Tutorial 2 (Solution) 1. An electron is confined to a one-dimensional, infinitely deep potential energy well of width L = 100 pm.

Transcription

1 Seester 007/008 SMS0 Modern Pysics Tutorial Tutorial (). An electron is confined to a one-diensional, infinitely deep potential energy well of widt L 00 p. a) Wat is te least energy te electron can ave? Te least energy of electron corresponds to te least quantu nuber, wic is n for te ground state of te electron. Tus, 8L or n 7.7 ev 8 4 ( s) ( ) ( 8)( 9. 0 kg)( 00 0 ) b) How uc energy ust be transferred to te electron if it is to ake a quantu jup fro its ground state to its second excited state? nergy transferred to te electron fro its ground state to its second excited state, 8L 8L ( ) () ( ) 7 8L or 0 ev

2 Seester 007/008 SMS0 Modern Pysics Tutorial c) If te electron gains te energy for te jup fro energy level to energy level by absorbing ligt, wat ligt wavelengt is required? λ c 4 8 ( s)(.0 0 /s) d) Once te electron as been excited to te second excited state, wat wavelengt of ligt can it eit by de-excitation? Te electron can jup directly to te ground state fro ligt of wavelengt λ by eitting te Te energy difference for te jups and are and Te wavelengt of ligt eitted in te first of tese jup fro n to n is ( ) λ Te wavelengt of ligt eitted in te first of tese jup fro n to n is ( ) λ.0 0 8

3 Seester 007/008 SMS0 Modern Pysics Tutorial. Consider te Bor odel of te doubly ionzed Litiu ato (Litiu as Z ) a) Wat are te tree lowest allowed energies of te electron? Call tese,,. Use ev units. Z ev.49 ev.4 ev, 0.6 ev,.6 ev b) Wat are te different energies of potons tat can be eitted wen te electron undergoes transitions starting fro te tird level, to lower energy levels?.6 (.4) 08 ev :.6 ( 0.6) 7 ev : 0.6 (.4) 9.8 ev c) Wic of te preceding transitions results in te longest wavelengt poton? Wat is tat longest wavelengt? Transition give te longest wavelengt of te poton c λ 40 ev.n 7 ev 7.9 n

4 Seester 007/008 SMS0 Modern Pysics Tutorial d) Wen tis longest wavelengt ligt is incident on a certain etal, te potoelectric effect occurs. Te work function of te etal is.9 ev. Wat is te stopping potential for tis case? ev stop K ax Φ ( 7.0.9) 5. ev ev or V stop 5. Volts. A olecule of ass.0 x 0-4 kg is oving along at a speed of 00 ±0 /sec, were te ±0 /sec is te statistical spread or uncertainty in deterination of its velocity or v 0 /sec. Te olecule is soewat localized in space and is described by a travelling wave packet solution to te Scrödinger wave equation centred around x 0 at t 0 and oving in te positive x direction. a) Wat is te iniu uncertainty x in te deterination of its positions? Te iniu uncertainty x ~ v x ~ 4 4.s x, (.0 0 kg)( 0 /s) x ~ 5.5 0

5 Seester 007/008 SMS0 Modern Pysics Tutorial b) Wat is te de Broglie wavelengt of tis olecule? de Broglie wavelengt, λ λ v 4 (.0 0 kg)( 00 /s) s c) Wat is te (non-relativistic) kinetic energy of tis olecule? Non-relativistic kinetic energy, K K v (.0 0 )( 00 /s) 0 d) Using part(c), deterine te frequency, f of tis travelling wave packet. Frequency of travelling wave packet f s 0 4. s e) Does te frequency. f in part(d) satisfy te relation λf v wit te given central value of v 00 /s. Sow your calculation. λ f 49.5 /s ~ 50 /s or / 00 /s

6 Seester 007/008 SMS0 Modern Pysics Tutorial wic is ½ of te central value of v f) Suppose tis electron as a wavefunction given at tie t 0 by ψ ( x,0) A sin( ( π / λ) x). Wat is te probability density at x λ 4? (Write your answer in ters of A and λ.) Probability density at x λ 4, ψ ( x λ 4,0) A A A π λ sin λ 4 π sin 4. In wat respect did classical pysics fail to account for te structure of te ato? Answer Classical pysics failed to account for te stability of an ato. According to classical pysics an electron revolving round te nucleus sould eit radiation (electroagnetic waves). Tus a revolving electron loses energy and oves into a saller orbit and ultiately it sould fall into te nucleus. 5. Can a ydrogen ato absorb a poton of energy greater tan te binding energy (B.) of te ato? Answer Te B. of te ydrogen ato is nuerically equal to te energy of te lowest state, i.e, -.6 ev. Tis is te energy, wic ust be supplied to it in order to reove its electron fro te lowest state (level) to a state of zero energy. We know tat te igest energy level of te ato corresponds to n and ence it is 0. Above tis level, tere are energy states of te syste consisting of unbound electron plus te ionised ato. Te total energy of an unbound electron is not quantized and is positive. Obviously, any energy > 0 is possible for te electron and te energy state fors a continuu. Wen a ydrogen ato receives a poton of any energy wic is greater tan its B.. of.6 ev, te ato absorbs te poton and te electron of te ato passes fro its discrete ground state to a continuu state.