Review

2.

The failures of Classical Physics:  Black-body radiations Medan elektromagnetic adalah kumpulan osilator harmonik. 1 osilator = 1 frekuensi sinar = c/  Berdasarkan dalil equipartisi, yaitu pada temperatur T, nilai rata-rata dari setiap suku kuadrat di dalam ungkapan untuk energi total adalah ½ kT

Diperbaiki oleh Planck:

Heat Capacity: Classical For a solid composed of N such atomic oscillators: Giving a total energy per mole of sample: So the heat capacity at constant volume per mole is: This law of Dulong and Petit (1819) is approximately obeyed by most solids at high T ( > 300 K). But by the middle of the 19 th century it was clear that C V  0 as T  0 for solids. So…what was happening?

Einstein Uses Planck’s Work Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies [later QM showed is actually correct] Einstein (1907): model a solid as a collection of 3N independent 1-D oscillators, all with constant , and use Planck’s equation for energy levels occupation of energy level n: (probability of oscillator being in level n) classical physics (Boltzmann factor) Average total energy of solid:

Some Nifty Summing Using Planck’s equation: Now let Which can be rewritten: Now we can use the infinite sum: To give: So we obtain:

At last…the Heat Capacity! Differentiating: Now it is traditional to define an “Einstein temperature”: Using our previous definition: So we obtain the prediction:

Limiting Behavior of C V (T) Low T limit: These predictions are qualitatively correct: C V  3R for large T and C V  0 as T  0: High T limit: 3R CVCV T/  E

But Let’s Take a Closer Look: High T behavior: Reasonable agreement with experiment Low T behavior: C V  0 too quickly as T  0 !

 Efek foto Listrik  Efek Compton  Difraksi Elektron ===================================================

Contoh:

Contoh: Penafsiran fungsi gelombang Fungsi gelombang elektron pada keadaan energi terendah atom hydrogen adalah Hitunglah peluang relatif menemukan elektron di dalam volume kecil yang besarnya 1,0 pm 3 yang terletak pada (a) inti, (b) jarak a 0 dari inti!

Contoh: Fungsi gelombang atom hydrogen adalah sebagai berikut, Normalisasikan fungsi gelombang tersebut bila koordinatnya adalah koordinat bola!