2.3.12 · D1Hydrogen atom — solving in spherical coordinates
Hydrogen atom mein ek single electron hai jo ek single proton ke pull se bana hua hai, aur yeh pull sirf distance par depend karta hai — isliye poore problem mein perfect spherical symmetry hai. Yeh deep dive — bilkul scratch se — har woh symbol build karta hai jo tumhe chahiye, taaki "spherical symmetry ⇒ equation ek distance problem aur do angle problems mein split ho jaati hai" magical nahi, obvious lage.
Yeh parent note ka prerequisite toolkit hai. Agar wahan koi symbol aisa laga jaise hawa mein se aaya ho, toh woh yahan define hai — order mein, har ek ek picture ke saath.
Koi bhi symbol aane se pehle, physical scene: ek chota sa negative electron, ek bhaari positive proton, aur unke beech electric attraction. Baaki sab kuch sirf electron kahan hai aur uski kitni energy hai ka hisaab hai.
Intuition Figure s01 — central pullPicture padhiye: red dot centre mein proton hai, blue dot electron hai, aur orange arrow force hai. Gaur karo — yeh arrow bilkul usi dashed grey line par pada hai jo donon ko jodin hai — yeh usi line ke along point karta hai , kabhi sideways nahi. Us arrow ki strength sirf ek cheez badal sakti hai: dashed line ki length r central kahenge, aur yahi sab kuch ki jad hai.
Pehle yeh batana hoga ki electron kahan hai . Aam tarika teen numbers x , y , z direction-friendly sawaal poochte hain.
Definition Spherical coordinates
r distance . Ek single non-negative number. Picture: centre se dot tak seedhi line ki length .θ polar angle , upar wale axis (z measure kiya jaata hai. Range 0 π ϕ azimuthal angle , vertical axis ke around measure kiya jaata hai. Range 0 2 π
Intuition Figure s02 — ek sphere par teen sawaalFigure mein blue arrow r Green arc θ vertical z khulta hai — yahi hai seedha-upar se tilt. Orange arc neeche floor plane mein baith kar ϕ around swing karta hai, measuring karta hai ki tum kis taraf ghoom rahe ho. Teen arcs, teen independent answers, aur saath mein yeh electron ko sphere par kahin bhi pin kar dete hain.
Intuition Yeh teen hi kyun,x , y , z r ( r , θ , ϕ ) r ( θ , ϕ )
ϕ full loop 0 2 π θ half 0 π ϕ ϕ θ
Wavefunction ki baat karne se pehle, ek notation ki zaroorat hai jis par woh rely karti hai: kisi number ki magnitude (modulus ya absolute value bhi kehte hain), jo do vertical bars ∣ ⋅ ∣
∣ z ∣ ∣ z ∣ size, sign ignore karke : ∣ − 3 ∣ = 3 ∣5∣ = 5 0 ≥ 0 complex hote hain: z = a + ib i 2 = − 1 i z a b ∣ z ∣ = a 2 + b 2 origin se distance . Isko square karne par, ∣ z ∣ 2 = a 2 + b 2 ≥ 0
Intuition Yeh kyun chahiye hogaProbability kabhi negative nahi ho sakti, lekin jo quantity hum milne wale hain (ψ ∣ ⋅ ∣ kisi bhi aisi number ko ek honest non-negative size mein badalta hai — exactly wahi jo probability demand karti hai.
ψ ψ psi ) ek number hai jo theory space ke har point par attach karti hai. Iska matlab: ∣ ψ ∣ 2 ψ likely paoge. Picture: ek cloud jo jahan ∣ ψ ∣ ∣ ψ ∣
Intuition "Magnitude squared" kyun,ψ ψ ∣ ψ ∣ 2 ∣ ⋅ ∣ ∣ ψ ∣ 2 ≥ 0 ψ ∣ ψ ∣ 2 measure karte hain.
Kyunki position ( r , θ , ϕ ) ψ ( r , θ , ϕ ) r θ ϕ
Definition Volume elementd V ∣ ψ ∣ 2 per unit volume (density) hai, apne aap mein probability nahi. Actual probability nikalne ke liye volume ka ek chhota chunk d V d V = r 2 sin θ d r d θ d ϕ . Integral sign ∫ kahiin hona require karne par normalization rule milta hai
∫ ∣ ψ ∣ 2 d V = 1.
r 2 sin θ d r d θ d ϕ d θ d ϕ bahut chhota patch sweep karta hai, lekin equator ke paas wahi angular step, axis se door, ek bada patch sweep karta hai. Factor r 2 sin θ r 2 sin θ
Equation poochti hai ki ψ derivative hai.
d r df f slope jaise r r f Second derivative d r 2 d 2 f curvature : kya graph upar mur raha hai (valley) ya neeche (hill)?
Intuition Figure s03 — slope vs curvatureBlue curve ek sample f ( r ) r Green straight line ek point par sirf us curve ko chhooti hai — uski steepness hi wahan df / d r Orange note dikhata hai poori curve ka upar ki taraf ek katori ki tarah jhukna: yeh upar ki taraf jhukna positive curvature hai, second derivative. Slope tilt batata hai; curvature batata hai ki tilt kis tarah badal raha hai.
Definition Partial derivative∂ f ∂ θ ∂ f θ r ϕ ∂ doosre variables still hain.
Intuition Physics ko curvature kyun chahiyeQuantum mechanics mein electron ki kinetic energy is baat mein encode hoti hai ki ψ bend karti hai. Bahut zyada wiggle karne wali wavefunction (high curvature) = high energy; gentle wali = low energy. Isliye Schrödinger's equation mein second derivatives har jagah aate hain — woh literally "yahan kitni motion energy hai" hain.
Kinetic energy ko ek saath teeno directions mein curvature chahiye. Directions bundle karne ke liye pehle nabla milta hai, phir isko square karte hain.
∇ ∇ teen slopes ka collector hai: ∇ f = ( ∂ x ∂ f , ∂ y ∂ f , ∂ z ∂ f ) f x y z
∇ 2 ∇ 2 ∇ us ka bhi har direction mein slope lo" — yani har direction mein second derivatives add karo:
∇ 2 f = ∂ x 2 ∂ 2 f + ∂ y 2 ∂ 2 f + ∂ z 2 ∂ 2 f . f total curvature hai, har direction mein sum kiya hua. Picture: har point par, "f
Intuition Second derivatives sum karna kyun sahi haiKinetic energy slope ke baare mein nahi hai (ek smoothly tilting wave phir bhi chalti hai), yeh bending ke baare mein hai, aur ek direction mein bending doosre ko cancel nahi karni chahiye — unhe add karna chahiye. Teen second derivatives sum karna exactly yahi karta hai. ∇
Intuition Spherical coordinates mein yeh itna ugly kyun lagta haiPlain x , y , z ( r , θ , ϕ ) r 2 sin θ geometry hain — yeh account karte hain ki "ϕ
Payoff: spherical form mein, ∇ 2 radial chunk (sirf r angular chunk (sirf θ , ϕ
E V ( r )
E energy (kinetic + potential), ek given state ke liye ek single number.V ( r ) potential energy , sirf distance r V ( r ) = − 4 π ε 0 r e 2
Intuition Figure s04 — the Coulomb wellRed curve V ( r ) V → 0 r = 0 V → − ∞ zero se neeche hai — iska matlab hai bound : bahar nikalne ke liye energy add karni padegi.
Definition Planck's constanth ℏ h Planck's constant hai, nature ka ek fixed number h ≈ 6.626 × 1 0 − 34 J⋅s × ℏ h ℏ = h /2 π ≈ 1.055 × 1 0 − 34 J⋅s 2 π = 2 π ℏ
Constants e ε 0 π
Solution ka ϕ e ⋅ i
e i
e ≈ 2.718 e x slope khud uske barabar hota hai .i i 2 = − 1 Saath mein, e i α = cos α + i sin α α unit circle par chalte point ko trace karta hai. Picture: 1 lambi ghadi ki sui jo ghoom rahi ho.
Intuition Ek exponential "which way around" kyun describe karta haiAngle ϕ hi hai, aur e i ϕ ϕ 0 2 π e i m ϕ m m ϕ 2 π khud mein wapas aaye exactly wahi hai jo §8 mein m
Definition Angular momentum operatorL ^ Orbital angular momentum measure karta hai ki koi object ek centre ke around kitna ghoom raha hai — sochiye electron proton ke around circulate kar raha hai, jaise ek planet sun ke orbit mein. (Yeh electron ka apna intrinsic "spin" nahi hai, jo alag property hai aur yahan hamare kaam ki nahi — yahan hum purely orbiting motion ki baat kar rahe hain.) Quantum mechanics mein yeh orbiting ek operator L ^ ψ ∇ 2 L ^ 2 Angular momentum operators .
Intuition Yeh Russian dolls ki tarah nest kyun karte hainn ℓ ℓ m ℓ "n limits ℓ, ℓ limits m" ) woh skeleton hai jis par parent note sab kuch tikaya hai — dekho Quantum numbers and the periodic table bhi.
Upar sab kuch ek random list nahi hai — har foundation apna output agle ko deta hai , energy levels aur orbital shapes tak pahunchte hue jo parent note compute karta hai. Neeche wala map woh hand-off chain hai; usse upar se neeche padho aur har node ko uske section se match karo.
central force depends only on r
choose spherical coords r theta phi
wavefunction psi of r theta phi
probability via volume element dV
derivatives and curvature
nabla and Laplacian del squared
Schrodinger equation E psi
potential V and constants h and hbar
separation into radial and angular parts
exponential loop rule e to i m phi
energy levels and orbital shapes
Top-left se shuru karo: symmetry coordinate choice force karti hai, jo ψ volume element ke through — ψ derivatives Laplacian banate hain; woh plus potential aur constants Schrödinger's equation assemble karte hain, jo separate ho jaati hai. Separated ϕ exponential loop-rule se milti hai aur baahir aate hain quantum numbers — energy levels aur orbital shapes ke final ingredients. Koi bhi arrow follow karo aur tum literally ek "yeh usse banane ke liye chahiye" dependency follow kar rahe ho.
Khud test karo — answer pehle apne dimag mein aane ke baad hi reveal karo.
r Distance (proton se electron tak).
θ Top (z 0 π
ϕ Vertical axis ke around; 0 2 π
∣ z ∣ ∣ z ∣ 2 ≥ 0 Size/distance from 0
∣ ψ ∣ 2 Probability density — electron wahan milne ki kitni likelihood hai (per unit volume).
ψ ∣ ψ ∣ 2 Probabilities negative ya complex nahi ho sakti; ∣ ψ ∣ 2 ≥ 0
Spherical volume element d V r 2 sin θ d V = r 2 sin θ d r d θ d ϕ
Normalization ∫ ∣ ψ ∣ 2 d V = 1 Electron definitely kahiin hai — total probability 1
Ek picture mein derivative df / d r Graph ki slope (steepness).
Second derivative kya represent karta hai? Curvature — slope kaise bend karta hai (valley vs hill).
Rounded ∂ Ek partial derivative: ek variable vary karo, baaki sab fixed rakho.
Nabla ∇ Teen directional slopes (gradient arrow).
Ek phrase mein, Laplacian ∇ 2 ψ
∇ 2 Sphere ki pure geometry (r 2 sin θ
V ( r ) State bound hai; electron ko free karne ke liye energy add karni padegi.
r → ∞ V → 0 Door jaane par attraction khatam ho jaati hai — escape karne ke liye kuch nahi bacha.
h ℏ h ℏ = h /2 π
α e i α Unit circle par chalta hua ek point.
Loop exponent m Wavefunction ko ϕ 2 π
Kya L ^ Nahi — yeh orbital (going-around) angular momentum hai, intrinsic spin nahi.
n , ℓ , m ℓ n = 1 , 2 , … ℓ = 0 , … , n − 1 m ℓ = − ℓ , … , + ℓ
