Time-Independent Perturbation Theory

Time-independent perturbation theory extracts spectral information from a Hamiltonian that is close, in a precisely stated sense, to one whose eigenproblem is understood. It is the standard language for weak level shifts, induced state mixing, symmetry-resolved splittings, polarizabilities, and low-energy effective Hamiltonians.

The method is not the act of writing

$$H=H_0+\lambda V.$$

Its substance is choosing a useful $H_0$, identifying which spectral subspace must be treated together, finding dimensionless coupling-to-gap ratios, and deciding what the truncated series is entitled to claim.

This chapter owns the static method. Full atomic Stark and Zeeman spectroscopy, molecular structure, many-body expansions, and rigorous analytic perturbation theory retain their own canonical homes.

The Central Eigenproblem

Let

$$H(\lambda) = H_0+\lambda V$$

and consider

$$H(\lambda) \lvert n(\lambda)\rangle = E_n(\lambda) \lvert n(\lambda)\rangle.$$

For an isolated reference level, the Rayleigh–Schrödinger ansatz is

$$\begin{aligned} E_n(\lambda) &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} +\cdots, \\ \lvert n(\lambda)\rangle &= \lvert n^{(0)}\rangle + \lambda\lvert n^{(1)}\rangle + \lambda^2\lvert n^{(2)}\rangle +\cdots. \end{aligned}$$

The zeroth-order pair obeys

$$H_0\lvert n^{(0)}\rangle = E_n^{(0)} \lvert n^{(0)}\rangle.$$

The expansion is local in $\lambda$. It is also local in spectral structure: a formula derived for an isolated eigenvalue is not valid inside a degenerate multiplet or at a continuum threshold.

Assumption Ledger

A mature perturbative calculation records at least the following.

Question	What must be stated
Reference problem	Which eigenvalues, eigenstates, projectors, or resolvents of $H_0$ are known?
Operator setting	Are $H_0$ and $H(\lambda)$ self-adjoint on a common domain, or is a quadratic-form treatment intended?
Target	Is the desired quantity an energy, state, expectation value, response coefficient, or effective Hamiltonian?
Spectral isolation	Is the reference eigenvalue isolated, degenerate, nearly degenerate, or embedded in a continuum?
Control	Which dimensionless matrix-element-to-gap ratios are small?
Symmetry	Which matrix elements vanish and which subspaces cannot mix?
Normalization	Is intermediate or unit normalization used for state corrections?
Truncation	Through which order is the answer retained, and how is the remainder assessed?

In a finite-dimensional Hilbert space, domain questions are invisible. For unbounded operators they are not. Textbook formulas are formal unless the perturbation preserves a suitable operator or form domain and the spectral object being followed has the required isolation. Kato’s analytic perturbation theory gives rigorous conditions; the elementary formulas here should not be presented as universal theorems for arbitrary unbounded $V$.

What Small Means

The parameter $\lambda$ may count orders, but physical smallness is set by dimensionless ratios. For an isolated state,

$$\epsilon_{mn} = \frac{ \lvert\lambda V_{mn}\rvert }{ \lvert E_n^{(0)}-E_m^{(0)}\rvert }, \qquad V_{mn} = \langle m^{(0)}\rvert V \lvert n^{(0)}\rangle.$$

The relevant ratios must be small for states that couple materially to $\lvert n^{(0)}\rangle$. A tiny perturbing energy can mix two nearly degenerate levels strongly. A large operator norm can be irrelevant if symmetry forbids its matrix elements in the target sector.

A single maximum over all states may be too crude or infinite. Practical control can require:

• matrix-element-weighted gap estimates;
• relative form bounds for unbounded perturbations;
• convergence of sums over high-energy states;
• a separated cluster rather than one separated level;
• numerical comparison over the parameter interval of interest.

Use Small Parameters and Error Estimates before interpreting an order label as an error estimate.

Intermediate Normalization

For a nondegenerate reference state, this chapter commonly uses

$$\langle n^{(0)} \vert n(\lambda)\rangle = 1.$$

Consequently,

$$\langle n^{(0)} \vert n^{(r)}\rangle = 0, \qquad r\ge1.$$

This convention fixes the component of every correction parallel to the reference state. It is algebraically convenient, but the resulting series is not unit normalized term by term. When expectation values are computed beyond leading order, either restore unit normalization or retain the normalization denominator explicitly.

The phase of the exact eigenstate remains conventional. Smoothly fixing that phase is part of differentiating states with respect to $\lambda$.

Order-by-Order Equations

Insert the series into the eigenvalue equation and equate powers of $\lambda$. At order $r\ge1$,

$$\begin{aligned} \bigl(H_0-E_n^{(0)}\bigr) \lvert n^{(r)}\rangle ={}& -V\lvert n^{(r-1)}\rangle \\ &+ \sum_{s=1}^{r} E_n^{(s)} \lvert n^{(r-s)}\rangle. \end{aligned}$$

Projecting with $\langle n^{(0)}\rvert$ and using intermediate normalization gives

$$E_n^{(r)} = \langle n^{(0)}\rvert V \lvert n^{(r-1)}\rangle.$$

This recursion assumes that $H$ contains only $H_0+\lambda V$. If the Hamiltonian itself has $\lambda^2$ or higher terms, additional contributions enter at the corresponding orders.

Projectors and the Reduced Resolvent

Define

$$P_n = \lvert n^{(0)}\rangle \langle n^{(0)}\rvert, \qquad Q_n=I-P_n,$$

and

$$R_n' \equiv Q_n \frac{1} {E_n^{(0)}-H_0} Q_n.$$

The inverse is taken only on the complementary subspace. Under intermediate normalization,

$$\begin{aligned} \lvert n^{(r)}\rangle = R_n' \Bigg[ V\lvert n^{(r-1)}\rangle - \sum_{s=1}^{r-1} E_n^{(s)} \lvert n^{(r-s)}\rangle \Bigg]. \end{aligned}$$

For $r=1$ this yields

$$\lvert n^{(1)}\rangle = \sum_{m\ne n} \frac{V_{mn}} {E_n^{(0)}-E_m^{(0)}} \lvert m^{(0)}\rangle.$$

The reduced-resolvent form exposes the central assumption: $E_n^{(0)}-H_0$ must be invertible on the relevant complementary subspace. A degenerate state violates that assumption by construction.

The First Three Spectral Results

For an isolated normalized reference state,

$$E_n^{(1)} = \langle n^{(0)}\rvert V \lvert n^{(0)}\rangle.$$

The first-order state correction is

$$\lvert n^{(1)}\rangle = \sum_{m\ne n} \frac{ \langle m^{(0)}\rvert V \lvert n^{(0)}\rangle }{ E_n^{(0)}-E_m^{(0)} } \lvert m^{(0)}\rangle.$$

The second-order energy correction is

$$E_n^{(2)} = \sum_{m\ne n} \frac{ \lvert \langle m^{(0)}\rvert V \lvert n^{(0)}\rangle \rvert^2 }{ E_n^{(0)}-E_m^{(0)} }.$$

These formulas are derived together on Nondegenerate Perturbation Theory. Focused pages in this chapter separate their physical interpretations, symmetry tests, normalization issues, and worked examples.

Energy Corrections and State Corrections

$E_n^{(1)}$ measures the diagonal response of the energy to the perturbation. It can vanish by symmetry even while the state changes at first order.

$\lvert n^{(1)}\rangle$ records induced mixing. It determines leading corrections to observables that are not already diagonal in the reference basis.

For a $\lambda$-independent observable $A$ and a unit-normalized state through first order,

$$\begin{aligned} \langle A\rangle_\lambda ={}& \langle n^{(0)}\rvert A \lvert n^{(0)}\rangle \\ &+ 2\lambda\, \operatorname{Re} \langle n^{(0)}\rvert A \lvert n^{(1)}\rangle + O(\lambda^2). \end{aligned}$$

A state correction is not itself observable. Its phase and component along the reference state depend on convention; observable matrix elements do not when all terms are treated consistently.

$E_n^{(2)}$ measures virtual mixing with other reference states. For a nondegenerate ground state,

$$E_0^{(2)} = \sum_{m\ne0} \frac{\lvert V_{m0}\rvert^2} {E_0^{(0)}-E_m^{(0)}} \le0,$$

because every denominator is negative. For excited states, contributions from lower and higher states have opposite signs.

Symmetry Before Summation

If a unitary symmetry $U$ commutes with $H_0$, reference states can be chosen in symmetry sectors. A perturbation with definite transformation properties connects only allowed sectors.

For parity,

$$\mathcal P V\mathcal P^{-1} = \pi_V V,$$

and parity eigenstates satisfy

$$V_{mn} = \pi_m\pi_V\pi_n V_{mn}.$$

If $\pi_m\pi_V\pi_n=-1$, the matrix element vanishes. Consequences include:

• a vanishing first-order energy for an odd perturbation in a parity eigenstate;
• sparse state corrections;
• a changed leading order for an observable;
• protected degeneracies when symmetry forbids mixing.

Selection rules reduce work and change error counting. They should be applied before a truncated sum is evaluated.

Isolated, Degenerate, and Nearly Degenerate Regimes

Spectral situation	Correct starting point	Main object
One isolated simple eigenvalue	Nondegenerate Rayleigh–Schrödinger theory	Reduced resolvent $R_n'$
Exact finite degeneracy	Degenerate perturbation theory	$PVP$ in the degenerate subspace
Cluster with small internal splittings	Quasi-degenerate or effective-subspace treatment	$PHP$ plus induced corrections
Level crossing with allowed coupling	Local two-state diagonalization	Avoided-crossing Hamiltonian
State near a continuum threshold	Resolvent or scattering treatment	Boundary-valued Green function
Strongly preferred model space	Brillouin–Wigner or Feshbach projection	Energy-dependent $H_{\mathrm{eff}}(E)$

For an exactly degenerate eigenspace, first diagonalize

$$W=PVP.$$

Its eigenvectors define the good zeroth-order combinations and its eigenvalues give the first-order splittings. Corrections from $Q=I-P$ are computed only after this internal problem is solved. The canonical derivation is Degenerate Perturbation Theory.

Exact degeneracy is not required for failure. If internal splittings are comparable to off-diagonal couplings, put the whole cluster in $P$ and diagonalize an effective Hamiltonian there. The boundary of the cluster must itself be controlled:

$$\frac{ \lVert QVP\rVert }{ \Delta_{\mathrm{out}} } \ll1.$$

Here $\Delta_{\mathrm{out}}$ is the separation from omitted states, interpreted in a norm or matrix-element sense appropriate to the problem.

Rayleigh–Schrödinger and Brillouin–Wigner

The standard order-by-order expansion removes the unknown exact energy from denominators by expanding consistently in $\lambda$. Brillouin–Wigner theory retains $E$ inside a projected resolvent.

Feature	Rayleigh–Schrödinger	Brillouin–Wigner
Eigenvalue problem	Linear at each perturbative order	Nonlinear and self-consistent in $E$
Denominators	Reference energies	Exact or partially resummed energy
Order counting	Explicit	Can mix orders unless expanded carefully
Normalization	Intermediate normalization is natural	Projected-state normalization needs special care
Root selection	Follows a labeled reference branch	Multiple self-consistent roots can appear
Effective-theory bridge	Obtained order by order	Direct through an energy-dependent resolvent

For projectors $P$ and $Q=I-P$,

$$H_{\mathrm{eff}}(E) = PHP + PHQ \frac{1}{E-QHQ} QHP.$$

This is exact where the inverse and projection setup are meaningful. Expanding its energy dependence recovers Rayleigh–Schrödinger coefficients; truncating it another way defines a different reorganization that requires its own error check.

See Brillouin–Wigner Perturbation Theory.

Parameter Derivatives

For a differentiable normalized eigenstate of a nondegenerate branch,

$$\frac{dE_n(\lambda)}{d\lambda} = \left\langle n(\lambda) \left\vert \frac{\partial H(\lambda)} {\partial\lambda} \right\vert n(\lambda) \right\rangle.$$

For $H(\lambda)=H_0+\lambda V$,

$$\left. \frac{dE_n}{d\lambda} \right|_{\lambda=0} = \langle n^{(0)}\rvert V \lvert n^{(0)}\rangle = E_n^{(1)}.$$

This relation is exact under its assumptions. Hellmann–Feynman Theorem gives the full proof and treats degenerate branches, parameter-dependent bases, approximate states, generalized forces, and changing domains.

Sums Over Intermediate States

Completeness gives exact identities such as

$$\sum_m \lvert \langle m\rvert A\lvert n\rangle \rvert^2 = \langle n\rvert A^\dagger A \lvert n\rangle,$$

provided the full discrete and continuum resolution is included.

Energy denominators generally prevent completeness from collapsing a perturbative sum directly. Alternatives include:

• commutator sum rules;
• an inhomogeneous Dalgarno–Lewis equation;
• an explicitly justified closure approximation;
• a resolvent matrix element;
• numerical summation with convergence and exact-sum-rule checks.

The canonical guide is Sum Rules and Completeness Tricks.

Worked Benchmark: A Linear Force on the Oscillator

Consider

$$H_0 = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2, \qquad V=F x.$$

Parity gives

$$E_n^{(1)} = F\langle n\rvert x\lvert n\rangle = 0.$$

Using

$$x = \sqrt{\frac{\hbar}{2m\omega}} \bigl(a+a^\dagger\bigr),$$

only $m=n\pm1$ contribute at second order:

$$\begin{aligned} E_n^{(2)} ={}& \frac{ F^2\hbar n/(2m\omega) }{ \hbar\omega } \\ &+ \frac{ F^2\hbar(n+1)/(2m\omega) }{ -\hbar\omega } \\ ={}& -\frac{F^2}{2m\omega^2}. \end{aligned}$$

Completing the square gives the exact check:

$$\begin{aligned} \frac{1}{2}m\omega^2x^2 + \lambda F x ={}& \frac{1}{2}m\omega^2 \left( x+\frac{\lambda F}{m\omega^2} \right)^2 \\ &- \frac{\lambda^2F^2} {2m\omega^2}. \end{aligned}$$

Therefore

$$E_n(\lambda) = \hbar\omega \left(n+\frac{1}{2}\right) - \frac{\lambda^2F^2} {2m\omega^2}.$$

The energy series terminates at second order even though the displaced eigenstate contains contributions at all orders in the original oscillator basis. This separates energy accuracy from state accuracy and shows how symmetry, selection rules, and an exact transformation validate the result.

Anharmonicity as the Opposite Benchmark

For

$$H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2 + g x^4, \qquad g\gt0,$$

the natural control parameter is

$$\epsilon = \frac{g\hbar} {m^2\omega^3}.$$

Low orders are useful for small $\epsilon$, but the full energy expansion is generally asymptotic rather than a terminating polynomial. Variational, WKB, and numerical methods provide independent comparisons in other regimes.

Use Anharmonic Oscillator for the method comparison rather than duplicating it here.

Convergence, Analyticity, and Asymptotic Use

The existence of every formal coefficient does not prove convergence.

In finite-dimensional analytic matrix families, an isolated eigenvalue is analytic until it encounters a complex-parameter singularity such as an exceptional point. The nearest such singularity can set the Taylor-series radius even when no singularity lies on the physical real interval.

For infinite-dimensional Hamiltonians, high-order coefficients often grow factorially. The quartic oscillator is the standard example: low orders can be accurate while the infinite Rayleigh–Schrödinger series diverges. The correct question is then how the optimally truncated series, resummation, and nonperturbative sectors reconstruct the observable.

Diagnostics include:

• ratio and growth of successive coefficients;
• stability under truncation order;
• comparison with exact limits or numerical diagonalization;
• analytic information about nearby level collisions;
• variational bounds;
• symmetry and sum-rule checks;
• sensitivity to basis cutoff and high-energy tails.

Do not call a series convergent because its first three terms decrease.

Numerical Perturbation Benchmarks

Direct diagonalization of

$$H_N(\lambda) = P_N H(\lambda)P_N$$

in a growing basis provides a layered check:

1. convergence of the truncated eigenvalue as $N$ grows;
2. agreement of finite differences or fitted coefficients with perturbation theory;
3. stability under the chosen fitting interval in $\lambda$;
4. contamination by nearby avoided crossings;
5. model error from omitted continuum or boundary physics.

A robust coefficient check uses symmetric parameter values where symmetry permits, multiple fit windows, and reference states tracked by overlap rather than only by sorted eigenvalue index.

See Perturbation Theory Benchmarks.

What the Method Computes Well

Time-independent perturbation theory is especially effective for:

• shifts of isolated bound-state energies;
• weak mixing and corrected expectation values;
• symmetry-resolved splitting of finite multiplets;
• static susceptibilities and polarizabilities;
• low-energy effective Hamiltonians generated by virtual transitions;
• derivatives with respect to weak static parameters;
• systematic comparisons with exact or numerical spectra.

What It Does Not Automatically Control

The method should be reorganized or replaced for:

• exact or near degeneracy not included in the model space;
• states embedded in or approaching a continuum;
• tunneling effects beyond every algebraic order;
• strong coupling at the physical parameter value;
• long-time transition probabilities under a drive;
• spectral rearrangements that change the identity of the tracked state;
• observables dominated by high-energy tails absent from the chosen basis;
• divergent series summed beyond their useful truncation.

Failure means the original expansion point or retained subspace is no longer the right local description, not that the Hamiltonian has become inaccessible.

Reading Path

Goal	Route
Derive the standard isolated-level formulas	Nondegenerate Perturbation Theory
Interpret and evaluate a linear energy shift	First-Order Energy Corrections
Compute mixing and linear observable response	First-Order State Corrections
Analyze quadratic shifts, signs, and polarizability	Second-Order Energy Corrections
Generate and diagnose terms beyond second order	Higher-Order Structure
Resolve an exact multiplet	Degenerate Perturbation Theory
Treat a cluster with small internal splittings	Quasi-Degenerate Perturbation Theory
Keep exact energy in projected denominators	Brillouin–Wigner Perturbation Theory
Study the named energy-independent order expansion	Rayleigh–Schrödinger Perturbation Theory
Convert exact energy derivatives into expectation values	Hellmann–Feynman Theorem
Simplify or validate intermediate-state sums	Sum Rules and Completeness Tricks
Compare linear, quadratic, cubic, and quartic oscillator perturbations	Perturbation Theory for the Harmonic Oscillator
Compare perturbative, variational, WKB, and numerical regimes	Anharmonic Oscillator
Choose between quadratic, degenerate, and crossover DC Stark treatments	Stark Effect as a Perturbation Example
Choose the correct weak-, intermediate-, or strong-field Zeeman basis	Zeeman Effect as a Perturbation Example
Diagnose a suspicious expansion	Common Failure Modes
Check signs and normalization	Notation and Conventions

First-Order Energy Corrections, First-Order State Corrections, Second-Order Energy Corrections, Higher-Order Structure, Quasi-Degenerate Perturbation Theory, Rayleigh–Schrödinger Perturbation Theory, Hellmann–Feynman Theorem, Perturbation Theory for the Harmonic Oscillator, Stark Effect as a Perturbation Example, and Zeeman Effect as a Perturbation Example form the current focused sequence. Every planned leaf in this chapter now has a mature draft. Their canonical role is depth and reusable derivation, not repetition of this guide.

Canonical Boundaries

This chapter owns the static approximation method. It relies on but does not duplicate:

• spectral decomposition, operators, and measurement from Core Formalism;
• exact oscillator, hydrogenic, and two-level solutions from Wave Mechanics and Model Systems;
• symmetry multiplets and selection rules from Symmetry, Angular Momentum, and Spin;
• interaction-picture transition amplitudes from Quantum Dynamics;
• full atomic, molecular, and condensed-matter applications;
• rigorous operator-analytic theorems and many-body linked-cluster theory.

Stark, Zeeman, oscillator, and effective-Hamiltonian examples appear here only to teach method selection and validity.

Common Mistakes

• Calling $V$ small without comparing its matrix elements with relevant gaps.
• Applying isolated-level formulas inside a degenerate or nearly degenerate cluster.
• Diagonalizing $V$ globally instead of $PVP$ in the intended subspace.
• Setting $\lambda=1$ before order counting and nondimensionalization are complete.
• Mixing the sign conventions for $\Delta_{nm}$ and $\Delta_{mn}$.
• Treating intermediate normalization as unit normalization.
• Reading a state correction as a directly measurable quantity.
• Forgetting continuum states in second-order sums or completeness relations.
• Assuming a vanishing first-order energy means the state is unchanged.
• Comparing Rayleigh–Schrödinger and Brillouin–Wigner truncations without matching orders.
• Trusting a few decreasing coefficients as proof of convergence.
• Fitting numerical eigenvalues without tracking the same branch through avoided crossings.

Exercises

Derive the first-order energy

Starting from

$$\bigl(H_0-E_n^{(0)}\bigr) \lvert n^{(1)}\rangle = -V\lvert n^{(0)}\rangle + E_n^{(1)} \lvert n^{(0)}\rangle,$$

derive $E_n^{(1)}$.

Solution

Left-multiply by $\langle n^{(0)}\rvert$. The left side vanishes because $\langle n^{(0)}\rvert H_0=E_n^{(0)}\langle n^{(0)}\rvert$. Hence

$$0 = -\langle n^{(0)}\rvert V \lvert n^{(0)}\rangle + E_n^{(1)},$$

so

$$E_n^{(1)} = \langle n^{(0)}\rvert V \lvert n^{(0)}\rangle.$$

Sign of the ground-state second order

Show that $E_0^{(2)}\le0$ for a nondegenerate ground state. When can equality hold?

Solution

Every denominator $E_0^{(0)}-E_m^{(0)}$ is negative and every numerator $\lvert V_{m0}\rvert^2$ is nonnegative. Therefore every term is nonpositive. Equality holds if $V$ has no matrix element from the ground state to any other reference eigenstate, subject to completeness and domain assumptions.

Complete the oscillator benchmark

For $V=Fx$, compute the first-order state correction to $\lvert n\rangle$.

Solution

Only $\lvert n-1\rangle$ and $\lvert n+1\rangle$ contribute. The result is

$$\begin{aligned} \lvert n^{(1)}\rangle ={}& \frac{ F\sqrt{\hbar/(2m\omega)} \sqrt n }{ \hbar\omega } \lvert n-1\rangle \\ &- \frac{ F\sqrt{\hbar/(2m\omega)} \sqrt{n+1} }{ \hbar\omega } \lvert n+1\rangle. \end{aligned}$$

The signs follow from $E_n-E_{n-1}=+\hbar\omega$ and $E_n-E_{n+1}=-\hbar\omega$.

Decide whether to enlarge the subspace

A target state couples to a nearby state with $\lvert V_{mn}\rvert=3\,\mathrm{meV}$ and separation $5\,\mathrm{meV}$. Other coupled states are at least $0.5\,\mathrm{eV}$ away with comparable matrix elements. What is the natural organization?

Solution

The nearby ratio is $3/5=0.6$, which is not small. Retain and diagonalize the two nearby states together. Coupling to the distant complement is of order

$$\frac{3\,\mathrm{meV}}{0.5\,\mathrm{eV}} = 0.006,$$

so ordinary corrections from those states may be controlled.

Correct an observable

Let $A$ be independent of $\lambda$ and $\langle n^{(0)}\vert n^{(1)}\rangle=0$. Derive the first-order correction to $\langle A\rangle$.

Solution

Expanding the bra and ket gives

$$\begin{aligned} \langle A\rangle_\lambda ={}& \langle n^{(0)}\rvert A \lvert n^{(0)}\rangle \\ &+ \lambda \left[ \langle n^{(1)}\rvert A \lvert n^{(0)}\rangle + \langle n^{(0)}\rvert A \lvert n^{(1)}\rangle \right] + O(\lambda^2) \\ ={}& A_{nn}^{(0)} + 2\lambda\, \operatorname{Re} \langle n^{(0)}\rvert A \lvert n^{(1)}\rangle + O(\lambda^2). \end{aligned}$$

If $A$ depends explicitly on $\lambda$, its derivative supplies another first-order term.

Cross-Links

• Approximation Volume Overview
• Notation and Conventions
• Nondegenerate Perturbation Theory
• First-Order Energy Corrections
• First-Order State Corrections
• Second-Order Energy Corrections
• Higher-Order Structure
• Degenerate Perturbation Theory
• Quasi-Degenerate Perturbation Theory
• Brillouin–Wigner Perturbation Theory
• Rayleigh–Schrödinger Perturbation Theory
• Hellmann–Feynman Theorem
• Sum Rules and Completeness Tricks
• Perturbation Theory for the Harmonic Oscillator
• Anharmonic Oscillator
• Stark Effect as a Perturbation Example
• Zeeman Effect as a Perturbation Example
• Perturbation Theory Benchmarks
• Symmetry Constraints on Hamiltonians
• Projectors

References

• B. Zwiebach, Quantum Physics III, MIT OpenCourseWare 8.06, Chapter 1, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vols. 1–2, Wiley, 1977.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• T. Kato, Perturbation Theory for Linear Operators, corrected printing of the 2nd ed., Springer, 1995.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.
• C. M. Bender and T. T. Wu, “Anharmonic oscillator,” Physical Review 184, 1231–1260, 1969.