Physical Chemistry · Part 4 of 9 · Free

Exactly Solvable Systems II — Oscillator & Rotor — formula sheet

Every key expression and definition from Quantum Chemistry, Part 4, on one page. Free to read, no sign-in.

Key expressions

Hooke's law and the harmonic potential
F = −kx ⇒ V(x) = −∫F dx = ½kx^2

Taylor expansion of a bond potential
V(r) = V(r_e) + (dV/dr)_re(r − r_e) + ½(d^2V/dr^2)_re(r − r_e)^2 + …

the force constant of a chemical bond
k = (d^2V/dr^2)_r = re — the curvature of the potential at the bottom of the well

the reduced mass of a diatomic
μ = m_1m_2/(m_1 + m_2) or equivalently 1/μ = 1/m_1 + 1/m_2

exact separation of a two-body kinetic energy
−(ℏ^2/2m_1)∂^2/∂x_1^2 − (ℏ^2/2m_2)∂^2/∂x_2^2 = −(ℏ^2/2M)∂^2/∂X^2 − (ℏ^2/2μ)∂^2/∂x^2

the harmonic-oscillator Schrödinger equation
−(ℏ^2/2μ) d^2ψ/dx^2 + ½kx^2ψ = Eψ

the oscillator equation in dimensionless form
d^2ψ/dy^2 + (ε − y^2)ψ = 0

extracting the asymptotic behaviour
ψ(y) = f(y) e^−y2/2

Hermite's equation
f″ − 2y f′ + (ε − 1) f = 0

the power-series ansatz
f(y) = ∑_j=0^∞ a_j y^j = a_0 + a_1y + a_2y^2 + …

the Hermite recursion relation
a_j+2 = [(2j + 1 − ε)/((j+1)(j+2))] a_j

the asymptotic ratio of successive coefficients
a_j+2/a_j → 2j/j^2 = 2/j as j → ∞

why an infinite series is not acceptable
ψ ~ e^y2 × e^−y2/2 = e^+y2/2 → ∞

the termination condition
2n + 1 − ε = 0 ⇒ ε = 2n + 1 ⇒ 2E/ℏω = 2n + 1

the harmonic-oscillator energy levels
E_n = (n + ½) ℏω = (n + ½)hν , n = 0, 1, 2, 3, …

the Rodrigues formula for Hermite polynomials
H_n(y) = (−1)^n e^y2 (d^n/dy^n) e^−y2

the Hermite recurrence relation
H_n+1(y) = 2y H_n(y) − 2n H_n−1(y) , H_0 = 1, H_1 = 2y

the Hermite generating function
e^2yt − t2 = ∑_n H_n(y) t^n/n!

the normalised harmonic-oscillator wavefunctions
ψ_n(x) = (1/√(2^nn!)) (α/π)^1/4 H_n(α^1/2x) e^−αx2/2 , α = μω/ℏ = √(μk)/ℏ

the two lowest oscillator states, written out
ψ_0(x) = (α/π)^1/4 e^−αx2/2 , ψ_1(x) = (4α^3/π)^1/4 x e^−αx2/2

the zero-point energy
E_0 = ½ℏω = ½hν ≠ 0

penetration of the classically forbidden region
P(|y| > y_tp) = 2∫_ytp^∞|ψ_n|^2 dy , for n = 0 this is erfc(1) = 0.15730

the annihilation and creation operators
â = (2ℏμω)^−1/2(μωx̂ + ip̂) â^† = (2ℏμω)^−1/2(μωx̂ − ip̂)

the fundamental commutator of the ladder operators
[â, â^†] = 1

the Hamiltonian in terms of the number operator
Ĥ = ℏω(â^†â + ½) = ℏω(N̂ + ½) , N̂ ≡ â^†â

the ladder commutators
[Ĥ, â^†] = +ℏω â^† [Ĥ, â] = −ℏω â

the raising operator makes a new eigenstate one quantum higher
Ĥ(â^†|ψ⟩) = (â^†Ĥ + ℏωâ^†)|ψ⟩ = (E + ℏω)(â^†|ψ⟩)

positivity of a norm bounds the energy from below
⟨ψ|â^†â|ψ⟩ = ‖â|ψ⟩‖^2 ≥ 0 ⇒ ⟨N̂⟩ ≥ 0 ⇒ E ≥ ½ℏω

the ground state and the zero-point energy, from the algebra alone
â|0⟩ = 0 ⇒ N̂|0⟩ = 0 ⇒ E_0 = ½ℏω

the normalised ladder relations
â^†|n⟩ = √(n+1) |n+1⟩ , â|n⟩ = √n |n−1⟩ , N̂|n⟩ = n|n⟩

position and momentum as ladder operators
x̂ = (ℏ/2μω)^1/2(â + â^†) , p̂ = i(ℏμω/2)^1/2(â^† − â)

the position matrix elements of the oscillator
⟨m|x̂|n⟩ = (ℏ/2μω)^1/2[√n δ_m,n−1 + √(n+1) δ_m,n+1]

expansion of the dipole moment in the displacement
μ(x) = μ_0 + (dμ/dx)_0 x + ½(d^2μ/dx^2)_0 x^2 + …

the Morse potential
V(r) = D_e[1 − e^−a(r−r_e)]^2 , a = ω(μ/2D_e)^1/2

Morse vibrational term values
G(v) = ω̃_e(v + ½) − ω̃_ex_e(v + ½)^2

classical rotation: I is to rotation what mass is to translation
E = ½Iω^2 = J_z^2/2I , I = mr^2 , J_z = Iω

the particle-on-a-ring equation
−(ℏ^2/2I) d^2Φ/dφ^2 = EΦ

the general solution, before any boundary condition
Φ(φ) = A e^imφ , E = m^2ℏ^2/2I , m = √(2IE)/ℏ

the cyclic boundary condition quantises m
Φ(φ + 2π) = Φ(φ) ⇒ e^2πim = 1 ⇒ m = 0, ±1, ±2, ±3, …

the two-dimensional rotor: energies and normalised eigenfunctions
E_m = m^2ℏ^2/2I , m = 0, ±1, ±2, … , Φ_m(φ) = (2π)^−1/2e^imφ

the z component of angular momentum in polar coordinates
L̂_z = −iℏ ∂/∂φ

L_z is sharp for a particle on a ring
L̂_z e^imφ = −iℏ(im)e^imφ = mℏ e^imφ

the particle-on-a-sphere equation
−(ℏ^2/2I) Λ^2Y(θ,φ) = E Y , Λ^2 = (1/sinθ) ∂/∂θ (sinθ ∂/∂θ) + (1/sin^2θ) ∂^2/∂φ^2

the φ equation and its cyclic boundary condition
d^2Φ/dφ^2 = −m^2Φ ⇒ Φ_m = (2π)^−1/2e^imφ , m = 0, ±1, ±2, …

the particle on a sphere: energies and the range of m
E_l = l(l+1)ℏ^2/2I , l = 0, 1, 2, … , m = −l, −l+1, …, 0, …, l−1, l

the degeneracy of a rotational level
degeneracy of the level l = 2l + 1

the spherical harmonics
Y_l^m(θ,φ) = Θ_l,m(θ) Φ_m(φ) = N_l,m P_l^|m|(cosθ) e^imφ

the associated Legendre equation
(1 − x^2)P″ − 2xP′ + [l(l+1) − m^2/(1−x^2)]P = 0

Legendre polynomials: Rodrigues formula and recursion
P_l(x) = (1/2^ll!) (d^l/dx^l)(x^2 − 1)^l , (l+1)P_l+1 = (2l+1)xP_l − lP_l−1

the associated Legendre functions (Condon–Shortley phase)
P_l^m(x) = (−1)^m(1 − x^2)^m/2 (d^m/dx^m) P_l(x)

orthonormality of the spherical harmonics
∫_0^2π∫_0^π Y_l′^m′* Y_l^m sinθ dθ dφ = δ_ll′δ_mm′

the real p orbitals as combinations of spherical harmonics
p_x ∝ (Y_1^−1 − Y_1^+1)/√2 , p_y ∝ i(Y_1^−1 + Y_1^+1)/√2 , p_z = Y_1^0

the moment of inertia of a diatomic
I = μr^2 , μ = m_1m_2/(m_1+m_2)

rotational energy levels of a rigid diatomic
E_J = J(J+1)ℏ^2/2I , J = 0, 1, 2, … , degeneracy 2J+1

the rotational constant B in wavenumbers
F(J) = B J(J+1) , B = ℏ^2/(2hcI) = h/(8π^2cI) , units cm^−1

the position of a rotational line
ν̃(J → J+1) = F(J+1) − F(J) = B[(J+1)(J+2) − J(J+1)] = 2B(J+1)

rotational term values with centrifugal distortion
F(J) = B J(J+1) − D J^2(J+1)^2

line positions with centrifugal distortion
ν̃(J → J+1) = 2B(J+1) − 4D(J+1)^3

the Kratzer relation between distortion and vibration
D ≈ 4B^3/ω̃_e^2

the classical components of angular momentum
L_x = y p_z − z p_y , L_y = z p_x − x p_z , L_z = x p_y − y p_x

the z component, in Cartesian and in polar coordinates
L̂_z = −iℏ(x ∂/∂y − y ∂/∂x) = −iℏ ∂/∂φ

the angular-momentum commutation relations
[L̂_x, L̂_y] = iℏL̂_z , [L̂_y, L̂_z] = iℏL̂_x , [L̂_z, L̂_x] = iℏL̂_y

the total angular momentum operator
L̂^2 = L̂_x^2 + L̂_y^2 + L̂_z^2

L̂^2 commutes with every component
[L̂^2, L̂_x] = [L̂^2, L̂_y] = [L̂^2, L̂_z] = 0

expanding the commutator term by term
[L̂_x, L̂_y] = [ŷp̂_z, ẑp̂_x] − [ŷp̂_z, x̂p̂_z] − [ẑp̂_y, ẑp̂_x] + [ẑp̂_y, x̂p̂_z]

the first surviving term
[ŷp̂_z, ẑp̂_x] = ŷp̂_x[p̂_z, ẑ] = −iℏ ŷp̂_x

and the answer is the third component, as claimed
[L̂_x, L̂_y] = iℏ(x̂p̂_y − ŷp̂_x) = iℏ L̂_z

L̂^2 is the legendrian of D.5, times −ℏ^2
L̂^2 = −ℏ^2Λ^2 = −ℏ^2[(1/sinθ)∂/∂θ(sinθ ∂/∂θ) + (1/sin^2θ)∂^2/∂φ^2]

the spherical harmonics are the simultaneous eigenfunctions
L̂^2Y_l^m = l(l+1)ℏ^2Y_l^m , L̂_zY_l^m = mℏ Y_l^m

the uncertainty relation between two components
σ_Lxσ_Ly ≥ ½|⟨[L̂_x, L̂_y]⟩| = ½ℏ|⟨L_z⟩| = ½ℏ^2|m|

the eigenvalues, so far completely unknown
Ĵ^2|λ,μ⟩ = λℏ^2|λ,μ⟩ , Ĵ_z|λ,μ⟩ = μℏ|λ,μ⟩

the raising and lowering operators
Ĵ_± = Ĵ_x ± i Ĵ_y

the ladder commutators
[Ĵ_z, Ĵ_±] = ±ℏ Ĵ_± , [Ĵ^2, Ĵ_±] = 0

Ĵ_+ raises μ by exactly one
Ĵ_z(Ĵ_+|λ,μ⟩) = (Ĵ_+Ĵ_z + ℏĴ_+)|λ,μ⟩ = (μ+1)ℏ (Ĵ_+|λ,μ⟩)

the positivity condition at the top of the ladder
‖Ĵ_+|λ,μ⟩‖^2 = ⟨λ,μ|Ĵ_−Ĵ_+|λ,μ⟩ = ℏ^2[λ − μ(μ+1)] ≥ 0

the two ends of the ladder
λ = j(j+1) (from the top) and λ = j′(j′−1) (from the bottom) ⇒ j′ = −j

the complete angular-momentum spectrum, from the commutators alone
Ĵ^2|j,m⟩ = j(j+1)ℏ^2|j,m⟩ , Ĵ_z|j,m⟩ = mℏ|j,m⟩j = 0, ½, 1, 3/2, 2, … m = −j, −j+1, …, j−1, j (2j+1 values)

the normalised ladder relations
Ĵ_±|j,m⟩ = ℏ√[j(j+1) − m(m±1)] |j,m±1⟩ = ℏ√[(j∓m)(j±m+1)] |j,m±1⟩

the magnitude of a quantum angular momentum
|J| = √(j(j+1)) ℏ

the cone angle in the vector model
cosα = m / √(j(j+1))

the spin of an electron, numerically
|S| = √(s(s+1))ℏ = √(¾)ℏ = 0.8660ℏ , S_z = ±½ℏ , angle to z = 54.74°

the electron spin magnetic moment and the Bohr magneton
μ_z = −g_eμ_Bm_s , μ_B = eℏ/2m_e = 9.27401 × 10^−24 J T^−1 , g_e = 2.002319

the Clebsch–Gordan series for adding two angular momenta
J = j_1+j_2, j_1+j_2−1, j_1+j_2−2, …, |j_1−j_2| , M = m_1 + m_2

Definitions worth memorising

Force constant, k: the second derivative of the potential energy with respect to displacement, evaluated at equilibrium. Units N m^−1. It measures the stiffness of a bond, not its strength: a bond's strength is its dissociation energy D_e, the depth of the whole well, while k is only the curvature at the very bottom. The two usually correlate but they are different quantities and are routinely confused in examinations.

Where the quantisation came from: not from the Schrödinger equation, which has perfectly good solutions for every ε, but from the boundary condition at infinity — the requirement that ψ remain finite and normalisable. This is the same mechanism as in the particle in a box, where quantisation came from the requirement that ψ vanish at the walls. In every exactly soluble problem, quantisation is a boundary condition, never an axiom.

The virial theorem for the oscillator: ⟨T⟩ = ⟨V⟩ = ½E_n. Kinetic and potential energy are exactly equal on average, in every state. The general statement is 2⟨T⟩ = ⟨x · dV/dx⟩, which for V ∝ x^s gives 2⟨T⟩ = s⟨V⟩; the oscillator has s = 2 so ⟨T⟩ = ⟨V⟩, while the Coulomb potential has s = −1 so 2⟨T⟩ = −⟨V⟩, which is the version used constantly in Part 5.

Vibrational selection rules (harmonic, electric dipole): (i) Δv = ±1, from ⟨f|x̂|i⟩ ≠ 0 only for adjacent levels; and (ii) (dμ/dx)_0 ≠ 0 — the dipole moment must change during the vibration. This is the gross selection rule for infrared activity. Note carefully that the molecule need not have a permanent dipole: CO_2 has none, yet its antisymmetric stretch and both bends are IR active because the dipole changes. Conversely N_2 and O_2 are IR inactive, which is why they are not greenhouse gases.

The two conditions, and where each comes from: (i) m must be an integer, from the cyclic boundary condition in φ — the same argument as D.4. (ii) l must be a non-negative integer and |m| ≤ l, from the requirement that the Legendre solutions remain finite at the poles — the same kind of argument as the termination of the Hermite series in D.2. Both are boundary conditions; neither is an extra postulate.

Rotational constant, B: B = h/8π^2cI in cm^−1, or B = h/8π^2I in Hz if the c is omitted. Both conventions are in use and both appear in question papers, so check the units of the answer. B is inversely proportional to the moment of inertia: a heavy, long molecule has a small B and closely spaced levels; a light, short one has a large B. For ^12C^16O, B = 1.9314 cm^−1; for H–^35Cl, B = 10.593 cm^−1; for H–^127I, B = 6.51 cm^−1 — all computed here from tabulated bond lengths.

Rotational selection rules: (i) ΔJ = ±1, from the transition-moment integral over the spherical harmonics; and (ii) the molecule must have a permanent electric dipole moment — the gross selection rule. This is why HCl, CO and H_2O give microwave spectra while H_2, N_2, O_2, CO_2 and CH_4 give none. Note the contrast with the vibrational case, where a changing dipole is required but a permanent one is not; here a permanent one is required, because a rotating molecule presents an oscillating dipole to the radiation field only if it has a dipole to rotate.

Two observables can have simultaneous eigenfunctions — can both be sharp in the same state — if and only if their operators commute. Applied here: L̂^2 and any one component can be sharp together, so a state can have a definite magnitude of angular momentum and a definite projection on one chosen axis. But no two components can be sharp together, so once L_z is known exactly, L_x and L_y are completely undetermined.

The half-integer surprise. The algebra permits j = ½, 3/2, 5/2, … as well as the integers. The differential equation of D.5 cannot: there, single-valuedness of e^imφ round a full turn forced m to be an integer. Both statements are correct, and they are not in conflict — the resolution is that half-integer angular momenta exist but are not associated with motion through space, so no wavefunction of φ is involved and the single-valuedness argument does not apply. That is spin, and D.9 is about it.

Spin: an intrinsic angular momentum carried by a particle, with no classical analogue and no associated spatial coordinate. It obeys the same algebra as orbital angular momentum — [Ŝ_x, Ŝ_y] = iℏŜ_z, Ŝ^2|s,m_s⟩ = s(s+1)ℏ^2|s,m_s⟩ — but for a given particle s is fixed, a permanent property like mass or charge. For the electron, proton and neutron s = ½; for the photon s = 1; for a spin-0 nucleus such as ^12C, s = 0.

Where these come from

This sheet is distilled from Quantum Chemistry, Part 4 — 9 sections that derive every one of these results and show you how to use them.

Read Part 4 All formula sheets