Foundations — Fermi-Dirac statistics — fermions, Fermi energy

2.4.17 · D1 · Physics › Thermodynamics & Statistical Mechanics (Advanced) › Fermi-Dirac statistics — fermions, Fermi energy

Ye page har symbol, word, aur picture ko build karta hai jis par parent note depend karta hai — un cheezon se shuru karke jo ek curious 12-saal-ka baccha pehle se jaanta hai (seats, temperature, probability) aur bilkul wahin khatam karta hai jahan parent note shuru hota hai. Yahan kuch bhi assume nahi kiya gaya; agar parent ne kuch use kiya, to hum use pehle yahan earn karte hain.

0. Mental model: ek stadium mein seats

Ek stadium imagine karo jahan har seat ek quantum state hai — ek specific "slot" jisme ek particle baithne ki permission hai, jise uski momentum aur spin jaisi cheezein label karti hain. Seats height ke hisaab se arrange hain: neeche ki seats = low energy, upar ki seats = high energy.

Figure s01 — kya dekhna hai: har chhota box ek seat hai; page par uski height uski energy hai. Neeche ki teen rows lavender shaded hain aur har ek mein ek coral dot hai: woh seats filled hain. Upar ki pale grey rows empty hain. Dashed mint line water line hai — neeche full aur upar empty ke beech ki boundary. Dhyan karo ki sabse thande temperature par filling bilkul sharp cut hoti hai: full, full, full, phir achanak empty. Yahi sharp cut ek nazar mein Fermi energy ki poori kahani hai.

Do tarah ke guests hote hain, aur woh bilkul alag behave karte hain:

• Fermions antisocial hote hain — ek seat par zyada se zyada ek.
• Bosons gregarious hote hain — ek seat par koi bhi number pile ho sakta hai.

Hum yahan sirf fermions ki parwah karte hain. Yahi "one per seat" rule is poore topic ka beej hai.

1. Energy $\epsilon$ — ek seat ki height

Topic ko iska isiliye zarurat hai: fermions seats lowest energy se upar ki taraf fill karte hain, isliye hamen seats ko height ke hisaab se rank karna aana chahiye. Symbol $\epsilon$ bas "is waqt jis seat ki baat kar rahe hain uski height" hai.

2. Temperature $T$ aur Boltzmann constant $k_B$

Temperature sirf convert karne ke baad energy-per-jiggle hai. Yeh conversion ek constant ka kaam hai:

Room temperature $T=300$ K par, $k_{B}T\approx 0.025\text{eV}$ — copper ke $7\text{eV}$ se kaafi chhota. Yeh yaad rakhna: isiliiye metals fermion sense mein "cold" hain even 300 K par.

3. Rate of change — derivative ka concept

Exponential se milne se pehle hamen ek chhota vocabulary word chahiye, kyunki exponential ki sabse clean definition isi ka use karti hai.

Hamen iska yahan kyun zarurat hai: kuch functions kisi formula se nahi balki kitni fast grow karte hain — se best describe hote hain, aur exponential uska champion example hai.

4. Exponential $e^x$ — nature energies ko kaise weight deti hai

Aage ke formulas $e^{\text{something}}$ se bhare hain. Ise use karne se pehle, hum ise earn karte hain.

Figure s02 — kya dekhna hai: lavender curve $e^x$ hai. Teen markers matter karte hain. $x=0$ par coral dot height $1$ par baitha hai — the break-even point. Iske left (negative $x$) mein curve floor ko hug karti hai, $0$ ki taraf shrink hoti hai. Iske right (positive $x$) mein woh rocket ki tarah upar jaati hai. Key visual habit: sirf exponent ka sign batata hai ki value tiny hai ya huge, kuch compute karne se pehle.

Ab hum exponent ka sign padhna jaante hain: negative ⇒ $e^x\to 0$, positive ⇒ $e^x\to \infty$. Yeh habit yaad rakhna — agle do sections mein yeh on/off switch ban jaata hai is baat ka ki seat filled hai ya nahi.

5. Chemical potential $\mu$ — (moving) water line

Topic ko iska isiliye zarurat hai: ek fixed number of electrons ke saath, kuch cheez decide karni chahiye ki filling kitni upar jaati hai. Woh "kitni upar" $\mu$ hai. Poori kahani ke liye Chemical Potential dekhein. Hum $\mu$ se abhi milte hain, koi bhi occupation formula likhne se pehle, kyunki $\mu$ woh reference point hai jiske against har seat ki energy measure ki jaati hai.

6. Occupation probability $f(\epsilon )$ build karna

Ab hamare paas har ingredient hai — seat ki energy $\epsilon$, thermal energy $k_{B}T$, exponential $e^x$, aur water line $\mu$ — toh ab hum woh ek formula build kar sakte hain jis par poora topic tika hai.

Top aur bottom ko $e^{+(\epsilon -\mu )/k_{B}T}$ se multiply karo tidy karne ke liye:

Ab, $f$ defined aur derived hone ke baad, exponent $x=\frac{\epsilon -\mu }{k_{B}T}$ ke sign se iska behaviour padho — "seat water line se kitni door baithti hai, thermal shakes mein":

Toh: negative exponent ⇒ high occupancy; positive exponent ⇒ suppressed occupancy. Exponent ek "water line se upar distance" gauge hai, aur iska sign up/down switch hai is baat ka ki seat filled hai ya nahi.

7. Floating water line — $\mu$ $T$ ke saath kaise drift karta hai

Ab ki $f$ exist karta hai, hum precisely keh sakte hain ki water line kaise move karta hai.

8. Density of states $g(\epsilon )$ — har height par kitni seats

Seats height ke hisaab se evenly spread nahi hain. Ek stadium mein floor ke paas kum seats ho sakti hain aur mid-height par zyada.

Iska formula likhne se pehle hamen do plain physical quantities name karni hain jo isme hain:

Figure s03 — kya dekhna hai: lavender curve $\sqrt{\epsilon }$ ki tarah upar uth rahi hai, yeh $g(\epsilon )$ hai — higher energy par zyada seats. Mint curve $f(\epsilon )$ hai, left par $1$ ke paas aur dashed $\mu$ line se guzarke right par $0$ par drop karti hai. Coral shaded region unka product $fg$ hai — actually filled seats. Picture ko ek multiplication ke roop mein padho: jahan bhi zyada seats aur zyada fill hone ki chance dono overlap karein, waahan zyada electrons milenge; woh overlap coral hump hai.

9. Momentum labels: $\mathbf{p}$, $\mathbf{k}$, $k_F$

Parent note seats momentum space mein count karta hai. Yahan vocabulary hai.

Figure s04 — kya dekhna hai: har dot ek allowed momentum $\mathbf{k}$ hai (clarity ke liye 2-D mein dikhaya; actually 3-D). Lavender circle ke andar coral dots filled hain; bahar grey dots empty hain. Circle ki radius, slate arrow se marked, $k_F$ hai. $T=0$ par boundary ek crisp circle hai — wahi sharp cut jo stadium water line mein tha, ab momentum space mein draw kiya. Har dot secretly do electrons rakhta hai (spin up + down), jo section 8 ka $g_s=2$ hai.

10. Spin — woh label jo fermions ko fermions banata hai

Topic ko iska isiliye zarurat hai: exclusion rule ek add-on nahi hai; yeh electrons ki half-integer spin se forced hai jo unke combined wavefunction ko sign flip kara deta hai jab aap dono ko swap karo. Woh sign flip do-in-one-seat ko mathematically impossible banata hai.

Foundations topic ko kaise feed karte hain

Map ko teen stages ke roop mein padho, top se bottom tak arrows follow karte hue. Stage 1 (the rule): ek quantum state ek seat hai, aur half-integer spin Pauli exclusion force karta hai, jo har seat ko $n=0$ ya $n=1$ par cap karta hai. Stage 2 (the probability): energy, thermal energy $k_{B}T$, exponential, chemical potential aur woh occupation cap ek machine mein daalo, aur Fermi-Dirac probability $f$ nikalta hai. Stage 3 (the payoff): $f$ ko zero temperature par cool karo Fermi energy pane ke liye, aur $f$ ko density of states se multiply karo probabilities ko real electron count mein convert karne ke liye.

Equipment checklist

Khud test karo — sirf jawab dene ke baad reveal karo.