Zero-point Energy Minimum energy corresponds to n=1 n=0 => n(x)=0 for all x => P(x)=0 => no electron in the well zero-point energy never at rest! Uncertainty principle: if x=L/, then px > ħ/(x) =  ħ /L => E > (px )2/2m =  2 ħ 2/2mL2 = h2/(8mL2)

Problem An electron is trapped in an infinite well which is 250 pm wide and is in its ground state. How much energy must it absorb to jump up to the state with n=4? Solution: standing waves => n(/2)=L E=p2/2m= h2/2m 2 = n2h2/8mL2 E4-E1= (h2/8mL2)(42 -1) =15(6.63x10-34)2/[8(9.11x10-31)(250x10-12)2=90.3 eV

P(x) =|(x)|2 is probability/unit length Problem 16.107 L01 Mar 20/02 An electron is trapped in an infinite well that is 100 pm wide and is in its ground state. What is the probability that you can detect the electron in an interval of width x=5.0 pm centered at x= (a) 25 pm (b)50 pm (c)90 pm ? P(x) =|(x)|2 is probability/unit length P(x)dx = probability that electron is located in interval dx at x L=100 x 10-12 m (x)=(2/L)1/2 sin(x/L) P(x) x = (2/L)sin2(x/L) x =(1/10) sin2(x/L) a) P(x) x =.1 sin2(/4) = .05 b) P(x) x =.1 sin2(/2) = .1 c) P(x) x =.1 sin2(.9) = .0095

Problem A particle is confined to an infinite potential well of width L. If the particle is in its ground state, what is the probability that it will be found between (a) x=0 and x=L/3 (b) x=L/3 and x=2L/3 (c) x=2L/3 and x=L? If x is not small, we must integrate! (a) (b) (c)

Solution Total probability of finding the particle between x=a and x=b is

Solution (cont’d) Since cos(2y)= cos2(y)-sin2(y)=1-2sin2(y) we have sin2(y) = (1/2)(1 - cos(2y))

Solution(cont’d) For a=0, b=L we have L/L - (1/2)[sin(2)-sin(0)] = 1 (a) a=0, b=L/3 1/3 - (1/2)[sin(2/3)-sin(0)]= .195 (b) a=L/3, b=2L/3 1/3 - (1/2)[sin(4/3)-sin(2/3)]= .61 (c) a=2L/3, b=L 1/3 - (1/2)[sin(2)-sin(4/3)]= .195

Infinite Potential Well   Nodes at ends => standing waves (x)=0 outside

Electron in a Finite Well U0 <  L Only 3 levels with E < U0 states with energies E > U0 are not confined all energies for E > U0 are allowed since the electron has enough kinetic energy to escape to infinity Quantization of energy

No longer a node at x=0 and x=L Electron has small probability of penetrating the walls Area under each curve is 1 oscillation Exponential decay

Harmonic Oscillator Potential P(x) 0 everywhere