Quantum mechanics I

Difference b/w Classical and Quantum Mechanics

👉 Classical Mechanics

1. Macroparticles – Ye bade objects ke liye use hota hai (jaise car, planets, ball, etc).

2. Certainty – Yahan sab kuch exact predict kiya ja sakta hai. Agar aapko initial position aur velocity pata ho toh future motion predict kar sakte ho.

3. Continuous – Energy ya motion continuous mana jata hai (koi breaks nahi).

👉 Quantum Mechanics

1. Microparticles – Ye chhoti cheezon ke liye apply hota hai (jaise electrons, protons, atoms).

2. Uncertainty – Yahan exact position aur momentum ek sath predict karna possible nahi (Heisenberg’s Uncertainty Principle).

3. Discontinuous – Energy discrete hoti hai, packets (quanta/photons) ke form mein.

Blackbody kya hota hai?

Blackbody = ek aisi ideal body jo apne upar girne wali sari radiation (light, heat, etc.) ko absorb kar leti hai.

Matlab wo kuch reflect nahi karti, kuch transmit nahi karti, sirf absorb karti hai.

Baad mein wo absorbed energy ko apne temperature ke hisaab se emit karti hai.

👉 Is emitted radiation ko hi blackbody radiation bolte hain.

📌 Example socho: ek perfect black box jisme chhota sa hole ho → jo bhi light andar jayegi, wo phas jaayegi aur absorb ho jayegi. Bahar aane ka chance almost zero hai. Ye ek ideal blackbody ka model maana jata hai.

Experiment by Lummer & Pringsheim

Ye dono German scientists the (1890s ke around).

Unhone blackbody radiation ko measure karne ke liye ek accurate setup banaya.

Ek hollow cavity blackbody banayi jisme chhota hole tha. Is hole se jo radiation bahar aati thi usko study kiya.

Phir unhone intensity of radiation vs wavelength ke graphs (curves) nikale different temperatures par.

👉 Result kya mila?

1. Har temperature par ek specific wavelength pe radiation ka maximum hota hai.

2. Jaise-jaise temperature badhta hai, maximum wali wavelength chhoti hoti jati hai (yaani hotter body → shorter wavelength → zyada bluish light).

Yahi baad mein Planck, Wien aur Rayleigh-Jeans laws ko samjhane ke liye foundation bana.

---

⚡ Easy Formula yaad rakhne ke liye (jo Lummer–Pringsheim curves se mila):

Higher T → peak shift towards shorter λ

(Example: garam loha lal chamakta hai, aur aur zyada heat karne pe sa

fed/bluish chamakta hai).

Graph Explanation (Lummer & Pringsheim)

Graph ka axis:

Vertical (Y-axis) → Energy radiated (Eλ)

Horizontal (X-axis) → Wavelength (λ)

Curves: T₃ > T₂ > T₁ (yaani T₃ sabse zyada temperature hai, T₁ sabse kam).

---

🔑 Characteristics of the Curves

1. For each temperature → ek specific wavelength hoti hai jahan radiation maximum hota hai.

Matlab har curve ka ek peak point hota hai (maximum energy).

Isko λmax (wavelength of maximum emission) bolte hain.

2. Position of maxima (peak) shift karta hai:

Agar temperature decrease hota hai → maxima longer wavelength pe shift karta hai.

Agar temperature increase hota hai → maxima shorter wavelength pe shift karta hai.

👉 Ye directly Wien’s Displacement Law ko support karta hai:

λ_{\text{max}} \times T = \text{constant}

3. Higher temperature → curve ke peaks zyada pronounced aur broad hote hain.

Matlab zyada temperature pe energy distribution zyada uniform hoti hai across wavelengths.

Lower temperature pe curve narrow hota hai, energy concentration ek chhote wavelength range me hoti hai.

---

🔥 Real-Life Example:

Jab lohe ka tukda garam karte ho:

• Pehle red chamakta hai (longer wavelength, lower T).

• Fir zyada garam karoge toh orange → yellow → white → bluish chamkega (shorter wavelength, higher T).

• Yehi curve shift ka real example hai.

---

👉 Easy Hack to Remember:

T ↑ → λmax ↓ (temperature badhe toh wavelength chhoti hoti hai).

Hotter objects look bluer, colder objects look redder.

⚡Failure of Classical Theory on Blackbody Radiation

Classical physicists ne Blackbody Radiation ko explain karne ke liye alag–alag formulas diye. Do main attempts the:

---

1. Wien’s Law

Wien ne ek expression derive kiya thermodynamic considerations se.

Uska formula tha:

E_\lambda d\lambda = A_1 \lambda^{-5} e^{-A_2/\lambda T} d\lambda

jahan A₁ aur A₂ constants hain.

👉 Kya mila?

Ye formula shorter wavelength (lower λ) region mein achhe se kaam karta tha.

Lekin longer wavelength region mein ye poora fail ho gaya.

---

2. Rayleigh–Jeans Law

Rayleigh aur Jeans ne energy distribution ke liye equipartition of energy principle apply kiya.

Unhone blackbody ke radiation ka ek formula nikala:

E_\lambda d\lambda = \frac{8 \pi k T}{\lambda^4} d\lambda

👉 Kya mila?

Ye expression longer wavelength region ke liye sahi aaya.

Lekin shorter wavelength region mein ye fail ho gaya.

---

⚠ Problem:

Shorter wavelength region (UV region) mein Rayleigh–Jeans formula kehta tha ki energy infinite ho jayegi.

Isko bola jata hai “Ultraviolet Catastrophe” 🚨

---

🔑 Conclusion (Notes ka main point):

Classical expressions partial solutions dete the (koi sirf long λ pe sahi, koi sirf short λ pe sahi).

Lekin ek bhi formula pura spectrum (all wavelengths) explain nahi kar paaya.

Isi wajah se kaha gaya:

“Blackbody radiation cannot be explained by classical methods.”

Aur yahin se Quantum Mechanics ka janm hua, jab Planck ne energy quantization ka concept diya.

---

👉 Easy Memory Trick:

• Wien → Short λ (works for small range, fail for long).

• Rayleigh–Jeans → Long λ (works for long, fail for short → infinite energy problem)

Planck’s Quantum Theory (1900)

Ye theory ne classical physics ki failings ko solve kiya, specially black body radiation ka problem jisko classical laws explain nahi kar pa rahe the.

---

Main Postulates of Planck’s Quantum Theory

1. Energy ka emission/absorption continuous nahi hota, balki discrete packets (bundles) ke form mai hota hai.

• Ye discrete packets ko quanta kehte hain.

• Light ke case mai ise hum photon bolte hain.

(ii) – Energy of a Quantum

Planck ne bola ki ek radiation ke ek bundle (quantum) ki energy directly uski frequency (ν) pe depend karti hai.

Formula:

E = h\nu

E = ek quantum (photon) ki energy

h = Planck’s constant (6.626 × 10⁻³⁴ Js)

= frequency of radiation

---

🔑 Matlab kya hua?

Agar frequency zyada hai → har quantum ki energy bhi zyada hogi.

Agar frequency kam hai → energy bhi kam hogi.

Yaani light ka rang (frequency) decide karta hai ki ek photon kitni “powerful” hai.

---

🔥 Example:

Red light (ν kam) → photon ki energy kam hoti hai.

Blue/UV light (ν high) → photon ki energy bahut zyada hoti hai.

Isiliye UV rays skin ko damage kar dete hain, par red light harmless hoti hai.

3.Body energy ko sirf poore quanta ke multiples mai absorb/emit kar sakti hai.

• Matlab energy levels quantized hain:

• E_n = nh\nu, \quad n = 0, 1, 2, 3, …

4. Ye quantization ka idea hi black body radiation ka spectrum sahi se explain karta hai.

Short wavelength region (high frequency) aur long wavelength region dono match kar gaye Planck ke equation se.

---

👉 Simple analogy socho:

Classical physics kehti thi ke energy ek smooth river ki tarah hai — har level pe flow karegi.

Planck ne bola: “No bhai, ye toh coins ki tarah hai — sirf poore coins (discrete packets) hi diye jaa sakte hain, change (fractions) nahi milega!”

Isi wajah se quantum mechanics ki

foundation rakhi gayi. 🚀

Planck radiation formula

Wiens law using planck radiation law

de broglie

Davissson- German exp

🔹 Background

De Broglie ne bola tha ki electrons ka dual nature hota hai → particle bhi hai aur wave bhi.

Agar electron wave hai, toh uska diffraction (tirchha bikharna) aur interference possible hona chahiye (jaise X-rays crystals se diffract hote hain).

Is baat ko experimentally prove kiya Davisson aur Germer ne 1927 mai.

---

🔹 Experiment Setup

1. Electron source:

Ek heated filament se electrons nikale.

Unko ek potential difference (40V–68V) se accelerate kiya → iska matlab unhe energy mili.

2. Collimator:

Yeh ek slit hota hai jo electron beam ko straight aur narrow banata hai.

3. Nickel Crystal Plate:

Electron beam ko nickel plate pe focus kiya gaya.

Nickel crystal ke atoms ke layers ek tarah ke "diffraction grating" banate hain.

4. Detector (Movable Collector + Galvanometer):

Nickel plate se jo electrons reflect hote hain, unko alag–alag angles pe measure kiya detector se.

Detector current (galvanometer) ke through electrons ke number ko record karta tha.

---

🔹 Observation

Electrons jab nickel plate se takraaye, toh simple reflection ke alawa unhone diffraction pattern dikhaya.

Matlab kuch specific angles pe zyada intensity mili (jaise constructive interference hota hai).

Particularly, 54 V accelerating potential pe aur 50° angle pe maximum intensity mili.

---

🔹 Explanation with De Broglie + Bragg’s Law

De Broglie ke hisaab se electron ka wavelength:

\lambda = \frac{h}{mv}

Experiment ke hisaab se jo λ nikla (≈1.65 Å) woh same tha jo Bragg’s diffraction law se aata hai for nickel crystal planes:

2d \sin \theta = n\lambda

Yani electron truly wave behave kar raha tha, jaise X-rays karte hain.

🔹 Conclusion

✔ Experiment ne de Broglie hypothesis ko prove kar diya → electron waves bhi hote hain.

✔ Yahi se Wave–Particle Duality solid ho gaya.

✔ Isi wajah se Electron diffraction ek accepted fact hai.

---

🔹 Bonus Info

Davisson aur G.P. Thomson ne 1937 ka Nobel Prize share kiya electron diffraction discover karne ke liye.

Interestingly, J.J. Thomson (G.P. Thomson ke father) ko pehle hi electron discovery ke liye Nobel mila tha! Matlab father ne electron ko "particle" dikhaya aur beta ne "wave" dikhaya 😄.