Particle in a box — solving TISE, energy levels, wavefunctions: Foundations

2.3.10 · D1 · Physics › Modern Physics › Particle in a box — solving TISE, energy levels, wavefunctio

Yeh page assume karta hai ki tumne pehle kuch bhi nahi dekha. Parent note ki central equation ko padhne se pehle, hume usmein har letter ko kamana padega. Hum symbol by symbol jayenge, har ek ke saath simple words, ek picture, aur woh reason jiske liye topic ko uski zaroorat hai. Hum woh equation tab tak nahi likhenge jab tak uske har symbol ko pehle se build na kar liya ho — yahi is page ka poora point hai.

0. Hum kya build kar rahe hain?

Is page ke end tak tum parent note ki master equation ko bina kisi unknown symbol ke padh paoge. Woh equation aath ideas ko ek saath bandhti hai, jinhe hum is order mein build karte hain: box mein position, woh wave $ψ$ jo wahan rehti hai, curvature ka idea (woh wave kitni tezi se bend karti hai), potential $V$ jo trap ko shape karta hai, physical dials mass $m$ aur Planck's constant $ℏ$ , energy $E$ , aur finally walls par boundary conditions. Hum inhe ek ek karke milte hain, ek aisa order mein jahan har ek pehle wale par tikta hai, aur sirf final section mein — jab sab aath define ho jaate hain — hum inhe assemble karte hain.

Notice karo: humne abhi tak koi formula nahi likha, aur humne jaanbujhkar in ideas ko plain words mein nama kiya hai (koi derivative symbol nahi, koi Greek soup nahi) precisely kyunki kuch bhi define nahi hua tha. Har ek ko neeche apna section milta hai.

Figure — Particle in a box — solving TISE, energy levels, wavefunctions

Figure dekho: $x = 0$ se $x = L$ tak ek seedha corridor, do solid walls ke saath. Jo kuch bhi hum karte hain woh is picture par rehta hai.

1. Position $x$ — box mein kahan ho

Picture: upar wali figure mein horizontal ruler. $x = 0$ left wall hai, $x = L$ right wall hai.

Topic ko iska kyun zaroorat hai: baaki sab cheez — wave height, potential — ek function hai is baat ka ki tum kahan ho, isliye hume "kahan" ke liye ek naam chahiye.

2. Box ki length $L$ — trap kitna wide hai

Picture: corridor ki poori width.

Topic ko iska kyun zaroorat hai: $L$ poore problem mein sirf ek size hai. Baad mein tum energies ko $\frac{1}{L ^{2}}$ se shrink hote dekhoge — bada box ek gentle trap hai. $L$ ke bina "main kitna confined hoon?" ka sawaal hi nahi hai.

3. Wavefunction $ψ (x)$ — ek wave, particle-dot nahi

Picture: agli figure mein smooth curve, corridor floor ke upar rise karti aur neeche dip karti.

Topic ko iska kyun zaroorat hai: poora problem hai "is box mein kaun si wave shapes allowed hain?" $ψ$ hi woh unknown hai jiske liye hum solve karte hain.

4. Probability density $∣ ψ ∣^{2}$ aur normalisation — particle kahan milne ki probability hai

Picture: upar wali figure mein, wave ke neeche pale-yellow humped curve $∣ ψ ∣^{2}$ hai — hamesha $\geq 0$ , sabse tall wahan jahan wave sabse wide swing karti hai.

Topic ko iska kyun zaroorat hai: yahi woh rule hai jo abstract wave ko kuch measurable se connect karta hai — electron actually kahan hai.

5. Derivative $\frac{d ψ}{d x}$ — wave ki steepness

Picture: imagine karo ki ek tiny surfer wave par khada hai. Derivative unke pair ke neeche zameen ka tilt hai.

Yahan kyun aur kuch aur kyun nahi? Hume ek tool chahiye jo space mein change measure kare. "Slope" exactly wahi tool hai. "Kitni tezi se wave bend karti hai" ko kuch simple aur better capture nahi kar sakta.

6. Second derivative $\frac{d ^{2} ψ}{d x ^{2}}$ — curvature (yeh star hai)

Picture: figure mein, do waves — ek lazy one-hump wave (low curvature, blue) aur ek rapid three-hump wave (high curvature, pink). Pink wali bahut zyada hard bend karti hai.

Second derivative kyun aur first kyun nahi? Sines aur cosines mein yeh magical property hai ki unka second derivative same shape return karta hai, flip aur scaled karke: $\frac{d ^{2}}{d x ^{2}} sin (k x) = - k^{2} sin (k x)$ . Poora problem is "self-returning" property ka statement hai — sirf second derivative ise reveal karta hai.

7. Potential energy $V (x)$ — trap ki shape

Picture: $x = 0$ aur $x = L$ par infinitely tall cliffs, beech mein flat valley floor ( $V = 0$ ). Pehli figure ki walls dekho.

Topic ko iska kyun zaroorat hai: $V$ hi trap hai. $V$ badlo aur tumhe alag problem milti hai — ek spring (Quantum Harmonic Oscillator) ya ek leaky box (Quantum Tunnelling and Finite Well).

8. Mass $m$ aur Planck's constant $ℏ$ — physical dials

Picture: $ℏ$ ko "wave kitni wiggly hai" aur "uski kitni energy lagti hai" ke beech exchange rate samjho. Chota $ℏ$ = quantum world ek fine grain hai jise hum human scale par notice nahi karte.

Topic ko inki kyun zaroorat hai: $L$ ke saath milkar, yeh constants box mein har energy ki scale set karte hain — yeh woh dials hain jo wave-shape ko joules ki sankhya mein badal dete hain.

9. Energy $E$ — ek allowed wave ki total energy

Picture: ek horizontal line jo us wave ki energy represent karti height par kheenchi ho. Alag allowed waves alag heights par baithti hain — levels ki ek ladder.

Topic ko iska kyun zaroorat hai: $E$ hi ultimately solve karte hain. Poore topic ka headline result yeh hai ki sirf certain values of $E$ allowed hain.

10. Boundary conditions — woh rule jo walls impose karti hain

Picture: figure 2 mein do pink dots jahan wave exactly $x = 0$ aur $x = L$ par floor par touch karti hai — jaise ek jump rope dono ends par bandhi ho.

Topic ko iska kyun zaroorat hai: boundary condition ke bina equation ke infinitely many smooth solutions hain aur koi quantisation nahi. Walls referee hain.

11. Quantum number $n$ aur wavelength $λ$ — har allowed wave ko label karna

Picture: horizontal lines ki ek ladder, har $n$ ke liye ek. Kyunki energy $n^{2}$ ke saath badhegi, rungs $1, 4, 9, 16, \dots$ heights par baithte hain — jaise-jaise upar chadho, alag hote jaate hain.

Topic ko iska kyun zaroorat hai: $n$ hi quantisation ka symbol hai — woh single integer jo har allowed state ko index karta hai — aur $λ$ "kitne arches fit hote hain" aur "uski kitni curvature (hence energy) lagti hai" ke beech bridge hai.

12. Master equation assemble karna

Ab — aur sirf ab — har symbol define ho chuka hai, isliye hum finally parent note ki central equation padh sakte hain:

Tum ab parent note ko pehli line se padhne ke liye equipped ho.

Foundations topic ko kaise feed karte hain

Neeche ka map woh dependency order dikhata hai jise humne follow kiya. Ise top-down padho: geometry ( $x$ , $L$ ) wave $ψ$ ko host karti hai; wave ki curvature aur probability meaning hoti hai; trap $V$ boundary conditions impose karta hai; physical constants $m$ aur $ℏ$ scale set karte hain; yeh sab TISE mein assemble hote hain; ise solve karne par ek quantised label $n$ forced hota hai, jo allowed energies aur wave shapes deliver karta hai — poora topic. Agar tum har arrow trace kar sako aur har node naam le sako, tum ready ho.

Equipment checklist

Neeche har line ek sawaal hai uske baad uska jawab, ::: marker se separated (marker ke right taraf sab kuch cover karo aur khud test karo). Agar koi jawab fuzzy lage, toh parent note tackle karne se pehle exactly wahi piece upar se dobara padho.

x

symbol yahan kya matlab rakhta hai?

Box mein position, left wall se measure karke (

0

L

tak).

L

kya hai?

Box ki width — do walls ke beech distance; problem mein sirf ek length scale.

ψ (x)

kya hai, plain words mein?

Position

x

par particle ki wave ki height; positive ya negative ho sakti hai.

ψ

khud probability kyun nahi ho sakti?

Yeh negative (ya complex) ho sakti hai;

∣ ψ ∣^{2} \geq 0

paane ke liye tumhe uska size square karna hoga, phir ek region par integrate karna hoga.

∣ ψ ∣^{2}

general mein aur yahan kya matlab rakhta hai?

General mein

ψ^{*} ψ

(possibly complex number ka size-squared); yahan

ψ

real hai, isliye yeh bas

ψ^{2}

hai.

Normalisation condition kya kehti hai aur kyun?

\int_{0}^{L} ∣ ψ ∣^{2} d x = 1

— particle certainly box mein kahin hai.

\frac{d ^{2} ψ}{d x ^{2}}

kaun si physical quantity measure karta hai?

Curvature — wave kitni tezi se bend karti hai; zyada curvature matlab zyada kinetic energy.

Kinetic-energy term minus sign kyun carry karta hai?

Sine/cosine axis ki taraf wapas curve kaarte hain, isliye

ψ^{''}

ka sign

ψ

se opposite hota hai; minus sign use flip karke positive energy cost banata hai.

Sine/cosine ke liye second derivative special kyun hai?

\frac{d ^{2}}{d x ^{2}} sin (k x) = - k^{2} sin (k x)

— yeh same shape flip aur scaled karke return karta hai.

Infinite square well ke liye

V (x)

kya hai?

Andar

0

(

0 < x < L

) aur walls par ya unse bahar

\infty

, isliye

V (0) = V (L) = \infty

aur

ψ = 0

wahan.

Boundary conditions kya hain aur woh kyun matter karti hain?

ψ (0) = ψ (L) = 0

; woh zyaadatar shapes ko reject karti hain aur wahan quantisation paida hoti hai.

Wavelength

λ

kya hai aur box mein fit hone se iska kya relation hai?

Ek full wave ki length; box puri half-waves ki whole number hold karta hai,

L = nλ /2

Quantum number

n

kya count karta hai?

Allowed wave mein arches ki sankhya;

n = 1, 2, 3, \dots

(kabhi

0

ya fraction nahi).

h

aur

ℏ

kya hain?

h = 6.63 \times 1 0^{- 34}

J·s Planck's constant hai;

ℏ = h /2 π

TISE mein use hone wala reduced version hai.

Kya tum TISE term by term padh sakte ho?

Kinetic (

- \frac{ℏ ^{2}}{2 m} ψ^{''}

) + potential (

V ψ

) = total (

E ψ

), ek wave ke liye likha gaya.

Connections

The Hinglish parent note
Schrödinger Equation (TISE) — woh master equation jise yeh symbols assemble karte hain.
Wavefunction and Born Rule — $ψ$ , $∣ ψ ∣^{2}$ aur normalisation ka matlab.
Standing Waves on a String — "sirf certain waves fit hoti hain" idea ke peeche classical picture.
Heisenberg Uncertainty Principle — kyun ek trapped particle kabhi perfectly still nahi ho sakta.
Quantum Harmonic Oscillator — ek alag trap $V (x)$ , alag levels.
Quantum Tunnelling and Finite Well — kya badal jaata hai jab walls finite hoti hain.
Quantum Dots — ek real box jise tum build kar sakte ho, exactly inhi levels ka use karke.