Physical Chemistry · Part 2 of 9 · Free

Operators, Eigenvalues & the Schrödinger Equation — formula sheet

Every key expression and definition from Quantum Chemistry, Part 2, on one page. Free to read, no sign-in.

Key expressions

the sum and product of two operators
(Â + B̂)f = Âf + B̂f , (ÂB̂)f = Â(B̂f) — rightmost operator acts first

why operator order matters
x̂(d/dx)f = x f′ but (d/dx)(x̂f) = (d/dx)(xf) = f + x f′

the definition of a linear operator
Â(c_1f_1 + c_2f_2) = c_1Âf_1 + c_2Âf_2 for all f_1, f_2 and all constants c_1, c_2

the exponential of an operator
e^Â ≡ 1̂ + Â + Â²/2! + Â³/3! + … = ∑_n=0^∞ Â^n/n!

Baker–Campbell–Hausdorff, to first order
e^Âe^B̂ = e^Â+B̂+½[Â,B̂]+… , equal to e^Â+B̂ only when [Â,B̂] = 0

the matrix element of an operator in a basis
A_mn = ∫ ψ_m* Â ψ_n dτ ≡ ⟨m|Â|n⟩

a projection operator and its defining idempotency
P̂_nφ = ψ_n ∫ψ_n*φ dτ = c_nψ_n , P̂_n² = P̂_n

the integration by parts that proves p̂ Hermitian
∫_−∞^∞ f* (−iℏ dg/dx) dx = −iℏ [f*g]_−∞^∞ + iℏ ∫ (df*/dx) g dx = ∫ (−iℏ df/dx)* g dx

step 1 of the reality proof
∫ ψ* Âψ dτ = ∫ ψ* a ψ dτ = a ∫ ψ*ψ dτ = a (for normalised ψ)

step 2 — the conjugated equation
∫ (Âψ)* ψ dτ = a*

step 1 — Hermiticity applied to two different eigenfunctions
∫ψ_m*(Âψ_n)dτ = ∫(Âψ_m)*ψ_ndτ

step 2 — the key line of the orthogonality proof
(a_n − a_m) ∫ψ_m*ψ_n dτ = 0

the TDSE
iℏ ∂Ψ(r,t)/∂t = ĤΨ(r,t) — the time-dependent Schrödinger equation (TDSE)

separating the variables
iℏ ψ(r) df/dt = f(t) Ĥψ(r) ⇒ iℏ (1/f) df/dt = (1/ψ) Ĥψ

the separated equations: a trivial one in time and the TISE in space
(i) iℏ df/dt = E f ⇒ f(t) = e^−iEt/ℏ ; (ii) Ĥψ = Eψ

the general separated solution
Ψ(r,t) = ψ(r) e^−iEt/ℏ — a stationary state

the general solution of the time-dependent Schrödinger equation
Ψ(r,t) = ∑_n c_n ψ_n(r) e^−iE_nt/ℏ , c_n = ∫ψ_n*Ψ(r,0) dτ

the time-evolution operator and the conservation of probability
Û(t) = e^−iĤt/ℏ , Û†Û = 1̂ ⇒ ∫|Ψ(t)|²dτ is constant in time

the expectation value
⟨A⟩ = ∫ ψ* Â ψ dτ (normalised ψ) , ⟨A⟩ = ∫ψ*Âψdτ / ∫ψ*ψdτ (general)

the uncertainty (standard deviation) of an observable
σ_A² = ⟨(A − ⟨A⟩)²⟩ = ⟨A²⟩ − ⟨A⟩² , σ_A = √(⟨A²⟩ − ⟨A⟩²)

the expectation value as a probability-weighted mean of the eigenvalues
⟨A⟩ = ∫(∑_mc_mψ_m)* Â(∑_nc_nψ_n) dτ = ∑_m∑_n c_m*c_n a_n ∫ψ_m*ψ_ndτ = ∑_n |c_n|² a_n

expansion in a complete orthonormal set, and the recipe for the coefficients
φ = ∑_n c_nψ_n with c_n = ∫ψ_n* φ dτ

the expansion coefficients as probability amplitudes
P(result = a_n) = |c_n|² = |∫ψ_n*φdτ|² , ∑_n|c_n|² = 1

convergence in the mean — the precise sense in which an expansion is complete
lim_N→∞ ∫ |φ − ∑_n=1^N c_nψ_n|² dτ = 0

the closure (completeness) relation
∑_n ψ_n*(x′) ψ_n(x) = δ(x − x′) ≡ ∑_n|n⟩⟨n| = 1̂

the momentum-space wavefunction as a Fourier transform
φ(x) = (2πℏ)^−1/2 ∫ φ̃(p) e^ipx/ℏ dp , φ̃(p) = (2πℏ)^−1/2 ∫ φ(x) e^−ipx/ℏ dx

step 1 — differentiate under the integral
d⟨A⟩/dt = ∫ (∂Ψ*/∂t) ÂΨ dτ + ∫ Ψ* (∂Â/∂t) Ψ dτ + ∫ Ψ* Â (∂Ψ/∂t) dτ

step 2 — move Ĥ across by Hermiticity, and collect
−(1/iℏ)∫(ĤΨ)*ÂΨdτ + (1/iℏ)∫Ψ*ÂĤΨdτ = (1/iℏ)∫Ψ*(ÂĤ − ĤÂ)Ψdτ

the Ehrenfest theorem (general form)
d⟨A⟩/dt = (i/ℏ) ⟨[Ĥ, Â]⟩ + ⟨∂Â/∂t⟩

Ehrenfest 1 — the quantum analogue of p = mv
m d⟨x⟩/dt = ⟨p⟩

Ehrenfest 2 — the quantum analogue of Newton's second law
d⟨p⟩/dt = −⟨dV/dx⟩ = ⟨F⟩

the commutator
[Â, B̂] = ÂB̂ − B̂Â , [Â, B̂] = 0 ⇔ ÂB̂ = B̂Â

a worked commutator, done with a test function
[x̂², d/dx] f = x²f′ − (x²f)′ = x²f′ − (2xf + x²f′) = −2xf ⇒ [x̂², d/dx] = −2x̂

commutators of powers, from the product rule
[x̂, p̂²] = [x̂, p̂]p̂ + p̂[x̂, p̂] = iℏp̂ + p̂(iℏ) = 2iℏp̂

the general canonical commutators
[x̂, g(p̂)] = iℏ dg/dp , [f(x̂), p̂] = iℏ df/dx

the derivation of the canonical commutator
[x̂, p̂_x]f = x(−iℏ ∂f/∂x) − (−iℏ ∂/∂x)(xf) = −iℏ x f′ + iℏ(f + x f′) = iℏ f

the canonical commutation relations
[x̂, p̂_x] = iℏ , [ŷ, p̂_y] = iℏ , [ẑ, p̂_z] = iℏ ; all other pairs commute

the angular-momentum operators
L̂_x = ŷp̂_z − ẑp̂_y , L̂_y = ẑp̂_x − x̂p̂_z , L̂_z = x̂p̂_y − ŷp̂_x

the angular-momentum commutation relations (cyclic in x, y, z)
[L̂_x, L̂_y] = iℏL̂_z , [L̂_y, L̂_z] = iℏL̂_x , [L̂_z, L̂_x] = iℏL̂_y

L̂² commutes with every component
[L̂², L̂_x] = [L̂², L̂_y] = [L̂², L̂_z] = 0

the ladder-operator commutators
[L̂_z, L̂_±] = ±ℏL̂_± , [L̂², L̂_±] = 0

if a complete set of simultaneous eigenfunctions exists, the operators commute
ÂB̂ψ = Â(bψ) = baψ , B̂Âψ = B̂(aψ) = abψ ⇒ [Â,B̂]ψ = (ba − ab)ψ = 0

the Schwarz (Cauchy–Schwarz) inequality
|⟨f|g⟩|² ≤ ⟨f|f⟩⟨g|g⟩ — the Schwarz inequality

step 1 — the variances as squared lengths
σ_A² = ⟨ψ|ΔÂ²|ψ⟩ = ⟨ΔÂψ|ΔÂψ⟩ = ⟨f|f⟩ with f = ΔÂψ, g = ΔB̂ψ

step 2 — the inequality applied
σ_A²σ_B² = ⟨f|f⟩⟨g|g⟩ ≥ |⟨f|g⟩|²

step 3 — keep only the imaginary part
σ_A²σ_B² ≥ [(⟨f|g⟩ − ⟨g|f⟩)/2i]²

step 4 — the commutator appears
⟨f|g⟩ − ⟨g|f⟩ = ⟨[ΔÂ, ΔB̂]⟩ = ⟨[Â, B̂]⟩

THE GENERALISED UNCERTAINTY RELATION
σ_A σ_B ≥ ½ |⟨[Â, B̂]⟩|

the position–momentum uncertainty relation
σ_x σ_px ≥ ℏ/2

the minimum-uncertainty condition as a differential equation
(−iℏ d/dx − ⟨p⟩)ψ = λ(x − ⟨x⟩)ψ

the energy–time uncertainty relation
ΔE Δt ≥ ℏ/2

the Mandelstam–Tamm form of the energy–time relation
σ_E σ_Q ≥ (ℏ/2)|d⟨Q⟩/dt| ⇒ σ_E τ_Q ≥ ℏ/2 with τ_Q = σ_Q/|d⟨Q⟩/dt|

Definitions worth memorising

Operator: a rule Â which, applied to a function f, produces another function g, written Âf = g. The caret (the ‘hat’) distinguishes the operator Â from the number or observable A. An operator is meaningless until it is given something to act on: the symbol Â standing alone is an instruction waiting for an argument, exactly like the symbol √ standing alone.

Dirac notation, in one paragraph. Write the state ψ_n as a ket |n⟩ and its complex conjugate as a bra ⟨n|. The overlap integral ∫ψ_m*ψ_ndτ becomes the bracket ⟨m|n⟩, the matrix element ∫ψ_m*Âψ_ndτ becomes ⟨m|Â|n⟩, and the expectation value becomes ⟨ψ|Â|ψ⟩. Nothing new is being said; the notation simply removes the integral signs and the dummy variables, and it makes the coordinate system invisible, which is why it is universal in the research literature. This book writes integrals in the beginner layers and brackets in the advanced ones, and you should be able to move between them without thinking.

Hermitian operator: Â is Hermitian if, for every pair of well-behaved functions f and g vanishing suitably at the boundaries, ∫ f* (Âg) dτ = ∫ (Âf)* g dτ. Equivalently, in Dirac notation, ⟨f|Âg⟩ = ⟨Âf|g⟩, and in matrix form A_mn = A_nm*, i.e. the matrix equals its own conjugate transpose. Older texts call such operators self-adjoint.

Degeneracy and Gram–Schmidt. If g independent eigenfunctions share one eigenvalue a, that eigenvalue is g-fold degenerate, and every linear combination of those g functions is also an eigenfunction with the same eigenvalue (check it: Â(c₁ψ₁ + c₂ψ₂) = a(c₁ψ₁ + c₂ψ₂)). One may therefore choose an orthogonal set within the degenerate subspace, and the Gram–Schmidt procedure constructs it: φ₁ = ψ₁; φ₂ = ψ₂ − φ₁⟨φ₁|ψ₂⟩/⟨φ₁|φ₁⟩; and so on. So the eigenfunctions of a Hermitian operator can always be taken orthonormal — automatically when the eigenvalues differ, by construction when they do not.

Adjoint (Hermitian conjugate): Â† is defined by ⟨f|Âg⟩ = ⟨Â†f|g⟩ for all f, g. An operator is Hermitian when Â† = Â, anti-Hermitian when Â† = −Â, and unitary when Â† = Â^−1.

Eigenvalue equation: Âψ = aψ, where a is a number. A function ψ satisfying this is an eigenfunction of Â and the number a is the corresponding eigenvalue. The German eigen means ‘own’ or ‘characteristic’: ψ is a function that the operator leaves alone apart from a scale factor.

The correspondence rules. (1) Write the classical expression for the observable in terms of Cartesian coordinates and momenta. (2) Replace each coordinate q by multiplication by q, and each momentum component p_q by −iℏ∂/∂q. (3) The result is the quantum operator. If the classical expression contains a product of a coordinate and its own conjugate momentum, replace it by the symmetrised Hermitian combination.

Stationary state: a state whose wavefunction is a single energy eigenfunction times the phase factor e^−iEt/ℏ. Because |e^−iEt/ℏ|^2 = 1, the probability density |Ψ|^2 = |ψ|^2 is independent of time, and so is the expectation value of every operator that does not itself contain t. Nothing observable changes — that, and only that, is what ‘stationary’ means.

Expectation value: for a system in the normalised state ψ, the mean value of the observable A over many measurements on identically prepared systems is ⟨A⟩ = ∫ ψ* Â ψ dτ. If ψ is not normalised, divide by ∫ψ*ψdτ.

Superposition principle: if ψ_1 and ψ_2 are possible states of a system, then so is ψ = c_1ψ_1 + c_2ψ_2 for any complex constants c_1, c_2. The system in such a state is not ‘in state 1 or state 2 and we do not know which’; it is in a new state with properties of its own, and the interference terms in |ψ|^2 are what distinguish the two situations experimentally.

Collapse postulate: if a measurement of A on a system in the state φ = ∑c_nψ_n returns the eigenvalue a_n, then immediately after the measurement the state of the system is ψ_n. The superposition is destroyed; the other amplitudes are gone, not merely small. A second measurement made immediately returns a_n again with probability 1.

Constants of the motion. If ∂Â/∂t = 0 and [Ĥ, Â] = 0, then d⟨A⟩/dt = 0: the expectation value of A is conserved, and so is the whole probability distribution of A. Energy is conserved because [Ĥ, Ĥ] = 0 identically; angular momentum is conserved in a central field because [Ĥ, L̂^2] = [Ĥ, L̂_z] = 0; parity is conserved in a symmetric potential. Every conservation law in quantum mechanics is a vanishing commutator, which via Noether's theorem is a symmetry.

Commutator: [Â, B̂] ≡ ÂB̂ − B̂Â. It is itself an operator. If [Â, B̂] = 0 the two operators commute, meaning the order of application does not matter; if not, they do not commute, and the commutator measures by how much.

The rule of thumb. [Â, B̂] = 0 ⇒ A and B are compatible: they can both be sharp at once, and there is a common set of eigenfunctions. [Â, B̂] ≠ 0 ⇒ A and B are incompatible: there is a trade-off between how sharply each can be known, quantified by the uncertainty relation of B.9.

The modern definition. An angular momentum is defined as any triple of Hermitian operators satisfying [Ĵ_x, Ĵ_y] = iℏĴ_z and cyclic permutations. Orbital angular momentum, spin, and their vector sums all satisfy it; spin has no differential representation and no classical analogue, so this algebraic definition is the only one available for it. Everything in Part 4 — term symbols, Clebsch–Gordan coupling, the Landé g factor — is a consequence of this algebra and nothing else.

Theorem. Two Hermitian operators Â and B̂ possess a complete set of simultaneous eigenfunctions if and only if they commute. Observables whose operators commute are called compatible; those whose operators do not are incompatible.

Where these come from

This sheet is distilled from Quantum Chemistry, Part 2 — 9 sections that derive every one of these results and show you how to use them.

Read Part 2 All formula sheets