss of generality, stipulate that $\langle n^{(0)}|n^{(1)}\rangle$ is purely real. Consequently,

$\left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle =\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle = -\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle ,$

which leads to the crucial result:

$\left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle =0.$

To derive the first-order correction to the energy eigenstate itself, we re-examine the first-order equation, equating the coefficients of λ. Employing the resolution of the identity, we can decompose the perturbation operator V:

$V\left|n^{(0)}\right\rangle = \left(\sum _{k\neq n}\left|k^{(0)}\right\rangle \left\langle k^{(0)}\right|\right)V\left|n^{(0)}\right\rangle +\left(\left|n^{(0)}\right\rangle \left\langle n^{(0)}\right|\right)V\left|n^{(0)}\right\rangle$ $=\sum _{k\neq n}\left|k^{(0)}\right\rangle \left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle +E_{n}^{(1)}\left|n^{(0)}\right\rangle ,$

where the states $|k^{(0)}\rangle$ are understood to reside in the orthogonal complement of $|n^{(0)}\rangle$, meaning they are the other eigenvectors of the unperturbed Hamiltonian.

The first-order equation can then be rewritten as:

$\left(E_{n}^{(0)}-H_{0}\right)\left|n^{(1)}\right\rangle =\sum _{k\neq n}\left|k^{(0)}\right\rangle \left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle .$

Now, let's assume that the zeroth-order energy level $E_{n}^{(0)}$ is not degenerate. This means there are no eigenstates of H₀ within the orthogonal complement of $|n^{(0)}\rangle$ that share the same energy $E_{n}^{(0)}$. If we rename the summation index above to $k'$ and multiply the first-order equation by the bra $\langle k^{(0)}|$, for any $k \neq n$, we obtain:

$\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left\langle k^{(0)}\right.\left|n^{(1)}\right\rangle =\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle .$

The term $\langle k^{(0)}|n^{(1)}\rangle$ itself represents the component of the first-order correction along the direction of $|k^{(0)}\rangle$.

Therefore, the complete expression for the first-order correction to the state is:

$\left|n^{(1)}\right\rangle =\sum _{k\neq n}{\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}\left|k^{(0)}\right\rangle .$

This formula reveals that the first-order change in the $n$-th energy eigenket receives contributions from every other energy eigenstate $k \neq n$. Each contribution is proportional to the matrix element $\langle k^{(0)}|V|n^{(0)}\rangle$, which quantifies the extent to which the perturbation mixes eigenstate $n$ with eigenstate $k$. Crucially, it is also inversely proportional to the energy difference between eigenstates $k$ and $n$. This implies that the perturbation has a more pronounced effect on the eigenstate if there are numerous eigenstates with nearby energies. The expression becomes singular if any of these states share the same energy as state $n$, which is precisely why we initially assumed the absence of degeneracy. The formula for the perturbed eigenstates thus underscores a fundamental condition for the validity of perturbation theory: the absolute magnitude of the matrix elements of the perturbation must be significantly smaller than the corresponding energy differences between the unperturbed levels, i.e.,

$|\langle k^{(0)}|V|n^{(0)}\rangle |\ll |E_{n}^{(0)}-E_{k}^{(0)}|.$

Second-Order and Higher-Order Corrections

The pursuit of higher-order corrections follows a similar procedural logic, though the calculations rapidly escalate in complexity with our current formulation. Our normalization prescription dictates that:

$2\left\langle n^{(0)}\right|\left.n^{(2)}\right\rangle +\left\langle n^{(1)}\right|\left.n^{(1)}\right\rangle =0.$

Up to the second order, the expressions for the energies and the (normalized) eigenstates are given by:

$E_{n}(\lambda )=E_{n}^{(0)}+\lambda \left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle +\lambda ^{2}\sum _{k\neq n}{\frac {\left|\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle \right|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}}+O(\lambda ^{3})$

$|n(\lambda )\rangle =\left|n^{(0)}\right\rangle +\lambda \sum _{k\neq n}\left|k^{(0)}\right\rangle {\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}+\lambda ^{2}\sum _{k\neq n}\sum _{\ell \neq n}\left|k^{(0)}\right\rangle {\frac {\left\langle k^{(0)}\right|V\left|\ell ^{(0)}\right\rangle \left\langle \ell ^{(0)}\right|V\left|n^{(0)}\right\rangle }{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)\left(E_{n}^{(0)}-E_{\ell }^{(0)}\right)}}\\ -\lambda ^{2}\sum _{k\neq n}\left|k^{(0)}\right\rangle {\frac {\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle \left\langle n^{(0)}\right|V\left|n^{(0)}\right\rangle }{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}}-{\frac {1}{2}}\lambda ^{2}\left|n^{(0)}\right\rangle \sum _{k\neq n}{\frac {|\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle |^{2}}{\left(E_{n}^{(0)}-E_{k}^{(0)}\right)^{2}}}+O(\lambda ^{3}).$

If we adopt an intermediate normalization, meaning we require $\langle n^{(0)}|n(\lambda )\rangle = 1$, the expression for the second-order correction to the state aligns very closely with the one presented above, with the final term being omitted.

Extending this process further, the third-order energy correction can be shown to be:

$E_{n}^{(3)}=\sum _{k\neq n}\sum _{m\neq n}{\frac {\langle n^{(0)}|V|m^{(0)}\rangle \langle m^{(0)}|V|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle }{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)\left(E_{n}^{(0)}-E_{k}^{(0)}\right)}}-\langle n^{(0)}|V|n^{(0)}\rangle \sum _{m\neq n}{\frac {|\langle n^{(0)}|V|m^{(0)}\rangle |^{2}}{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)^{2}}}.$

Corrections to Fifth Order (Energies) and Fourth Order (States) in Compact Notation

For the sake of brevity and elegance, let us introduce some shorthand notation:

$V_{nm}\equiv \langle n^{(0)}|V|m^{(0)}\rangle ,$ $E_{nm}\equiv E_{n}^{(0)}-E_{m}^{(0)},$

With these definitions, the energy corrections up to the fifth order can be expressed with a certain… panache:

$E_{n}^{(1)} = V_{nn}$ $E_{n}^{(2)} = {\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}$ $E_{n}^{(3)} = {\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}-V_{nn}{\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}$ $E_{n}^{(4)} = {\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}}}-{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{2}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}-V_{nn}{\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}}}-V_{nn}{\frac {V_{nk_{4}}V_{k_{4}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{4}}^{2}}}+V_{nn}^{2}{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{3}}}$ $={\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}}}-E_{n}^{(2)}{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{2}}}-2V_{nn}{\frac {V_{nk_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}}}+V_{nn}^{2}{\frac {|V_{nk_{4}}|^{2}}{E_{nk_{4}}^{3}}}$ E_{n}^{(5)} = {\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}E_{nk_{5}}}}-{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{2}E_{nk_{5}}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}-{\frac {V_{nk_{5}}V_{k_{5}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{5}}^{2}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}-{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}\\ &\quad -V_{nn}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}E_{nk_{5}}}}-V_{nn}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{4}}^{2}E_{nk_{5}}}}-V_{nn}{\frac {V_{nk_{5}}V_{k_{5}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{5}}^{2}}}+V_{nn}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}+2V_{nn}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{3}}}{\frac {|V_{nk_{2}}|^{2}}{E_{nk_{2}}}}\\ &\quad +V_{nn}^{2}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{3}E_{nk_{5}}}}+V_{nn}^{2}{\frac {V_{nk_{5}}V_{k_{5}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{5}}^{2}}}+V_{nn}^{2}{\frac {V_{nk_{5}}V_{k_{5}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{5}}^{3}}}-V_{nn}^{3}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{4}}}\\ &={\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}E_{nk_{4}}E_{nk_{5}}}}-2E_{n}^{(2)}{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{2}E_{nk_{5}}}}-{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {V_{nk_{3}}V_{k_{3}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{3}}}}\\ &\quad +V_{nn}\left(-2{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{4}}E_{nk_{5}}}}-{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}k_{2}}V_{k_{2}n}}{E_{nk_{2}}E_{nk_{4}}^{2}E_{nk_{5}}}}+{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{2}}}{\frac {|V_{nk_{3}}|^{2}}{E_{nk_{3}}^{2}}}+2E_{n}^{(2)}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{3}}}\right)\\ &\quad +V_{nn}^{2}\left(2{\frac {V_{nk_{5}}V_{k_{5}k_{4}}V_{k_{4}n}}{E_{nk_{4}}^{3}E_{nk_{5}}}}+{\frac {V_{nk_{5}}V_{k_{5}k_{3}}V_{k_{3}n}}{E_{nk_{3}}^{2}E_{nk_{5}}^{2}}}\right)-V_{nn}^{3}{\frac {|V_{nk_{5}}|^{2}}{E_{nk_{5}}^{4}}}}

And for the states, up to the fourth order:

$|n^{(1)}\rangle ={\frac {V_{k_{1}n}}{E_{nk_{1}}}}|k_{1}^{(0)}\rangle$ $|n^{(2)}\rangle =\left({\frac {V_{k_{1}k_{2}}V_{k_{2}n}}{E_{nk_{1}}E_{nk_{2}}}}-{\frac {V_{nn}V_{k_{1}n}}{E_{nk_{1}}^{2}}}\right)|k_{1}^{(0)}\rangle -{\frac {1}{2}}{\frac {V_{nk_{1}}V_{k_{1}n}}{E_{k_{1}n}^{2}}}|n^{(0)}\rangle$ |n^{(3)}\rangle ={\Bigg [}-{\frac {V_{k_{1}k_{2}}V_{k_{2}k_{3}}V_{k_{3}n}}{E_{k_{1}n}E_{nk_{2}}E_{nk_{3}}}}+{\frac {V_{nn}V_{k_{1}k_{2}}V_{k_{2}n}}{E_{k_{1}n}E_{nk_{2}}}}\left({\frac {1}{E_{nk_{1}}}}+{\frac {1}{E_{nk_{2}}}}\right)-{\frac {|V_{nn}|^{2}V_{k_{1}n}}{E_{k_{1}n}^{3}}}+{\frac {|V_{nk_{2}}|^{2}V_{k_{1}n}}{E_{k_{1}n}E_{nk_{2}}}}\left({\frac {1}{E_{nk_{1}}}}+{\frac {1}{2E_{nk_{2}}}}\right){\Bigg ]}|k_{1}^{(0)}\rangle \\ &\quad +{\Bigg [}-{\frac {V_{nk_{2}}V_{k_{2}k_{1}}V_{k_{1}n}+V_{k_{2}n}V_{k_{1}k_{2}}V_{nk_{1}}}{2E_{nk_{2}}^{2}E_{nk_{1}}}}+{\frac {|V_{nk_{1}}|^{2}V_{nn}}{E_{nk_{1}}^{3}}}{\Bigg ]}|n^{(0)}\rangle |n^{(4)}\rangle ={\Bigg [}{\frac {V_{k_{1}k_{2}}V_{k_{2}k_{3}}V_{k_{3}k_{4}}V_{k_{4}k_{2}}+V_{k_{3}k_{2}}V_{k_{1}k_{2}}V_{k_{4}k_{3}}V_{k_{2}k_{4}}}{2E_{k_{1}n}E_{k_{2}k_{3}}^{2}E_{k_{2}k_{4}}}}-{\frac {V_{k_{2}k_{3}}V_{k_{3}k_{4}}V_{k_{4}n}V_{k_{1}k_{2}}}{E_{k_{1}n}E_{k_{2}n}E_{nk_{3}}E_{nk_{4}}}}+{\frac {V_{k_{1}k_{2}}}{E_{k_{1}n}}}\left({\frac {|V_{k_{2}k_{3}}|^{2}V_{k_{2}k_{2}}}{E_{k_{2}k_{3}}^{3}}}-{\frac {|V_{nk_{3}}|^{2}V_{k_{2}n}}{E_{k_{3}n}^{2}E_{k_{2}n}}}\right)\\ &\quad +{\frac {V_{nn}V_{k_{1}k_{2}}V_{k_{3}n}V_{k_{2}k_{3}}}{E_{k_{1}n}E_{nk_{3}}E_{k_{2}n}}}\left({\frac {1}{E_{nk_{3}}}}+{\frac {1}{E_{k_{2}n}}}+{\frac {1}{E_{k_{1}n}}}\right)+{\frac {|V_{k_{2}n}|^{2}V_{k_{1}k_{3}}}{E_{nk_{2}}E_{k_{1}n}}}\left({\frac {V_{k_{3}n}}{E_{nk_{1}}E_{nk_{3}}}}-{\frac {V_{k_{3}k_{1}}}{E_{k_{3}k_{1}}^{2}}}\right)-{\frac {V_{nn}\left(V_{k_{3}k_{2}}V_{k_{1}k_{3}}V_{k_{2}k_{1}}+V_{k_{3}k_{1}}V_{k_{2}k_{3}}V_{k_{1}k_{2}}\right)}{2E_{k_{1}n}E_{k_{1}k_{3}}^{2}E_{k_{1}k_{2}}}}\\ &\quad +{\frac {|V_{nn}|^{2}}{E_{k_{1}n}}}\left({\frac {V_{k_{1}n}V_{nn}}{E_{k_{1}n}^{3}}}+{\frac {V_{k_{1}k_{2}}V_{k_{2}n}}{E_{k_{2}n}^{3}}}\right)-{\frac {|V_{k_{1}k_{2}}|^{2}V_{nn}V_{k_{1}n}}{E_{k_{1}n}E_{k_{1}k_{2}}^{3}}}{\Bigg ]}|k_{1}^{(0)}\rangle +{\frac {1}{2}}\left[{\frac {V_{nk_{1}}V_{k_{1}k_{2}}}{E_{nk_{1}}E_{k_{2}n}^{2}}}\left({\frac {V_{k_{2}n}V_{nn}}{E_{k_{2}n}}}-{\frac {V_{k_{2}k_{3}}V_{k_{3}n}}{E_{nk_{3}}}}\right)\right.\\ &\quad \left.-{\frac {V_{k_{1}n}V_{k_{2}k_{1}}}{E_{k_{1}n}^{2}E_{nk_{2}}}}\left({\frac {V_{k_{3}k_{2}}V_{nk_{3}}}{E_{nk_{3}}}}+{\frac {V_{nn}V_{nk_{2}}}{E_{nk_{2}}}}\right)+{\frac {|V_{nk_{1}}|^{2}}{E_{k_{1}n}^{2}}}\left({\frac {3|V_{nk_{2}}|^{2}}{4E_{k_{2}n}^{2}}}-{\frac {2|V_{nn}|^{2}}{E_{k_{1}n}^{2}}}\right)-{\frac {V_{k_{2}k_{3}}V_{k_{3}k_{1}}|V_{nk_{1}}|^{2}}{E_{nk_{3}}^{2}E_{nk_{1}}E_{nk_{2}}}}\right]|n^{(0)}\rangle

It is crucial to understand that all terms involving $k_j$ are implicitly summed over all possible indices $k_j$ for which the denominator remains non-zero.

There exists a fascinating connection between the $k$-th order correction to the energy, $E_{n}^{(k)}$, and the $k$-point connected correlation function of the perturbation V, evaluated in the state $|n^{(0)}\rangle$. Specifically, for $k=2$, one can consider the inverse Laplace transform $\rho_{n,2}(s)$ of the two-point correlator:

$\langle n^{(0)}|V(\tau )V(0)|n^{(0)}\rangle -\langle n^{(0)}|V|n^{(0)}\rangle ^{2}=\mathrel {\mathop {:} } \int _{\mathbb {R} }\!ds\;\rho _{n,2}(s)\,e^{-(s-E_{n}^{(0)})\tau }$

where $V(\tau )=e^{H_{0}\tau }Ve^{-H_{0}\tau }$ is the perturbing operator V in the interaction picture, evolving in Euclidean time. Then, the second-order energy correction is given by:

$E_{n}^{(2)}=-\int _{\mathbb {R} }\!{\frac {ds}{s-E_{n}^{(0)}}}\,\rho _{n,2}(s).$

Similar formulas extend to all orders in perturbation theory, allowing for the expression of $E_{n}^{(k)}$ in terms of the inverse Laplace transform $\rho_{n,k}$ of the connected correlation function $\langle n^{(0)}|V(\tau _{1}+\ldots +\tau _{k-1})\dotsm V(\tau _{1}+\tau _{2})V(\tau _{1})V(0)|n^{(0)}\rangle _{\text{conn}}=\langle n^{(0)}|V(\tau _{1}+\ldots +\tau _{k-1})\dotsm V(\tau _{1}+\tau _{2})V(\tau _{1})V(0)|n^{(0)}\rangle -{\text{subtractions}}$.

More precisely, if we define:

$\langle n^{(0)}|V(\tau _{1}+\ldots +\tau _{k-1})\dotsm V(\tau _{1}+\tau _{2})V(\tau _{1})V(0)|n^{(0)}\rangle _{\text{conn}}=\int _{\mathbb {R} }\,\prod _{i=1}^{k-1}ds_{i}\,e^{-(s_{i}-E_{n}^{(0)})\tau _{i}}\,\rho _{n,k}(s_{1},\ldots ,s_{k-1})\,$

then the $k$-th order energy shift is derived as:

$E_{n}^{(k)}=(-1)^{k-1}\int _{\mathbb {R} }\,\prod _{i=1}^{k-1}{\frac {ds_{i}}{s_{i}-E_{n}^{(0)}}}\,\rho _{n,k}(s_{1},\ldots ,s_{k-1}).$

Effects of Degeneracy

Now, what happens when two or more energy eigenstates of the unperturbed Hamiltonian find themselves sharing the same energy – when they are degenerate? The situation becomes considerably more complex. The first-order energy shift loses its straightforward definition because there is no unique way to select a basis of eigenstates for the unperturbed system. The various eigenstates that belong to a given degenerate energy level will perturb differently, or, in some cases, may not admit a continuous family of perturbations at all.

This mathematical awkwardness is mirrored in the calculation of the perturbed eigenstate: the operator $E_{n}^{(0)}-H_{0}$ ceases to have a well-defined inverse.

Let D represent the subspace spanned by these degenerate eigenstates. Regardless of how minuscule the perturbation might be, within this degenerate subspace D, the energy differences between the eigenstates of H are non-zero. This guarantees a complete mixing of at least some of these states. Typically, the eigenvalues will split, and the eigenspaces will simplify, often becoming one-dimensional, or at least smaller in dimension than D.

The perturbations that prove successful are not those that are "small" relative to an ill-chosen basis of D. Instead, we consider a perturbation "small" if the new eigenstate remains close to the subspace D. The new Hamiltonian must be diagonalized within D, or perhaps a slight variation of it. These perturbed eigenstates within D then serve as the foundation for the perturbation expansion:

$|n\rangle =\sum _{k\in D}\alpha _{nk}|k^{(0)}\rangle +\lambda |n^{(1)}\rangle .$

For the first-order perturbation, we are compelled to solve the perturbed Hamiltonian restricted to the degenerate subspace D:

$V|k^{(0)}\rangle =\epsilon _{k}|k^{(0)}\rangle +{\text{small}}\qquad \forall |k^{(0)}\rangle \in D,$

where $\epsilon _{k}$ represents the first-order corrections to the degenerate energy levels, and "small" denotes a vector of $O(\lambda)$ orthogonal to D. This process is akin to diagonalizing the matrix:

$V_{kl} = \langle k^{(0)}|V|l^{(0)}\rangle \qquad \forall \;|k^{(0)}\rangle ,|l^{(0)}\rangle \in D.$

This procedure is, by its nature, an approximation, as we are neglecting states outside the subspace D (the "small" terms). The splitting of the degenerate energies, $\epsilon _{k}$, is a general observation. While this splitting may be small, $O(\lambda)$, relative to the overall energy scale of the system, it is fundamentally important for understanding certain subtle details, such as the spectral lines observed in Electron Spin Resonance experiments.

Higher-order corrections arising from other eigenstates outside D can be calculated in a manner analogous to the non-degenerate case:

$\left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle =\sum _{k\not \in D}\left(\langle k^{(0)}|V|n^{(0)}\rangle \right)|k^{(0)}\rangle .$

Since the operator on the left-hand side does not become singular when applied to eigenstates outside D, we can express:

$|n^{(1)}\rangle =\sum _{k\not \in D}{\frac {\langle k^{(0)}|V|n^{(0)}\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}|k^{(0)}\rangle ,$

but the effect on the degenerate states themselves remains on the order of $O(\lambda)$.

Near-degenerate states warrant similar treatment. When the original Hamiltonian's splittings are not significantly larger than the perturbation within the near-degenerate subspace, a comparable approach is necessary. An illustrative application can be found in the nearly free electron model, where a careful treatment of near-degeneracy results in an energy gap, even in the presence of small perturbations. Other eigenstates, in contrast, will merely shift the absolute energy of all near-degenerate states in unison.

Degeneracy Lifted to First Order

Let us consider a scenario involving degenerate energy eigenstates and a perturbation that completely dismantles this degeneracy to the first order of correction.

The perturbed Hamiltonian is denoted as:

${\hat {H}}={\hat {H}}_{0}+\lambda {\hat {V}}\,,$

Here, ${\hat {H}}_{0}$ is the unperturbed Hamiltonian, ${\hat {V}}$ is the perturbation operator, and $0<\lambda <1$ is the parameter quantifying the strength of the perturbation.

Our focus is on the degeneracy associated with the $n$-th unperturbed energy, $E_{n}^{(0)}$. We shall denote the unperturbed states within this degenerate subspace as $|\psi _{nk}^{(0)}\rangle$, where $k$ serves as an index for the unperturbed state within the degenerate set. The other unperturbed states are denoted as $|\psi _{m}^{(0)}\rangle$, where $m \neq n$ encompasses all other energy eigenstates whose energies differ from $E_{n}^{(0)}$. It is important to note that any degeneracy among these "other" states (i.e., for $\forall m \neq n$) does not alter our fundamental arguments. All states $|\psi _{nk}^{(0)}\rangle$ sharing the same energy $E_{n}^{(0)}$ when there is no perturbation ($\lambda = 0$). The energies $E_{m}^{(0)}$ of the states $|\psi _{m}^{(0)}\rangle$ where $m \neq n$ are, by definition, distinct from $E_{n}^{(0)}$, though they are not necessarily unique amongst themselves.

Using $V_{nl,nk}$ and $V_{m,nk}$ to represent the matrix elements of the perturbation operator $\hat{V}$ in the basis of the unperturbed eigenstates, and assuming that the basis vectors $|\psi _{nk}^{(0)}\rangle$ within the degenerate subspace have been chosen such that the matrix elements $V_{nl,nk}\equiv \left\langle \psi _{nl}^{(0)}\right|{\hat {V}}\left|\psi _{nk}^{(0)}\right\rangle$ are diagonal. Furthermore, assuming that the degeneracy is entirely resolved to the first order of correction, meaning that $E_{nl}^{(1)}\neq E_{nk}^{(1)}$ if $l\neq k$, we arrive at the following formulae for the energy correction up to the second order in $\lambda$:

$E_{nk}=E_{n}^{0}+\lambda V_{nk,nk}+\lambda ^{2}\sum \limits _{m\neq n}{\frac {\left|V_{m,nk}\right|^{2}}{E_{n}^{(0)}-E_{m}^{(0)}}}+{\mathcal {O}}(\lambda ^{3})\,,$

And for the state correction, to the first order in $\lambda$:

$\left|\psi _{nk}^{(1)}\right\rangle =\left|\psi _{nk}^{(0)}\right\rangle +\lambda \sum \limits _{m\neq n}{\frac {V_{m,nk}}{E_{m}^{(0)}-E_{n}^{(0)}}}\left(-\left|\psi _{m}^{(0)}\right\rangle +\sum \limits _{l\neq k}{\frac {V_{nl,m}}{E_{nl}^{(1)}-E_{nk}^{(1)}}}\left|\psi _{nl}^{(0)}\right\rangle \right)+{\mathcal {O}}(\lambda ^{2})\,.$

A noteworthy observation here is that the first-order correction to the state is orthogonal to the unperturbed state:

$\left\langle \psi _{nk}^{(0)}|\psi _{nk}^{(1)}\right\rangle =0\,.$

Generalization to Multi-Parameter Case

The extension of time-independent perturbation theory to scenarios involving multiple small parameters, $x^{\mu }=(x^{1},x^{2},\cdots)$, in place of a single parameter like λ, can be approached more systematically. This is achieved by employing the framework of differential geometry, which inherently involves defining derivatives of quantum states and calculating perturbative corrections through iterative differentiation at the unperturbed point.

Hamiltonian and Force Operator

From a differential geometric perspective, a parameterized Hamiltonian is conceptualized as a function defined on a parameter manifold. This function maps each specific set of parameters $(x^{1},x^{2},\cdots)$ to an Hermitian operator $H(x^{\mu})$ acting on the Hilbert space. The parameters themselves can represent various physical influences, such as external fields, interaction strengths, or driving parameters in the context of a quantum phase transition. Let $E_{n}(x^{\mu})$ and $|n(x^{\mu})\rangle$ denote the $n$-th eigenenergy and eigenstate of $H(x^{\mu})$, respectively. Within the language of differential geometry, the collection of states $|n(x^{\mu})\rangle$ forms a vector bundle over the parameter manifold. Crucially, derivatives of these states can be defined on this bundle. Perturbation theory, in this context, seeks to answer a fundamental question: given the values of $E_{n}(x_{0}^{\mu})$ and $|n(x_{0}^{\mu})\rangle$ at an unperturbed reference point $x_{0}^{\mu}$, how can we accurately estimate the values of $E_{n}(x^{\mu})$ and $|n(x^{\mu})\rangle$ at points $x^{\mu}$ in the immediate vicinity of that reference point.

Without loss of generality, we can simplify the problem by shifting the coordinate system such that the reference point $x_{0}^{\mu}$ coincides with the origin, i.e., $x_{0}^{\mu}=0$. A commonly employed form for the parameterized Hamiltonian in this context is the linearly parameterized Hamiltonian:

$H(x^{\mu})=H(0)+x^{\mu}F_{\mu}.$

If we consider the parameters $x^{\mu}$ as generalized coordinates, then the operators $F_{\mu}$ must be identified as the generalized force operators conjugate to those coordinates. The different indices $\mu$ serve to distinguish the various forces acting along different directions within the parameter manifold. For instance, if $x^{\mu}$ represents an external magnetic field aligned along the $\mu$-direction, then $F_{\mu}$ would naturally correspond to the magnetization in that same direction.

Perturbation Theory as Power Series Expansion

The efficacy of perturbation theory hinges on the adiabatic assumption. This assumption posits that the eigenenergies and eigenstates of the Hamiltonian vary smoothly as functions of the parameters. Consequently, their values in a local region can be approximated using power series expansions, akin to a Taylor series, in terms of these parameters:

$E_{n}(x^{\mu}) =E_{n}+x^{\mu}\partial _{\mu }E_{n}+{\frac {1}{2!}}x^{\mu}x^{\nu}\partial _{\mu }\partial _{\nu }E_{n}+\cdots$ $\left|n(x^{\mu})\right\rangle =\left|n\right\rangle +x^{\mu}\left|\partial _{\mu }n\right\rangle +{\frac {1}{2!}}x^{\mu}x^{\nu}\left|\partial _{\mu }\partial _{\nu }n\right\rangle +\cdots$

Here, $\partial_{\mu}$ denotes differentiation with respect to $x^{\mu}$. When applied to the state $|\partial_{\mu}n\rangle$, it is essential to interpret this as the covariant derivative, particularly if the vector bundle is equipped with a non-vanishing connection. All terms on the right-hand side of these series expansions are evaluated at the reference point $x^{\mu}=0$. Thus, $E_{n} \equiv E_{n}(0)$ and $|n\rangle \equiv |n(0)\rangle$. This convention will be maintained throughout this section: any function without explicit parameter dependence is assumed to be evaluated at the origin. It is important to acknowledge that these power series may exhibit slow convergence, or even fail to converge altogether, when energy levels approach each other. The adiabatic assumption fundamentally breaks down in the presence of energy level degeneracy, rendering perturbation theory inapplicable in such cases.

Hellmann–Feynman Theorems

The power series expansions outlined above can be readily evaluated provided there exists a systematic method for calculating derivatives to arbitrary order. The Hellmann–Feynman theorems offer precisely this capability, enabling the calculation of single derivatives of both the energy and the state. The first Hellmann–Feynman theorem provides the derivative of the energy:

$\partial _{\mu }E_{n}=\langle n|\partial _{\mu }H|n\rangle$

The second Hellmann–Feynman theorem addresses the derivative of the state, which can be resolved by projecting onto the complete basis of states, where $m \neq n$:

$\langle m|\partial _{\mu }n\rangle ={\frac {\langle m|\partial _{\mu }H|n\rangle }{E_{n}-E_{m}}},\qquad \langle \partial _{\mu }m|n\rangle ={\frac {\langle m|\partial _{\mu }H|n\rangle }{E_{m}-E_{n}}}.$

For a linearly parameterized Hamiltonian, the term $\partial_{\mu}H$ simply corresponds to the generalized force operator $F_{\mu}$.

These theorems are derived straightforwardly by applying the differential operator $\partial_{\mu}$ to both sides of the Schrödinger equation, $H|n\rangle =E_{n}|n\rangle$:

$\partial _{\mu }H|n\rangle +H|\partial _{\mu }n\rangle =\partial _{\mu }E_{n}|n\rangle +E_{n}|\partial _{\mu }n\rangle .$

By taking the inner product with the state $\langle m|$ from the left and again employing the Schrödinger equation in the form $\langle m|H = \langle m|E_{m}$, we obtain:

$\langle m|\partial _{\mu }H|n\rangle +E_{m}\langle m|\partial _{\mu }n\rangle =\partial _{\mu }E_{n}\langle m|n\rangle +E_{n}\langle m|\partial _{\mu }n\rangle .$

Given that the eigenstates of the Hamiltonian form an orthonormal basis, $\langle m|n\rangle = \delta_{mn}$, we can analyze the cases $m=n$ and $m \neq n$ separately. The former case leads directly to the first Hellmann–Feynman theorem, while the latter, upon rearrangement of terms, yields the second theorem. With these differential rules provided by the Hellmann–Feynman theorems, the perturbative corrections to both energies and states can be calculated in a systematic manner.

Correction of Energy and State

To the second order of approximation, the energy correction is expressed as:

$E_{n}(x^{\mu})=\langle n|H|n\rangle +\langle n|\partial _{\mu }H|n\rangle x^{\mu }+\Re \sum _{m\neq n}{\frac {\langle n|\partial _{\nu }H|m\rangle \langle m|\partial _{\mu }H|n\rangle }{E_{n}-E_{m}}}x^{\mu }x^{\nu }+\cdots ,$

where $\Re$ denotes the real part. The first-order derivative, $\partial_{\mu}E_{n}$, is directly obtained from the first Hellmann–Feynman theorem. To determine the second-order derivative, $\partial_{\mu}\partial_{\nu}E_{n}$, we simply apply the differential operator $\partial_{\mu}$ to the result of the first-order derivative, $\langle n|\partial_{\nu}H|n\rangle$:

$\partial _{\mu }\partial _{\nu }E_{n}=\langle \partial _{\mu }n|\partial _{\nu }H|n\rangle +\langle n|\partial _{\mu }\partial _{\nu }H|n\rangle +\langle n|\partial _{\nu }H|\partial _{\mu }n\rangle .$

It is important to note that for a linearly parameterized Hamiltonian, the second derivative $\partial_{\mu}\partial_{\nu}H$ vanishes at the operator level. By resolving the state derivative through the insertion of a complete set of basis states:

$\partial _{\mu }\partial _{\nu }E_{n}=\sum _{m}\left(\langle \partial _{\mu }n|m\rangle \langle m|\partial _{\nu }H|n\rangle +\langle n|\partial _{\nu }H|m\rangle \langle m|\partial _{\mu }n\rangle \right),$

all constituent parts can be calculated using the Hellmann–Feynman theorems. In terms of Lie derivatives, it holds that $\langle \partial_{\mu}n|n\rangle = \langle n|\partial_{\mu}n\rangle = 0$ according to the definition of the connection for the vector bundle. Consequently, the case $m=n$ can be excluded from the summation, thereby circumventing the singularity associated with the energy denominator. This same computational strategy can be extended to higher-order derivatives, yielding higher-order corrections.

The same computational framework is equally applicable to the correction of states. The result, up to the second order, is as follows:

$\left|n\left(x^{\mu }\right)\right\rangle =|n\rangle +\sum _{m\neq n}{\frac {\langle m|\partial _{\mu }H|n\rangle }{E_{n}-E_{m}}}|m\rangle x^{\mu }\\ +\left(\sum _{m\neq n}\sum _{l\neq n}{\frac {\langle m|\partial _{\mu }H|l\rangle \langle l|\partial _{\nu }H|n\rangle }{(E_{n}-E_{m})(E_{n}-E_{l})}}|m\rangle -\sum _{m\neq n}{\frac {\langle m|\partial _{\mu }H|n\rangle \langle n|\partial _{\nu }H|n\rangle }{(E_{n}-E_{m})^{2}}}|m\rangle -{\frac {1}{2}}\sum _{m\neq n}{\frac {\langle n|\partial _{\mu }H|m\rangle \langle m|\partial _{\nu }H|n\rangle }{(E_{n}-E_{m})^{2}}}|n\rangle \right)x^{\mu }x^{\nu }+\cdots .$

Both energy derivatives and state derivatives are indispensable in these derivations. Whenever a state derivative is encountered, it is resolved by inserting a complete basis set, at which point the Hellmann–Feynman theorem becomes applicable. Since differentiation can be systematically performed, this series expansion approach to perturbative corrections can be readily implemented computationally using symbolic processing software such as Mathematica.

Effective Hamiltonian

Consider a Hamiltonian $H^{(0)}$ that is strictly confined, either to a low-energy subspace ${\mathcal {H}}_{L}$ or a high-energy subspace ${\mathcal {H}}_{H}$. This confinement implies that there are no matrix elements within $H^{(0)}$ that connect the low- and high-energy subspaces, meaning $\langle m|H(0)|l\rangle =0$ if $m\in {\mathcal {H}}_{L}$ and $l\in {\mathcal {H}}_{H}$. Let $F_{\mu} = \partial_{\mu}H$ represent the coupling terms that bridge these subspaces. When the high-energy degrees of freedom are effectively integrated out, the resulting effective Hamiltonian within the low-energy subspace takes the form:

$H_{mn}^{\text{eff}}\left(x^{\mu }\right)=\langle m|H|n\rangle +\delta _{nm}\langle m|\partial _{\mu }H|n\rangle x^{\mu }+{\frac {1}{2!}}\sum _{l\in {\mathcal {H}}_{H}}\left({\frac {\langle m|\partial _{\mu }H|l\rangle \langle l|\partial _{\nu }H|n\rangle }{E_{m}-E_{l}}}+{\frac {\langle m|\partial _{\nu }H|l\rangle \langle l|\partial _{\mu }H|n\rangle }{E_{n}-E_{l}}}\right)x^{\mu }x^{\nu }+\cdots .$

Here, $m$ and $n$ are restricted to the low-energy subspace ${\mathcal {H}}_{L}$. This expression can be derived by applying a power series expansion to $\langle m|H(x^{\mu})|n\rangle$.

Formally, it is indeed possible to define an effective Hamiltonian that precisely yields the low-lying energy states and wavefunctions.[11] In practical applications, however, some form of approximation, typically perturbation theory, is generally required.

Time-Dependent Perturbation Theory

Method of Variation of Constants

Time-dependent perturbation theory, a framework pioneered by Paul Dirac and further elaborated by John Archibald Wheeler, Richard Feynman, and Freeman Dyson,[12] investigates the consequences of a time-dependent perturbation $V(t)$ acting upon a time-independent Hamiltonian $H_0$.[13] It stands as an exceptionally valuable tool for calculating the properties of virtually any physical system. Its applications span an astonishing range, from the quantitative description of phenomena as diverse as proton-proton scattering, the photo-ionization of materials, the scattering of electrons off lattice defects in conductors, neutron scattering off nuclei, the electric susceptibilities of materials, to neutron absorption cross-sections in nuclear reactors, and much, much more.[12]

Given that the perturbed Hamiltonian itself is time-dependent, its energy levels and eigenstates naturally inherit this time dependence. Consequently, the objectives of time-dependent perturbation theory diverge slightly from those of its time-independent counterpart. The quantities of primary interest are:

• The time-dependent expectation value of some observable $A$, given a specific initial state.
• The time-dependent expansion coefficients (relative to a given time-dependent state) of the basis states that correspond to energy eigenkets (eigenvectors) in the unperturbed system.

The first quantity is significant because it directly relates to the classical outcome of an $A$ measurement performed on a macroscopic ensemble of perturbed system copies. For instance, if we select $A$ to represent the displacement in the x-direction of an electron within a hydrogen atom, its expectation value, when multiplied by an appropriate coefficient, yields the time-dependent dielectric polarization of the hydrogen gas. By choosing a suitable perturbation, such as an oscillating electric potential, this allows for the calculation of the gas's AC permittivity.

The second quantity focuses on the time-dependent probability of occupying each eigenstate. This is particularly relevant in laser physics, where understanding the populations of different atomic states in a gas subjected to a time-dependent electric field is crucial. These probabilities also play a role in calculating the "quantum broadening" of spectral lines (refer to line broadening) and in analyzing particle decay within the domains of particle physics and nuclear physics.

Let us briefly examine the foundational method of Dirac's formulation of time-dependent perturbation theory. We begin by selecting an energy basis $|n\rangle$ for the unperturbed system. (We omit the $(0)$ superscripts for the eigenstates, as referring to energy levels and eigenstates of the perturbed system becomes less meaningful in this context.)

If, at time $t=0$, the unperturbed system is in an eigenstate (of the Hamiltonian) $|j\rangle$, its state at subsequent times evolves solely by a phase factor (within the Schrödinger picture, where state vectors evolve in time and operators remain constant):

$|j(t)\rangle =e^{-iE_{j}t/\hbar }|j\rangle ~.$

Now, we introduce a time-dependent perturbing Hamiltonian $V(t)$. The Hamiltonian governing the perturbed system is:

$H=H_{0}+V(t) ~.$

Let $|\psi (t)\rangle$ represent the quantum state of the perturbed system at time $t$. This state adheres to the time-dependent Schrödinger equation:

$H|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ~.$

At any given instant, the quantum state can be expressed as a linear combination of the complete eigenbasis $|n\rangle$:

$|\psi (t)\rangle =\sum _{n}c_{n}(t)e^{-iE_{n}t/\hbar }|n\rangle ~,$

where the coefficients $c_n(t)$ are complex functions of $t$ that we need to determine. These are often referred to as amplitudes (though, strictly speaking, they are amplitudes in the Dirac picture).

We have deliberately factored out the exponential phase factors $e^{-iE_{n}t/\hbar}$ on the right-hand side. This is a matter of convention and can be done without loss of generality. The rationale behind this step is that if the system commences in the state $|j\rangle$ and no perturbation is present, the amplitudes exhibit a convenient property: for all $t$, $c_j(t) = 1$ and $c_n(t) = 0$ if $n \neq j$.

The square of the absolute value of an amplitude, $|c_n(t)|^2$, directly represents the probability that the system resides in state $n$ at time $t$, since:

$\left|c_{n}(t)\right|^{2}=\left|\langle n|\psi (t)\rangle \right|^{2}~.$

Substituting this expansion into the Schrödinger equation and utilizing the product rule for the time derivative, we obtain:

$\sum _{n}\left(i\hbar {\frac {dc_{n}}{dt}}-c_{n}(t)V(t)\right)e^{-iE_{n}t/\hbar }|n\rangle =0~.$

By resolving the identity preceding $V$ and multiplying the entire equation from the left by the bra $\langle n|$, we can reduce this to a system of coupled differential equations for the amplitudes:

${\frac {dc_{n}}{dt}}={\frac {-i}{\hbar }}\sum _{k}\langle n|V(t)|k\rangle \,c_{k}(t)\,e^{-i(E_{k}-E_{n})t/\hbar }~$

where we have used equation (1) to evaluate the sum over $n$ in the second term, and subsequently employed the relation $\langle k|\Psi (t)\rangle =c_{k}(t)e^{-iE_{k}t/\hbar}$.

The matrix elements of $V$ play a role analogous to their counterparts in time-independent perturbation theory, being directly proportional to the rate at which amplitudes are transferred between states. However, it is crucial to note that the direction of this transfer is modulated by the exponential phase factor. Over time intervals significantly longer than the energy difference $E_k - E_n$, this phase factor completes numerous cycles, effectively averaging to zero. If the time dependence of $V$ is sufficiently gradual, this can induce oscillatory behavior in the state amplitudes. (Such oscillations are, for instance, instrumental in managing radiative transitions within a laser.)

Up to this point, no approximations have been made; the system of differential equations derived is exact. By providing appropriate initial conditions, $c_n(t)$, one could, in principle, ascertain a non-perturbative, exact solution. This becomes remarkably manageable when dealing with only two energy levels ($n=1, 2$), and this specific solution proves quite useful for modeling systems such as the ammonia molecule.

However, exact solutions become elusive when the number of energy levels increases. In such cases, one typically resorts to perturbative solutions. These can be obtained by rewriting the equations in integral form:

$c_{n}(t)=c_{n}(0)-{\frac {i}{\hbar }}\sum _{k}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle \,c_{k}(t')\,e^{-i(E_{k}-E_{n})t'/\hbar }~$

Repeatedly substituting this expression for $c_n$ back into the right-hand side generates an iterative solution:

$c_{n}(t)=c_{n}^{(0)}+c_{n}^{(1)}+c_{n}^{(2)}+\cdots$

where, for example, the first-order term is:

$c_{n}^{(1)}(t)={\frac {-i}{\hbar }}\sum _{k}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle \,c_{k}^{(0)}\,e^{-i(E_{k}-E_{n})t'/\hbar }~$

To this same level of approximation, the summation can be eliminated, as in the unperturbed state, $c_{k}^{(0)}=\delta _{kn}$. This simplifies the expression to:

$c_{n}^{(1)}(t)={\frac {-i}{\hbar }}\int _{0}^{t}dt'\;\langle n|V(t')|k\rangle \,e^{-i(E_{k}-E_{n})t'/\hbar }~$

Several subsequent results derive from this foundation, including Fermi's golden rule, which establishes a relationship between the rate of transitions between quantum states and the density of states at specific energies, and the Dyson series, obtained by applying the iterative method to the time evolution operator, serving as a foundational element for the method of Feynman diagrams.

Method of Dyson Series

Time-dependent perturbations can be systematically reorganized using the technique of the Dyson series. The Schrödinger equation:

$H(t)|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}$

admits a formal solution:

$|\psi (t)\rangle =T\exp {\left[-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'H(t')\right]}|\psi (t_{0})\rangle ~,$

where $T$ denotes the time-ordering operator:

$TA(t_{1})A(t_{2})={\begin{cases}A(t_{1})A(t_{2})&t_{1}>t_{2}\\A(t_{2})A(t_{1})&t_{2}>t_{1}\end{cases}}~.$

Consequently, the exponential term expands into the following Dyson series:

$|\psi (t)\rangle =\left[1-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt_{1}H(t_{1})-{\frac {1}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}H(t_{1})H(t_{2})+\ldots \right]|\psi (t_{0})\rangle ~.$

It is important to note that the $1/2!$ factor in the second term precisely cancels the double contribution arising from the time-ordering operator, and so forth for subsequent terms.

Consider the perturbation problem:

$[H_{0}+\lambda V(t)]|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}~$

assuming $\lambda$ is small and that the problem $H_{0}|n\rangle =E_{n}|n\rangle$ has already been solved.

We perform a unitary transformation to the interaction picture (also known as the Dirac picture):

$|\psi (t)\rangle =e^{-{\frac {i}{\hbar }}H_{0}(t-t_{0})}|\psi _{I}(t)\rangle ~.$

This transformation simplifies the Schrödinger equation to:

$\lambda e^{{\frac {i}{\hbar }}H_{0}(t-t_{0})}V(t)e^{-{\frac {i}{\hbar }}H_{0}(t-t_{0})}|\psi _{I}(t)\rangle =i\hbar {\frac {\partial |\psi _{I}(t)\rangle }{\partial t}}~$

The solution is then obtained via the aforementioned Dyson series:

$|\psi _{I}(t)\rangle =\left[1-{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}e^{{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}-{\frac {\lambda ^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}e^{{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\frac {i}{\hbar }}H_{0}(t_{1}-t_{0})}e^{{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-{\frac {i}{\hbar }}H_{0}(t_{2}-t_{0})}+\ldots \right]|\psi (t_{0})\rangle ~.$

Utilizing the solution of the unperturbed problem, $H_{0}|n\rangle =E_{n}|n\rangle$, and the completeness relation $\sum _{n}|n\rangle \langle n|=1$ (assuming, for simplicity, a pure discrete spectrum), we obtain, to first order:

$|\psi _{I}(t)\rangle =\left[1-{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}\sum _{m}\sum _{n}\langle m|V(t_{1})|n\rangle e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle \langle n|+\ldots \right]|\psi (t_{0})\rangle ~.$

Thus, the system, initially in the unperturbed state $|\alpha \rangle =|\psi (t_{0})\rangle$, can transition to another state $|\beta \rangle$ due to the perturbation. The corresponding transition probability amplitude, to first order, is given by:

$A_{\alpha \beta }=-{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}\langle \beta |V(t_{1})|\alpha \rangle e^{-{\frac {i}{\hbar }}(E_{\alpha }-E_{\beta })(t_{1}-t_{0})}~$

as detailed in the preceding section. The corresponding transition probability to a continuum is then furnished by Fermi's golden rule.

As a side note, it is worth mentioning that time-independent perturbation theory can also be systematically derived from this time-dependent perturbation theory framework, specifically from the Dyson series. By writing the unitary evolution operator, obtained from the Dyson series, as:

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 considering the perturbation $V$ to be time-independent, we can proceed.

Using the identity $\sum _{n}|n\rangle \langle n|=1$ and $H_{0}|n\rangle =E_{n}|n\rangle$ for a pure discrete spectrum, we can express $U(t)$ as:

$U(t)=1 - \left[{\frac {i\lambda }{\hbar }}\int _{t_{0}}^{t}dt_{1}\sum _{m}\sum _{n}\langle m|V|n\rangle e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle \langle n|\right] - \left[{\frac {\lambda ^{2}}{\hbar ^{2}}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\sum _{m}\sum _{n}\sum _{q}e^{-{\frac {i}{\hbar }}(E_{n}-E_{m})(t_{1}-t_{0})}\langle m|V|n\rangle \langle n|V|q\rangle e^{-{\frac {i}{\hbar }}(E_{q}-E_{n})(t_{2}-t_{0})}|m\rangle \langle q|\right] + \cdots$

It becomes evident that, to second order, a summation over all intermediate states is required. Assuming $t_0=0$ and considering the asymptotic limit for large times (which implies introducing a multiplicative factor $e^{-\epsilon t}$ for arbitrarily small $\epsilon$ in the integrands), the integration process eliminates oscillating terms while preserving secular ones. Consequently, the integrals become computable. Separating the diagonal terms from the others yields:

$U(t)=1 -{\frac {i\lambda }{\hbar }}\sum _{n}\langle n|V|n\rangle t -{\frac {i\lambda ^{2}}{\hbar }}\sum _{m\neq n}{\frac {\langle n|V|m\rangle \langle m|V|n\rangle }{E_{n}-E_{m}}}t -{\frac {1}{2}}{\frac {\lambda ^{2}}{\hbar ^{2}}}\sum _{m,n}\langle n|V|m\rangle \langle m|V|n\rangle t^{2}+\cdots \\ +\lambda \sum _{m\neq n}{\frac {\langle m|V|n\rangle }{E_{n}-E_{m}}}|m\rangle \langle n| + \lambda^{2}\sum _{m\neq n}\sum _{q\neq n}\sum _{n}{\frac {\langle m|V|n\rangle \langle n|V|q\rangle }{(E_{n}-E_{m})(E_{q}-E_{n})}}|m\rangle \langle q|+\cdots$

Here, the time-dependent secular series yields the eigenvalues of the perturbed problem, obtained recursively, while the time-independent portion provides the corrections to the stationary eigenfunctions, also as described previously ($|n(\lambda )\rangle =U(0;\lambda )|n\rangle$). The unitary evolution operator remains applicable to arbitrary eigenstates of the unperturbed problem, yielding a secular series valid for short times.

Strong Perturbation Theory

In a manner analogous to the treatment of small perturbations, it is possible to formulate a theory for strong perturbations. Consider, as is standard, the Schrödinger equation:

$H(t)|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}$

The pertinent question then becomes whether a dual Dyson series exists that is applicable in the limit of an increasingly large perturbation. This question can be answered affirmatively[14], and the resulting series is the well-known adiabatic series.[15] This approach is remarkably general and can be demonstrated as follows. Consider the perturbation problem:

$[H_{0}+\lambda V(t)]|\psi (t)\rangle =i\hbar {\frac {\partial |\psi (t)\rangle }{\partial t}}$

in the limit where $\lambda \to \infty$. Our objective is to find a solution of the form:

$|\psi \rangle =|\psi _{0}\rangle +{\frac {1}{\lambda }}|\psi _{1}\rangle +{\frac {1}{\lambda ^{2}}}|\psi _{2}\rangle +\ldots$

However, a direct substitution of this series into the equation proves unproductive. This predicament can be rectified by rescaling the time variable such that $\tau = \lambda t$, leading to the following meaningful equations:

$V(t)|\psi _{0}\rangle =i\hbar {\frac {\partial |\psi _{0}\rangle }{\partial \tau }}$ $V(t)|\psi _{1}\rangle +H_{0}|\psi _{0}\rangle =i\hbar {\frac {\partial |\psi _{1}\rangle }{\partial \tau }}$ $\vdots$

These equations can be solved sequentially, provided we first ascertain the solution to the leading-order equation. In this context, the adiabatic approximation becomes applicable. If $V(t)$ is time-independent, we recover the Wigner-Kirkwood series, a formulation frequently employed in statistical mechanics. Indeed, in this scenario, we introduce a unitary transformation:

$|\psi (t)\rangle =e^{-{\frac {i}{\hbar }}\lambda V(t-t_{0})}|\psi _{F}(t)\rangle$

which defines a "free picture," as our intention is to eliminate the interaction term. Now, in a manner dual to the approach for small perturbations, we must solve the Schrödinger equation:

$e^{{\frac {i}{\hbar }}\lambda V(t-t_{0})}H_{0}e^{-{\frac {i}{\hbar }}\lambda V(t-t_{0})}|\psi _{F}(t)\rangle =i\hbar {\frac {\partial |\psi _{F}(t)\rangle }{\partial t}}$

Observe that the expansion parameter $\lambda$ appears solely within the exponential term. This implies that the corresponding Dyson series, which we term a dual Dyson series, is meaningful for large values of $\lambda$. This is because we have derived this series by simply interchanging the roles of $H_0$ and $V$, a transformation that can be applied to move between the two formulations. This is known as the duality principle in perturbation theory. Choosing $H_{0}=p^{2}/2m$ yields, as previously mentioned, a Wigner-Kirkwood series, which is essentially a gradient expansion. The Wigner-Kirkwood series represents a semiclassical expansion whose eigenvalues are precisely those obtained from the WKB approximation.[16]

Examples

Example of First-Order Perturbation Theory – Ground-State Energy of the Quartic Oscillator

Consider a quantum harmonic oscillator subjected to a quartic potential perturbation. The Hamiltonian is given by:

$H=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}x^{2}}{2}}+\lambda x^{4}.$

The ground state of the unperturbed harmonic oscillator is:

$\psi _{0}=\left({\frac {\alpha }{\pi }}\right)^{\frac {1}{4}}e^{-\alpha x^{2}/2}$

(where $\alpha =m\omega /\hbar$), and its unperturbed ground state energy is:

$E_{0}^{(0)}={\tfrac {1}{2}}\hbar \omega$

Applying the formula for the first-order correction, we obtain:

$E_{0}^{(1)}=\lambda \left({\frac {\alpha }{\pi }}\right)^{\frac {1}{2}}\int e^{-\alpha x^{2}/2}x^{4}e^{-\alpha x^{2}/2}dx=\lambda \left({\frac {\alpha }{\pi }}\right)^{\frac {1}{2}}{\frac {\partial ^{2}}{\partial \alpha ^{2}}}\int e^{-\alpha x^{2}}dx,$

which simplifies to:

$E_{0}^{(1)}=\lambda {\frac {3}{4}}{\frac {1}{\alpha ^{2}}}={\frac {3}{4}}{\frac {\hbar ^{2}\lambda }{m^{2}\omega ^{2}}}.$

Example of First- and Second-Order Perturbation Theory – Quantum Pendulum

Let us examine the quantum-mathematical pendulum, characterized by the Hamiltonian:

$H=-{\frac {\hbar ^{2}}{2ma^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}-\lambda \cos \phi$

Here, the potential energy term, $-\lambda \cos \phi$, is treated as the perturbation, so $V=-\cos \phi$. The unperturbed, normalized quantum wave functions correspond to those of a rigid rotor and are given by:

$\psi _{n}(\phi )={\frac {e^{in\phi }}{\sqrt {2\pi }}},$

with corresponding energies:

$E_{n}^{(0)}={\frac {\hbar ^{2}n^{2}}{2ma^{2}}}.$

The first-order energy correction to the rotor state, induced by the potential energy perturbation, is calculated as:

$E_{n}^{(1)}=-{\frac {1}{2\pi }}\int e^{-in\phi }\cos \phi e^{in\phi }d\phi = -{\frac {1}{2\pi }}\int \cos \phi d\phi = 0.$

Employing the formula for the second-order correction, we derive:

$E_{n}^{(2)}={\frac {ma^{2}}{2\pi ^{2}\hbar ^{2}}}\sum _{k}{\frac {\left|\int e^{-ik\phi }\cos \phi e^{in\phi }\,d\phi \right|^{2}}{n^{2}-k^{2}}},$

which simplifies to:

$E_{n}^{(2)}={\frac {ma^{2}}{2\hbar ^{2}}}\sum _{k}{\frac {\left|\left(\delta _{n,1-k}+\delta _{n,-1-k}\right)\right|^{2}}{n^{2}-k^{2}}},$

or further:

$E_{n}^{(2)}={\frac {ma^{2}}{2\hbar ^{2}}}\left({\frac {1}{2n-1}}+{\frac {1}{-2n-1}}\right)={\frac {ma^{2}}{\hbar ^{2}}}{\frac {1}{4n^{2}-1}}.$

Potential Energy as a Perturbation

Consider the case of a free particle moving in one dimension, with kinetic energy $E$. The solution to the Schrödinger equation for the unperturbed state is:

$\nabla ^{2}\psi ^{(0)}+k^{2}\psi ^{(0)}=0$

which corresponds to plane waves with wavenumber $k = \sqrt{2mE/\hbar^2}$. If a weak potential energy $U(x,y,z)$ is introduced into the system, the perturbed state, to a first approximation, is described by the equation:

$\nabla ^{2}\psi ^{(1)}+k^{2}\psi ^{(1)}={\frac {2mU}{\hbar ^{2}}}\psi ^{(0)},$

The particular integral of this equation is given by[17]:

$\psi ^{(1)}(x,y,z)=-{\frac {m}{2\pi \hbar ^{2}}}\int \psi ^{(0)}U(x',y',z'){\frac {e^{ikr}}{r}}\,dx'dy'dz',$

where $r^{2}=(x-x')^{2}+(y-y')^{2}+(z-z')^{2}$. In two dimensions, the solution takes the form:

$\psi ^{(1)}(x,y)=-{\frac {im}{2\hbar ^{2}}}\int \psi ^{(0)}U(x',y')H_{0}^{(1)}(kr)\,dx'dy',$

with $r^{2}=(x-x')^{2}+(y-y')^{2}$ and $H_{0}^{(1)}$ being the Hankel function of the first kind. In the one-dimensional case, the solution is:

$\psi ^{(1)}(x)=-{\frac {im}{\hbar ^{2}}}\int \psi ^{(0)}U(x'){\frac {e^{ikr}}{k}}\,dx',$

where $r=|x-x'|$.

Applications

• Rabi cycle
• Fermi's golden rule
• Muon spin spectroscopy
• Perturbed angular correlation