Physical Chemistry · Part 3 of 9 · Free

Exactly Solvable Systems I — Boxes, Steps & Barriers — formula sheet

Every key expression and definition from Quantum Chemistry, Part 3, on one page. Free to read, no sign-in.

Key expressions

the infinite square well (particle in a 1-D box)
V(x) = 0 for 0 < x < L ; V(x) = ∞ for x ≤ 0 and x ≥ L

the Schrödinger equation inside the box
−(ℏ^2/2m) d^2ψ/dx^2 = Eψ (0 < x < L)

d^2ψ/dx^2 = −k^2ψ , k = √(2mE)/ℏ

ψ(x) = A sin(kx) + B cos(kx)

ψ(0) = A sin 0 + B cos 0 = B = 0 ⇒ B = 0

A sin(kL) = 0

the quantisation condition for the 1-D box
sin(kL) = 0 ⇒ kL = nπ , n = 1, 2, 3, …

√(2mE)/ℏ = nπ/L ⇒ 2mE = n^2π^2ℏ^2/L^2 ⇒ E = n^2π^2ℏ^2/2mL^2

the energy levels of a particle in a 1-D box
E_n = n^2h^2 / 8mL^2 , n = 1, 2, 3, …

the box constant for an electron, a number worth memorising
h^2/8m_eL^2 = 37.60 eV / (L/Å)^2

ψ′(a+ε) − ψ′(a−ε) = (2m/ℏ^2) ∫_a−ε^a+ε [V(x) − E] ψ(x) dx

the normalisation condition
∫_−∞^∞ |ψ(x)|^2 dx = 1

A^2 ∫_0^L sin^2(nπx/L) dx = 1

∫_0^L sin^2(nπx/L) dx = ½∫_0^Ldx − ½∫_0^Lcos(2nπx/L) dx = L/2 − 0 = L/2

the normalised eigenfunctions of the 1-D box
ψ_n(x) = √(2/L) · sin(nπx/L) , 0 ≤ x ≤ L ; ψ_n(x) = 0 otherwise

orthonormality of the box eigenfunctions
∫_0^L ψ_m(x) ψ_n(x) dx = δ_mn = 1 if m = n, 0 if m ≠ n

(2/L)∫_0^Lsin(mπx/L)sin(nπx/L)dx = (1/L)∫_0^L[cos((m−n)πx/L) − cos((m+n)πx/L)]dx

the order-of-magnitude zero-point energy from the uncertainty principle
E ≈ ⟨p^2⟩/2m ≥ ℏ^2/8mL^2 = h^2/32π^2mL^2

⟨ψ_m|Âψ_n⟩ = a_n⟨ψ_m|ψ_n⟩ ; ⟨Âψ_m|ψ_n⟩ = a_m*⟨ψ_m|ψ_n⟩ = a_m⟨ψ_m|ψ_n⟩

Hermiticity ⇒ the two are equal ⇒ (a_n − a_m)⟨ψ_m|ψ_n⟩ = 0 ⇒ ⟨ψ_m|ψ_n⟩ = 0 whenever a_m ≠ a_n

expansion of an arbitrary box state; Parseval's theorem is the normalisation
Ψ(x) = ∑_n=1^∞ c_nψ_n(x) , c_n = ∫_0^Lψ_n(x)Ψ(x)dx , ∑_n|c_n|^2 = 1

time evolution of an arbitrary box state
Ψ(x,t) = ∑_n c_nψ_n(x) e^−iE_nt/ℏ

revival time of the infinite square well
T_rev = h/E_1 = 2πℏ/E_1 = 8mL^2/h = 4mL^2/πℏ

φ(x) = N x(L − x) , N = √(30/L^5)

⟨Ĥ⟩ = −(ℏ^2/2m)N^2∫_0^Lx(L−x)(−2)dx = (ℏ^2/m)(30/L^5)(L^3/6) = 5ℏ^2/mL^2

P_n(a,b) = (b−a)/L − [sin(2nπb/L) − sin(2nπa/L)]/(2nπ) → (b−a)/L as n → ∞

ΔE_n/E_n = (2n+1)/n^2 ≈ 2/n → 0

⟨x⟩ = (2/L)∫_0^L x sin^2(nπx/L) dx

mean position in a 1-D box
⟨x⟩ = L/2 for every n

mean square position in a 1-D box
⟨x^2⟩ = L^2[1/3 − 1/(2n^2π^2)]

variance of position in a 1-D box
σ_x^2 = ⟨x^2⟩ − ⟨x⟩^2 = L^2[1/12 − 1/(2n^2π^2)]

p̂ψ_n = −iℏ√(2/L)(nπ/L)cos(nπx/L)

mean momentum in a 1-D box
⟨p⟩ = −iℏ(2/L)(nπ/L)∫_0^Lsin(nπx/L)cos(nπx/L)dx = −iℏ(nπ/L^2)∫_0^Lsin(2nπx/L)dx = 0

mean square momentum in a 1-D box
⟨p^2⟩ = 2m⟨Ĥ⟩ = 2mE_n = 2m · n^2h^2/8mL^2 = n^2h^2/4L^2 = (nπℏ/L)^2

the uncertainty product for a particle in a 1-D box
σ_xσ_p = ℏ √(n^2π^2/12 − 1/2)

⟨x^2⟩ = (2/L)∫_0^Lx^2sin^2(nπx/L)dx = (1/L)∫_0^Lx^2dx − (1/L)∫_0^Lx^2cos(2nπx/L)dx

∫_0^Lx^2cos(ax)dx = [x^2sin(ax)/a + 2x cos(ax)/a^2 − 2 sin(ax)/a^3]_0^L

the one-dimensional ‘pressure’ (force) on the walls of a box
P = −dE_n/dL = 2n^2h^2/8mL^3 = 2E_n/L

φ_n(p) = (2πℏ)^−1/2∫_0^Lψ_n(x)e^−ipx/ℏdx

|φ_n(p)|^2 ∝ (nπ/L)^2 [1 − (−1)^ncos(kL)] ∕ [k^2 − k_n^2]^2

the 3-D box inside the walls
−(ℏ^2/2m)[∂^2ψ/∂x^2 + ∂^2ψ/∂y^2 + ∂^2ψ/∂z^2] = Eψ

ψ(x,y,z) = X(x) · Y(y) · Z(z)

−(ℏ^2/2m)[(1/X)X″ + (1/Y)Y″ + (1/Z)Z″] = E

separation of variables for the 3-D box
−(ℏ^2/2m)X″ = E_xX , −(ℏ^2/2m)Y″ = E_yY , −(ℏ^2/2m)Z″ = E_zZ , E = E_x + E_y + E_z

eigenfunctions of the 3-D box
ψ_nxn_yn_z(x,y,z) = √(8/abc) sin(n_xπx/a) sin(n_yπy/b) sin(n_zπz/c)

energy levels of the 3-D box
E_nxn_yn_z = (h^2/8m)[n_x^2/a^2 + n_y^2/b^2 + n_z^2/c^2] , n_x, n_y, n_z = 1, 2, 3, …

Ĥ(φ_1φ_2φ_3) = (Ĥ_1φ_1)φ_2φ_3 + φ_1(Ĥ_2φ_2)φ_3 + φ_1φ_2(Ĥ_3φ_3) = (ε_1+ε_2+ε_3)φ_1φ_2φ_3

cumulative number of translational states below E
N(≤E) ≈ (1/8)(4π/3)R^3 = (π/6)(8mL^2E/h^2)^3/2 = (4π/3)V(2mE)^3/2/h^3

the three-dimensional density of states
g(E) = dN/dE = 2πV(2m)^3/2E^1/2/h^3

the translational partition function and the thermal de Broglie wavelength
q_trans = V/Λ^3 , Λ = h/√(2πmkT)

E = (h^2/8m)[(n_x^2 + n_y^2)/a^2 + n_z^2/c^2]

the FEMO HOMO–LUMO gap
ΔE = E_N/2+1 − E_N/2 = [(N/2+1)^2 − (N/2)^2] h^2/8m_eL^2 = (N+1) h^2/8m_eL^2

the predicted absorption maximum
λ_max = hc/ΔE = 8m_ecL^2/[h(N+1)]

Hückel energies of a linear polyene
E_k = α + 2β cos[kπ/(N+1)] , k = 1 … N

the free particle
ψ_k(x) = A e^ikx , E = ℏ^2k^2/2m , k any real number

penetration into the classically forbidden region
ψ ∝ e^−κ|x| outside , κ = √(2m(V_0 − E))/ℏ

eigenvalue condition, even parity
z tan z = √(z_0^2 − z^2) (even states)

eigenvalue condition, odd parity
−z cot z = √(z_0^2 − z^2) (odd states)

A cos(kL/2) = B e^−κL/2 ; −Ak sin(kL/2) = −κB e^−κL/2

k tan(kL/2) = κ

the single bound state of a shallow 1-D well
E_1 ≈ V_0 − mV_0^2L^2/2ℏ^2 , binding energy ≈ mV_0^2L^2/2ℏ^2

ψ_I = e^ik_1x + r e^−ik_1x , ψ_II = t e^ik_2x , k_1 = √(2mE)/ℏ , k_2 = √(2m(E−V_0))/ℏ

1 + r = t ; ik_1(1 − r) = ik_2t

amplitude reflection and transmission coefficients at a step
r = (k_1 − k_2)/(k_1 + k_2) , t = 2k_1/(k_1 + k_2)

reflection and transmission coefficients at a step, E > V_0
R = |r|^2 = [(k_1−k_2)/(k_1+k_2)]^2 ; T = (k_2/k_1)|t|^2 = 4k_1k_2/(k_1+k_2)^2

the flux sum rule — check it on every scattering problem
R + T = [(k_1−k_2)^2 + 4k_1k_2]/(k_1+k_2)^2 = 1

total reflection below a step, with a phase shift
r = (k − iκ)/(k + iκ) , |r|^2 = (k^2+κ^2)/(k^2+κ^2) = 1

one-dimensional probability current density
j = (ℏ/m) Im(ψ* dψ/dx) = (ℏ/2mi)(ψ*ψ′ − ψψ*′)

phase shift on total reflection at a step
r = e^−2iδ , tanδ = κ/k , so δ = arctan[√((V_0−E)/E)]

exact transmission coefficient of a rectangular barrier, E < V_0
T = [1 + V_0^2 sinh^2(κa) / (4E(V_0 − E))]^−1

the Gamow / thick-barrier tunnelling probability
T ≈ [16E(V_0−E)/V_0^2] e^−2κa , κ = √(2m(V_0−E))/ℏ

transmission above a rectangular barrier
T = [1 + V_0^2 sin^2(k_2a) / (4E(E − V_0))]^−1 , k_2 = √(2m(E−V_0))/ℏ

2ik = t e^ika[(ik + κ)(1 + ik/κ)e^−κa + (ik − κ)(1 − ik/κ)e^κa] / 2 …

the WKB tunnelling probability for a barrier of arbitrary shape
T ≈ exp[−(2/ℏ) ∫_x1^x_2 √(2m(V(x) − E)) dx]

the semiclassical primary kinetic isotope effect
k_H/k_D = exp[(½hν_H − ½hν_D)/k_BT] , ν_D ≈ ν_H/√2

k_H/k_D = exp(5080 / (8.314 × 298)) = 7.76

the tunnelling current in an STM
I ∝ e^−2κd , κ = √(2m_eφ)/ℏ

the Wigner tunnelling correction, valid for small corrections
k_obs = κ(T) k_TST , κ_Wigner = 1 + (1/24)(hν^‡/k_BT)^2

the tunnelling splitting of a symmetric double well
E_± = ε_0 ∓ t , |±⟩ = (|L⟩ ± |R⟩)/√2 , ΔE = 2t

Definitions worth memorising

Where quantisation comes from: not from a postulate, but from imposing boundary conditions on a differential equation. A free particle has a continuous energy spectrum. The moment you demand that its wavefunction vanish at two fixed points, only a discrete set of wavelengths — and hence a discrete set of energies — survives. Every quantum number in chemistry has this origin: n from a radial boundary condition, ℓ and m from single-valuedness on a sphere, v from normalisability of the oscillator.

Node: a point inside the allowed region where the wavefunction passes through zero, so that the probability density there is exactly zero. The walls are not nodes: ψ is zero there because it is forced to be, not because it is oscillating through zero. State n of a 1-D box has n − 1 interior nodes.

Zero-point energy: the energy of the lowest allowed state of a bound system, which is not zero. For the 1-D box it is E_1 = h^2/8mL^2. It cannot be removed by cooling, by any measurement, or by any change of reference point, because it is not thermal energy — it is the minimum energy consistent with the particle being confined at all.

Expectation value: for an observable represented by the operator Â and a normalised state ψ, ⟨A⟩ = ∫ψ* Âψ dτ. It is the average of the results of many measurements of A on identically prepared systems. It is not the value you get from one measurement, and it need not even be one of the possible results.

Separation of variables: if the Hamiltonian can be written as a sum of terms each depending on a different coordinate, and the boundary conditions do not couple those coordinates, then the eigenfunctions are products of the one-dimensional eigenfunctions and the eigenvalues are sums of the one-dimensional eigenvalues. This is why the hydrogen atom, the rigid rotor and the 3-D harmonic oscillator are solvable at all.

Degeneracy: two or more linearly independent wavefunctions having the same energy eigenvalue. The degree of degeneracy g is the number of such independent states. A level with g = 1 is called non-degenerate or singly degenerate.

Accidental degeneracy: a coincidence of energies not required by any obvious geometrical symmetry of the system. In the cubic box it arises because different triples of squares can sum to the same integer. In the two-dimensional square box the first case is n^2 = 50, reached by (1,7) and by (5,5). The word ‘accidental’ is traditional; in several famous cases — the ℓ-degeneracy of the hydrogen atom is the standout example — the accident later turned out to reflect a hidden symmetry.

Penetration depth: 1/κ = ℏ/√(2m(V_0−E)), the distance over which the wavefunction falls by a factor e in the classically forbidden region. Note the two dependences: it is shorter for a heavier particle and for a taller barrier. For an electron with V_0−E = 1 eV it is 0.195 nm; for a proton in the same situation only 0.0046 nm, some 43 times smaller. That single ratio explains most of the chemistry in C.9.

Quantum-mechanical tunnelling: the passage of a particle through a region where its total energy is less than the potential energy, so that classically it could not be. It happens because the wavefunction decays rather than terminates in the forbidden region; if the region is thin enough, enough amplitude survives to the far side to give a measurable transmission probability. Nothing gains energy and nothing ‘jumps over’ anything.

Where these come from

This sheet is distilled from Quantum Chemistry, Part 3 — 9 sections that derive every one of these results and show you how to use them.

Read Part 3 All formula sheets