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Zero-point Energy

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!

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Zero-point Energy

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  1. 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 ħ2/2mL2 = h2/(8mL2)

  2. 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

  3. Problem • 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

  4. (b) (c) (a) 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!

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

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

  7. 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

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

  9. U0 <  0 L Electron in a Finite Well • 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

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

  11. Harmonic Oscillator Potential P(x) 0 everywhere

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