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

2.3.10 · D5 · Physics › Modern Physics › Particle in a box — solving TISE, energy levels, wavefunctio

Neeche har reveal Statement ::: reasoning format mein hai. Statement padho, decide karo, phir uncover karo.

True or false — justify

Wavefunction

ψ_{n} (x) = 2/ L sin (nπ x / L)

ek probability hai.

False —

ψ

ek amplitude hai; probability density

∣ ψ_{n} ∣^{2}

hoti hai aur actual probability pane ke liye tumhe ise kisi region par integrate karna padta hai.

Box ki length

L

ko double karne se har energy level double ho jaata hai.

False — kyunki

E_{n} \propto 1/ L^{2}

hai,

L

double karne se har level

4

se divide ho jaata hai (

2

se nahi); levels girte hain aur paas aa jaate hain.

Ground state

n = 1

mein zero energy hoti hai kyunki particle "abhi excited nahi hua hai."

False —

E_{1} = h^{2} /8 m L^{2} > 0

; ek confined particle kabhi perfectly still nahi beth sakta, yeh uncertainty ki maang hai, ise zero-point energy kehte hain.

Energy levels equally spaced hote hain, jaise ladder ki rungon ki tarah.

False — spacing badhti hai:

E_{n + 1} - E_{n} = (2 n + 1) E_{1}

, toh gaps

3 E_{1}, 5 E_{1}, 7 E_{1}, \dots

hote hain. Equal spacing Quantum Harmonic Oscillator mein hoti hai, box mein nahi.

ψ_{n}

ke adjacent nodes ke beech particle move karta hai; node par woh momentarily ruk jaata hai.

False — yeh classical thinking hai. Node woh jagah hai jahan amplitude zero hai, matlab particle wahan kabhi mila hi nahi jaata; koi "beech mein move karna" nahi hota —

∣ ψ ∣^{2}

ek static probability landscape describe karta hai.

Wavefunction

x = 0

aur

x = L

par zero honi chahiye lekin walls ke thoda bahar nonzero ho sakti hai.

False — infinite walls ke bahar

V = \infty

hai, isliye wahan

ψ = 0

har jagah hoti hai; boundary values

ψ (0) = ψ (L) = 0

us bahar-wali-zero ko continuously match karne se aate hain.

Bahut bade

n

ke liye, probability density flat ho jaati hai aur classical jaisi dikhne lagti hai.

True — bahut saare wiggles packed hone se,

∣ ψ_{n} ∣^{2}

ko kisi bhi chhoti window par average karo toh woh uniform classical distribution ke paas pahunch jaata hai (ek particle jo constant speed se bounce karta hai har jagah equally likely hota hai). Yeh correspondence principle hai.

Kyunki

ψ_{n}

negative ho sakta hai, kuch jagahon par "negative probability" hoti hai.

False — probability density

∣ ψ_{n} ∣^{2} \geq 0

hamesha hoti hai;

ψ

ka sign interference ko affect karta hai jab waves combine hoti hain lekin kabhi negative probability nahi deta.

Usi box mein ek bhaari particle ke energy levels zyada widely spaced hote hain.

False —

E_{n} \propto 1/ m

hai, isliye bhaare particle ke levels neeche aur zyada tightly packed hote hain; bhaari objects ke liye classical limit zyada jaldi approach hoti hai.

Spot the error

ψ (0) = A sin 0 + B cos 0 = 0

, toh

A = 0

Galat term ko haata diya.

sin 0 = 0

aur

cos 0 = 1

hai, isliye

ψ (0) = B

hai; boundary condition

B = 0

force karti hai,

A = 0

nahi.

A

ko rakhna hi hamein live particle deta hai.

sin (k L) = 0

deta hai

k L = nπ

for

n = 0, 1, 2, \dots

, toh ground state

n = 0

hai."

n = 0

include karna error hai —

k = 0

ψ \equiv 0

hota hai (particle kahin bhi nahi). Lowest physical state

n = 1

hai.

"Kyunki sine aur cosine dono

ψ^{''} = - k^{2} ψ

solve karte hain, final wavefunction mein dono terms rakhe jaate hain."

Boundary condition

ψ (0) = 0

cosine ko eliminate kar deti hai (

B = 0

). General solution mein dono hain, lekin is box ka physical solution mein nahi.

"Normalisation deta hai

A^{2} \cdot L = 1

, toh

A = 1/ L

Integral

\int_{0}^{L} sin^{2} (nπ x / L) d x = L /2

hai,

L

nahi, kyunki

sin^{2}

average hokar

\frac{1}{2}

deta hai. Correct result hai

A = 2/ L

E_{2} - E_{1} = 2 E_{1}

kyunki hum level 1 se level 2 jaate hain."

Energy

n^{2}

ke anusaar scale hoti hai, toh

E_{2} - E_{1} = (4 - 1) E_{1} = 3 E_{1}

. Gap

2^{2} - 1^{2} = 3

hai, labels ka difference nahi.

"Walls ko finite karne se sirf energies thodi si lower hongi aur kuch nahi badlega."

Finite walls ke saath wavefunction barrier mein leak karti hai (tunnelling), sirf finitely many bound states hote hain, aur

ψ

walls par exactly zero nahi rehti — yeh ek real qualitative change hai.

ψ_{3}

ke box ke andar 3 nodes hain."

ψ_{n}

n - 1

interior nodes hote hain, isliye

ψ_{3}

ke 2 interior nodes hain (

x = 0, L

par dono ends ke boundary zeros interior nahi maane jaate).

Why questions

Confinement energy ko quantise kyon force karta hai?

Walls

ψ = 0

dono ends par demand karti hain, aur sirf wahi sine waves is condition ko satisfy karti hain jinke half-wavelengths poori sankhya mein fit hote hain — jaise string par standing waves, har fit ek allowed

k

hai, isliye ek allowed energy.

Ground-state energy nonzero kyun hai jabki box ke andar

V = 0

hai?

Zero-energy state ke liye

ψ =

constant chahiye (koi curvature nahi), lekin woh dono walls par vanish nahi ho sakta; jo sabse chhoti wave fit hoti hai usmein pehle se curvature hai, aur curvature hi

ψ^{''}

term mein kinetic energy hai.

Geometrically, zyada

n

ka matlab zyada energy kyun hota hai?

Zyada

n

= zyada wiggles = tez curvature

ψ^{''}

, aur TISE bada

∣ ψ^{''} ∣

badi kinetic energy se joddta hai — toh zyada bends zyada energy maangti hai.

Negative

n

ko hum kyun phek dete hain?

sin (- nπ x / L) = - sin (nπ x / L)

hai, sirf overall sign flip; kyunki sirf

∣ ψ ∣^{2}

physical hai, negative label usi state ko describe karta hai.

n = 1

particle ko left third mein paane ki probability

1/3

se kam kyun hai?

Density

∣ ψ_{1} ∣^{2}

beech mein hump karti hai aur edges ke paas thin hoti hai, isliye bahar wala third apne "fair share" se kam hold karta hai jo ek flat distribution deti.

Box ko bada karne se particle zyada classically kyun behave karta hai?

Bada

L

levels ko shrink aur crowd karta hai (

E_{n} \propto 1/ L^{2}

), isliye discrete spacing typical energies ke muqable mein negligible ho jaati hai — ladder ek continuum mein blur ho jaata hai.

k

ko "wave number" kyun kehte hain, koi unnamed constant kyun nahi?

Kyunki

k = nπ / L

nikalta hai

2 π / λ_{n}

ke barabar, seedha count karta hai ki wave ke kitne radians per metre fit hote hain — yeh standing wave ki spatial frequency hi hai.

ψ

walls par continuous honi chahiye lekin slope jump karne ki permission kyun hai?

ψ

ki continuity zaroori hai taaki probability well-defined rahe; slope sirf isliye jump kar sakta hai kyunki potential infinite hai — finite well mein continuous slope bhi demand hota.

Edge cases

Har

n

ke liye

x = 0

aur

x = L

par exactly

ψ (x)

kya hai?

Yeh sab

n

ke liye zero hai — yeh enforced boundary nodes hain; har allowed standing wave in do zeros ko share karta hai.

Agar hum

n = 0

try karein, toh "wavefunction" kya describe karti hai?

Kuch nahi —

ψ \equiv 0

har jagah matlab particle ke kahin bhi hone ki koi probability nahi, yeh physical state nahi hai, isliye

n = 0

discard kar diya jaata hai.

Ek macroscopic ball (

m \sim

1 kg) ko 1 m box mein rakhein toh quantised energies kyun nahi dikhte?

Spacing

\sim h^{2} / (8 m L^{2})

absurdly tiny ho jaata hai (

\sim 1 0^{- 67}

J), kisi bhi measurable energy se bahut neeche, isliye levels perfectly continuous lagte hain.

n \to \infty

ki limit mein nodes ki sankhya aur classical resemblance ka kya hota hai?

Nodes (

n - 1

) bina kisi bound ke badhte rehte hain aur fine-scale density average hokar uniform ho jaati hai, ek classical particle jo fixed speed se bounce karta hai se match karti hai — correspondence principle in action.

Center

x = L /2

par exactly,

ψ_{2}

node hai ya antinode?

Node hai —

sin (2 π \cdot \frac{1}{2}) = sin π = 0

, isliye

n = 2

state ke beech mein zero probability hai, unlike

n = 1

jo wahan peak karta hai.

Agar

L \to \infty

(walls infinity tak hata di jaayein) toh ground state energy ka kya hoga?

E_{1} = h^{2} /8 m L^{2} \to 0

aur levels ek continuum mein merge ho jaate hain — particle free ho jaata hai, unbound plane-wave picture recover hoti hai bina kisi quantisation ke.

Connections

Particle in a box — solving TISE, energy levels, wavefunctions — woh parent derivation jise yeh traps test karte hain.
Wavefunction and Born Rule — kyun $∣ ψ ∣^{2}$ , $ψ$ nahi, probability hai.
Heisenberg Uncertainty Principle — woh reason ki ground state zero energy nahi ho sakti.
Quantum Harmonic Oscillator — equal-spacing contrast trap.
Quantum Tunnelling and Finite Well — jab walls finite hoti hain toh kya toot jaata hai.
Standing Waves on a String — quantisation ke liye classical analogue.
Quantum Dots — jahan real world mein box-jaisa confinement dikhta hai.