Question

We have potential of 0, 0 < x < a. V(x) = Joo, elsewhere. a Find the ground state energy and the first and second excited states, if an electron is enclosed in this potential of size a = 0.100 nm. b Find the ground state energy and the first and second excited states, if a 1 g metal sphere is enclosed in this potential of size a = 10.0 cm. c Are the quantum effects important for both systems? Explain why or why not. d Use the uncertainty principle to estimate the velocities of the electron and the metal sphere.

Answer 1

We have the Potential V(x) = o<xsa else where { WC Fig: Potential of infinite well Problem inside the write energy eigen value We can well E. Con (x) -(1) - Ĥ [wn] = 2 m of enifinities the forbidden -t? g? Wy(1) dx² outside the well we can not write the proper Eq" because We can still sel the values of wn(x) at boundaries ora. Physically we expect Wha=0 in region since forbidden region 4(x) = 0 In fact we know ein particle has zero probability of being there. Then if we weite 4(x) in terms of the energy eigenfunctions en Wn (x) at boundaries we have condition wn (0) = Wn(a)= o Sole of W) will be like Wn(x) = A sinlanx) +B cos(kha) Wh(0) = AXO + Bx1=0 B=0 4(x) = we have

Wn(a) = A sin(kna) = 0 In a =nT kna ni then inturns sets allowed values This condition of energy En À 12 22 am Ima² Thr En = an²E2 where to = ħ²712 ali zina2 ground state energy no quantum number gr E. a -10 10 m o.1 nm mass of electron -3) 9.1x10 kg. me= TS h = 6.62 x 1034 Ground state enersy of electron h2 (6.62 X10 342 ema? 8x9.1X100 x 10 1032 E. - -68+31+ 20 10 43.8244 72.8 17 J Eo = 0.6019 x 10 ( E. = 37.62 eV

n=2 first excited State of electron E,= 4 Eo Ei= 150.49 eV state E = 9 to second excuted n=3 = 338,61 eV b as 10 cm Oolm 103 kg. Eo = 2 Ground state of metal sphere h2 8ma? (6.62 x 16-24) 8x10 3x (0.1)? u 3.82 44 x 10 8 -63 J -68+3+2 こ -44 ev - 5.478 10 3:42 x 10 First excited state A=2 Ei= 13.6x1544 eV second excited state = 30.81x10 -u4 eV

here in this example they quantum effects are important in small dimensions size, and mass. are important in first case of electron. In second the energy is every low compared to other energy (gravitational) Second case con not be defined only by quantum physics. There classical forces atto Important, case metal sphese using d) Velocity of electron & Uncertainty for election from uncertainity Principle AXDP 3 h UTT -34 O b = CA a unox こ AP こ 6.62 x 10 GAT ( lolom 6.62 x 10 -34 +10 40 x 9.1x1031 34+10+31 0.0579 410 5.798 iOS Tov 105 m/s order of electron velouty Similarly very high for metal sphere sysh 6.62.810 -34 munAX 103 x 4X 3-14 x 10 - 0.5270 xlo DU 0.5270 X 1030 -3443+1 misee.