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PHYS 2D PROBLEM SESSION 2012/5/17 Quiz 3 is graded Pick up quiz 3 today or next Tuesday Regrade request: 1 week 6.1 Determine which wave functions are physical Conditions: Continuous (e) Single-valued (d) Value is finite (a) Valid ones: b & c 6.16 Electron in a box, L=0.3 nm (x=0 to L), find the probability Pn the particle is found within [0, 0.1nm] for n=1 & n=100 1/2 ψn(x)=Asin(nπx/L), A=(2/L) n=1, Pn=0.196; n=100, Pn=0.332; n=∞, Pn=1/3 Correspondence principle: we get classical result with large n Classical: same probability everywhere, P=1/3 6.23 Square well with V(x<0)=∞, V(0<x<L)=0, V(x>L)=U, particle has energy E, find ψ(x) Regions I, II, III for x<0, 0<x<L, x>L 1. Write out Schrodinger eq & ψ for each region Region I: ψ(x)=0 Region II: Free space Region III: 6.23 2. Apply boundary conditions Wave function is continuous at x=0 & x=L dψ/dx is continuous at x=L Wave function is finite at x=∞ At x=0: , so A=0 At x=∞: is finite, so C=0 At x=L: Dividing, we get tan(kL)=-k/k’ 6.23 tan(kL)=-k/k’ , 2 cot (kL)=(U-E)/E 2 2 cot (kL)+1=1/sin (kL) 2 2 2 (kL) /sin (kL)>1, so (kL) [(U-E)/E+1]>1 2 2 2 2 (kL) [(U-E)/E+1]=(kL) U/E=2mL U/ħ >1 2mL2U/ħ2>1 to have a solution 2 2 2 2 No solution when 2mL U/ħ <1, or U<ħ /2mL 6.24 Show that is a solution to Schrodinger’s equation for quantum oscillator , terms and Equate terms , ψ(x) is the n=1 state Normalization: We can find C in terms of ( ) 6.32 Find <x> & <x2> & Δx for ground state of quantum oscillator ,