Variational principle and perturbation theory (intro)

5.1.2 · Chemistry › Physical Chemistry (Advanced)

1. The Variational Principle

Hum ye kyun maante hain ki energy hamesha overestimate hoti hai?

Ise first principles se derive kaise karte hain

Formula

$E [ϕ] \geq E_{0}$ ki derivation True eigenstates ${ψ_{n}}$ jisme $\hat{H} ψ_{n} = E_{n} ψ_{n}$ ek complete orthonormal basis banate hain. Kisi bhi trial function ko expand karo: $ϕ = \sum_{n} c_{n} ψ_{n} .$ Ye step kyun? Completeness guarantee karti hai ki har function eigenstates ka sum hota hai.

Numerator: $⟨ ϕ ∣ \hat{H} ∣ ϕ ⟩ = \sum_{n, m} c_{n}^{*} c_{m} ⟨ ψ_{n} ∣ \hat{H} ∣ ψ_{m} ⟩ = \sum_{n, m} c_{n}^{*} c_{m} E_{m} δ_{nm} = \sum_{n} ∣ c_{n} ∣^{2} E_{n} .$ Kyun? $\hat{H} ψ_{m} = E_{m} ψ_{m}$ aur orthonormality cross terms ko khatam kar deti hai.

Denominator: $⟨ ϕ ∣ ϕ ⟩ = \sum_{n} ∣ c_{n} ∣^{2} .$

Toh $E [ϕ] = \frac{\sum _{n} ∣ c _{n} ∣ ^{2} E _{n}}{\sum _{n} ∣ c _{n} ∣ ^{2}} .$ Ye eigenenergies ka ek weighted average hai. Kyunki $E_{n} \geq E_{0}$ sabhi $n$ ke liye: $E [ϕ] = \frac{\sum _{n} ∣ c _{n} ∣ ^{2} E _{n}}{\sum _{n} ∣ c _{n} ∣ ^{2}} \geq \frac{\sum _{n} ∣ c _{n} ∣ ^{2} E _{0}}{\sum _{n} ∣ c _{n} ∣ ^{2}} = E_{0} . ■$

Toh strategy: apne trial function mein parameters $α$ daalo, $E (α)$ compute karo, phir minimize karo: $\frac{\partial E ( α )}{\partial α} = 0.$

Figure — Variational principle and perturbation theory (intro)

2. Perturbation Theory (non-degenerate)

Hamiltonian ko split kyun karte hain?

Formula

$E_{n}^{(1)}$ aur $E_{n}^{(2)}$ derive karna $\hat{H} ψ_{n} = E_{n} ψ_{n}$ mein plug karo aur $λ$ ki powers collect karo.

Order $λ^{0}$ : $\hat{H}^{(0)} ψ_{n}^{(0)} = E_{n}^{(0)} ψ_{n}^{(0)}$ ✓ (known problem).

Order $λ^{1}$ : $\hat{H}^{(0)} ψ_{n}^{(1)} + \hat{H}^{'} ψ_{n}^{(0)} = E_{n}^{(0)} ψ_{n}^{(1)} + E_{n}^{(1)} ψ_{n}^{(0)} .$ Dono sides par $⟨ ψ_{n}^{(0)} ∣$ lagao. Kyun? Ye us piece ko project out karta hai jo chahiye aur hermiticity use karta hai. $⟨ ψ_{n}^{(0)} ∣ \hat{H}^{(0)} ∣ ψ_{n}^{(1)} ⟩ + ⟨ ψ_{n}^{(0)} ∣ \hat{H}^{'} ∣ ψ_{n}^{(0)} ⟩ = E_{n}^{(0)} ⟨ ψ_{n}^{(0)} ∣ ψ_{n}^{(1)} ⟩ + E_{n}^{(1)} .$ Kyunki $\hat{H}^{(0)}$ left par act karke $E_{n}^{(0)} ⟨ ψ_{n}^{(0)} ∣$ deta hai, pehla term right side ke pehle se cancel ho jaata hai. Result: $E_{n}^{(1)} = ⟨ ψ_{n}^{(0)} ∣ \hat{H}^{'} ∣ ψ_{n}^{(0)} ⟩ .$ First-order correction bas perturbation ka unperturbed state mein average hai.

$ψ_{n}^{(1)} = \sum_{m \neq = n} c_{m} ψ_{m}^{(0)}$ expand karo aur order- $λ^{1}$ equation par $⟨ ψ_{m}^{(0)} ∣$ se project karo, milta hai $c_{m} = \frac{⟨ ψ _{m}^{(0)} ∣ H ^ ^{'} ∣ ψ _{n}^{(0)} ⟩}{E _{n}^{(0)} - E _{m}^{(0)}}$ , aur second-order energy: $E_{n}^{(2)} = m \neq = n \sum \frac{∣ ⟨ ψ _{m}^{(0)} ∣ H ^ ^{'} ∣ ψ _{n}^{(0)} ⟩ ∣ ^{2}}{E _{n}^{(0)} - E _{m}^{(0)}} .$ Ground state ke liye, har denominator negative hai ⇒ $E_{0}^{(2)} \leq 0$ : second order hamesha ground state ko neeche dhakelta hai.

Common mistake Steel-manned misconceptions

(a) "Kam variational energy matlab zyada accurate wavefunction." Sahi lagta hai kyunki energy hi wo cheez hai jo hum minimize karte hain. Kyun tempting lagta hai: energy fast converge hoti hai. Fix: energy truth ke paas stationary hoti hai (2nd-order accurate), toh wavefunctions phir bhi kharab ho sakti hain jabki energy achi lagti hai. Energy ke alawa doosri properties bahut galat ho sakti hain.

(b) "Variational energy $E_{0}$ se neeche aa sakti hai agar kismat ho." Kyun tempting lagta hai: approximations kabhi kabhi undershoot karti hain. Fix: theorem ise forbid karta hai — $E [ϕ] \geq E_{0}$ rigorously. Agar tumhe chhota number mila, toh arithmetic/normalization mein galti hui hai.

(c) Degenerate levels par ordinary (non-degenerate) $E_{n}^{(2)}$ use karna. Kyun tempting lagta hai: formula wahan hai. Fix: agar $E_{n}^{(0)} = E_{m}^{(0)}$ toh denominator blow up ho jaata hai — tumhe degenerate perturbation theory use KARNI CHAHIYE (pehle degenerate subspace mein $\hat{H}^{'}$ diagonalize karo).

Recall Ise ek 12-saal ke bache ko samjhao (Feynman)

Socho sachi answer ek valley ka sabse nichla point hai jo tum dekh nahi sakte. Variational method: tum balls (guesses) valley mein phenko; har ball kahin na kahin upar ya bottom par hi girti hai, kabhi neeche nahi — toh jo sabse nichli ball tum pate ho wo bottom ka tumhara best estimate hai, aur adjustable knobs add karne se ball aur neeche roll karne lagti hai. Perturbation method: tum ek nearby easy hill ko already perfectly jaante ho, aur asli hill almost wahi hai bas ek chhoti si extra bump ke saath; tum real height paane ke liye bas chhote corrections add karte ho — pehle average bump height, phir ek aur barik fix.

Flashcards

Variational theorem inequality batao.

Kisi bhi normalizable trial

ϕ

ke liye:

E [ϕ] = \frac{⟨ ϕ ∣ H ^ ∣ ϕ ⟩}{⟨ ϕ ∣ ϕ ⟩} \geq E_{0}

, equality tab jab

ϕ

true ground state ho.

Variational energy hamesha

\geq E_{0}

kyun hoti hai?

Ye eigenenergies ka weighted average

\frac{\sum ∣ c _{n} ∣ ^{2} E _{n}}{\sum ∣ c _{n} ∣ ^{2}}

hai, aur har

E_{n} \geq E_{0}

First-order energy correction kya hai?

E_{n}^{(1)} = ⟨ ψ_{n}^{(0)} ∣ \hat{H}^{'} ∣ ψ_{n}^{(0)} ⟩

— perturbation ki expectation value unperturbed state mein.

Second-order energy correction (non-degenerate) kya hai?

E_{n}^{(2)} = \sum_{m \neq = n} \frac{∣ ⟨ ψ _{m}^{(0)} ∣ H ^ ^{'} ∣ ψ _{n}^{(0)} ⟩ ∣ ^{2}}{E _{n}^{(0)} - E _{m}^{(0)}}

Ground-state second-order correction hamesha

\leq 0

kyun hoti hai?

Numerator

\geq 0

hai aur har denominator

E_{0}^{(0)} - E_{m}^{(0)} < 0

hai.

Particle in a box ke liye trial wavefunction mein kya hona chahiye?

Boundary conditions

ϕ (0) = ϕ (a) = 0

(aur normalizable/single-valued honi chahiye).

Non-degenerate perturbation theory kab fail hoti hai?

Jab unperturbed levels degenerate hon (

E_{n}^{(0)} = E_{m}^{(0)}

) — denominators diverge ho jaate hain; degenerate PT use karo.

PIB trial

ϕ = x (a - x)

ke liye exact se compare karke kya energy milti hai?

E = \frac{5 ℏ ^{2}}{m a ^{2}} = \frac{10 ℏ ^{2}}{2 m a ^{2}}

vs exact

\frac{9.8696 ℏ ^{2}}{2 m a ^{2}}

— sirf 1.3% zyada.

Connections

Schrödinger Equation
Particle in a Box
Quantum Harmonic Oscillator
Hydrogen Atom
Helium Atom (Two-Electron Problems)
Hartree–Fock & Self-Consistent Field
Basis Sets in Quantum Chemistry
Hermitian Operators & Completeness
Degenerate Perturbation Theory