Homework (Kittel 7.1) Density of orbitals in one and two dimensions.

Transcription

1 Hoework 8.. Kittel 7. esity of orbitals i oe a two iesios. a Sow tat te esity of orbitals of a free eletro i oe iesio is were is te legt of te lie. b. Sow tat i two iesios, for a square of area, iepeet of. a p ±. Hee p { { a so { b p pp Hee {

2 . Kittel 7. Eergy of a relativisti feri gas. or eletros wit a eergy >>, were is te rest ass of te eletro, te eergy is give by p, were p is te oetu. or eletros i a ube of volue te oetu is of te for ultiplie by z y x + +, exatly as for te orelativisti liit. a Sow tat i tis extree relativisti liit te eri eergy of a gas of eletros is give by were /. b Sow tat te total eergy of te grou state of te gas is e geeral proble is treate by. Jutter, Zeitsrift fur Pysik 7, 98. a p. Hee p p 8 b a. Hee Eq Eq

3 Eq a Eq give. Kittel 7. Pressure a etropy of egeerate eri gas. a Sow tat a feri eletro gas i te grou state exerts a pressure p I a uifor erease of te volue of a ube every orbital as its eergy raise: e eergy of a orbital is proportioal to or. b i a expressio for te etropy of a feri eletro gas i te regio τ <<. otie tat σ as τ. a K. Hee p b B k C C S

4 S C k B S k B. Kittel 7. Ceial potetial versus teperature. Explai grapially wy te iitial urvature of µ versus τ is upwar for a ferio gas i oe iesio a owwar i tree iesios igure 7.7. Hit: e a urves are ifferet, were is give i Proble. It will be fou useful to set up te itegral for, te uber of partiles, a to osier fro te graps te beavior of te itegra betwee zero teperature a a fiite teperature. e β µ + µ t K,. or K, below µ will be exite to > µ. µ will sift to esure te uber of partiles reove fro < µ equals te uber of partiles at > µ. If ostat, te sift of µ will be approxiately zero. If ireases as ireases, te uber of partiles is weigte ore for > µ ta < µ. erefore µ ereases. Siilarly, if ereases as ireases, te µ will sift upwar. or -i,, µ > µ. or -i,, µ < µ.. Kittel 7. iqui He as a feri gas. e ato He as spi I a is a ferio. a Calulate as i able 7. te feri spere paraeters v, a for He at absolute zero, viewe as a gas of oiteratig ferios. e esity of te liqui is.8g. b Calulate te eat apaity at low teperatures << a opare wit te experietal value C,89k B as observe for <.K by. C. erso, W.

5 Reese, a J. C. Weatley, Pys. Rev., 9 96; see also igure 7.8. Exellet surveys of te properties of liqui He are give by J. Wilks, Properties of liqui a soli eliu, Oxfor, 967, a by J. C. Weatley, "ilute solutios of He a He at low teperatures," eria Joural of Pysis, 6, e priiples of refrigerators base o He - He are reviewe i Capter o ryogeis; su refrigerators proue steay teperatures ow to.k i otiuously atig operatio. a b.67 7 g. g ρ.8..6 k k erg. k B K v k s. C k B k B.9 k B.k B 6. Kittel 7.6 ass-raius relatiosip for wite warfs. Cosier a wite warf of ass a raius R. et te eletros be egeerate but orelativisti. a Sow tat te orer of agitue of te gravitatioal self-eergy is G R, were G is te gravitatioal ostat. If te ass esity is ostat witi te spere of raius R, te exat potetial eergy is G R. b Sow tat te orer of agitue of te kieti eergy of te eletros i te grou state is

6 8 R H R were is te ass of a eletro a H is te ass of a proto. Sow tat if te gravitatioal a kieti eergies are of te sae orer o agitue as require by te virial teore of eais, R g. If te ass is equal to tat of te Su g, wat is te esity of te wite warf? e It is believe tat pulsars are stars opose of a ol egeerate gas of eutros. Sow tat for a eutro star R 7 g. Wat is te value of te raius for a eutro star wit a ass equal to tat of te Su? Express te result i k. gr r. Hee r < R : gr G R r r > R : gr G r a so r < R : G r R r G R r > R : r G r g ρ r R r G r G r R R R R G R R G R G R R b e e e e H H a R. Hee

7 e H HR R H. R H et g e. e G R. R H R. G H.8 G H g g. Hee R esity: R g, ie R k.. Hee ρ R g e or eutro stars, replae te eletro ass wit te proto ass H. a so G R R G H R H R H R G H H H G H Hee R for a eutro star is H tat for a wite warf. R R g. So R G H H H G H H 7 g k.