Foundations — Schrödinger equation — time-dependent, time-independent

2.3.9 · D1 · Physics › Modern Physics › Schrödinger equation — time-dependent, time-independent

Is page par assume kiya gaya hai ki aapne parent note ka koi bhi notation pehle nahi dekha. Hum har symbol ko earn karenge — ek ek karke, har ek ko ek picture se anchor karke, har ek ko why the topic needs it se explain karke. Upar se neeche padho; neeche koi bhi cheez aise symbol use nahi karti jo upar build nahi ki gayi ho.

0. Shuru ki picture: ek particle ek cloud hai

Equation ko ek second ke liye bhool jao. Jis duniya ko tum jaante ho, usme ek ball ki ek position hoti hai. Quantum physics mein ek particle ke paas iske bajaaye ek sambhavna ka cloud hota hai. Jahan cloud ghana ho, particle milne ki zyada sambhavna hai; jahan patla, kam.

Figure dekho: blue curve har position $x$ par cloud ki "motaai" hai. Particle na kahin hai na har jagah, jab tak tum dekho nahi — aur jab tum dekhte ho, toh woh kahin appear hota hai, zyada baar jahan blue curve unchi ho. Woh blue curve wahi hai jo poora topic compute aur predict karne ki koshish karta hai.

1. $x$ aur $t$ — position aur time

Why the topic needs them: cloud alag alag jagahon ($x$) par alag hota hai aur time bitne ke saath drift karta hai aur reshape hota hai ($t$). Isliye har cheez in do inputs ki function hai. Hum "ek cheez jo $x$ aur $t$ par depend karti hai" ko uske naam ke baad (x,t) likhte hain.

2. $i$ — imaginary unit

Figure mein, length 1 ka ek horizontal yellow arrow ($+1$ number) $i$ se multiply hone par vertical ho jaata hai, phir $i$ ke ek aur application ke baad peeche point karta hai ($-1$), phir neeche ($-i$), phir ghar. $i$ se chaar multiplications = ek poora turn.

3. Complex number = ek plane mein arrow; $|\cdot |$ uski length

Why the topic needs it: $\Psi$ har point par ek complex arrow hai. Length-squared $|\Psi {|}^2$ "swing ka konsa hissa" ki information throw away kar deta hai aur sirf "kitna bada" rakhta hai — aur woh size wahi ban jaata hai jo ek probability hai (hamesha ek positive real number). Complex phase wiggles karta hai; length still reh sakti hai. Woh distinction hi ek stationary state ka poora matlab hai.

4. $e^{i\theta }$ — spinning unit arrow

Yeh tool kyun, koi aur kyun nahi? Humein kuch aisi cheez chahiye jo smoothly hamesha ke liye rotate kare aur kabhi length na badale. Fixed length ka spinning arrow exactly wahi wave hai jo "phase mein wiggle karti hai bina apni probability badale." Figure mein green arrow ki length hamesha 1 hai, isliye $|e^{i\theta }{|}^2=1$: phase spins karta hai, size kabhi nahi hilta. (Baad mein, jab ek baar physics quantities ko naam de denge, hum parent ki stationary state se milenge — ek shape exactly aisi spinning arrow se multiply hoti hai.)

5. $k$, $\omega$, aur wavelength — ek wave ki shape

Why the topic needs them: yeh do numbers Section 4 se spinning-arrow wave ko poori tarah describe karte hain — ek uski wiggle space mein control karta hai, doosra uska ticking time mein. Inhe milao, right ki taraf travel karne wali wave likhi jaati hai $\Psi (x,t)=A{e}^{i(kx-\omega t)},$ jahan $A$ sirf uski overall size hai. Bilkul agle section mein hum physical meaning attach karenge ($k$ momentum carry karta hai, $\omega$ energy carry karta hai) — lekin pehle un physical quantities ko naam chahiye.

6. $\hslash$, $p$, $E$, $m$, $V$ — physics quantities

Why the topic needs them: Schrödinger equation sirf ek wave ke liye energy conservation hai: total energy = motion energy + landscape energy, yani $E=\frac{p^2}{2m}+V$. Har symbol us sentence mein ek word hai.

7. $\Psi$ vs $\psi$ — do wavefunctions

8. Parent ki equations ke do naam

9. $\partial$ — derivative, "rate of change" tool

Yeh do tools kyun, aur nahi? Parent ko wave se $E$ aur $p$ extract karne hain. Ek time-derivative ticking rate $\omega$ (hence $E=\hslash \omega$) ko exponential se neeche kheenchti hai; do space-derivatives $k^2$ (hence $p^2$) kheenchte hain. Derivative woh akela tool hai jo "yeh kitni tezi se change ho raha hai?" ka jawab deta hai — exactly woh question jo ek wave ko waapis energy aur momentum mein badal deta hai.

10. $\hat{H}$ — Hamiltonian operator

Why the topic needs it: yeh poori equation ko $\hat{H}\psi =E\psi$ tak shrink karne deta hai — "energy machine ko shape par apply karo, same shape times energy wapas milo." (Zyada Hamiltonian Operator mein.)

11. $\int$ aur normalization

Why the topic needs it: $|\Psi {|}^2$ ek probability density hai; actual probability pane ke liye ek region par cloud ko add (integrate) karo. Total cloud ka weight exactly 1 hona chahiye.

Yeh sab topic ko kaise feed karta hai

Equipment checklist

Right side cover karo aur dekho reveal karne se pehle answer de sakte ho ya nahi.

Connections

• Parent topic — jahan yeh symbols kaam mein laaye jaate hain.
• de Broglie Hypothesis — $p=\hslash k$ ka source jo waves ko momentum se jodhta hai.
• Wavefunction and Born Interpretation — $|\Psi {|}^2$ aur normalization ka matlab.
• Hamiltonian Operator — energy machine $\hat{H}$.
• Particle in a Box — pehli jagah jahan yeh tools real energies banate hain.
• Energy Quantization — discrete "menu" jo emerge hota hai.
• Quantum Harmonic Oscillator — spring potential $V$ ke saath TISE.
• Heisenberg Uncertainty Principle — wave picture ka ek consequence.

#flashcards/physics