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Quantum Mechanics: Tunneling

Quantum Mechanics: Tunneling. Physics 123. Wave Function. (Wave function Y of matter wave) 2 dV =probability to find particle in volume dV . In 1-dimentional case probability P to find particle between x 1 and x 2 is

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Quantum Mechanics: Tunneling

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  1. Quantum Mechanics:Tunneling Physics 123 Lecture XVI

  2. Wave Function • (Wave function Y of matter wave)2dV=probability to find particle in volume dV . In 1-dimentional case probability P to find particle between x1 and x2 is • Unitarity condition (probability to find particle somewhere is one): • Schrödinger equation predicts wave function for a system • System is defined by potential energy, boundary conditions Lecture XVI

  3. Properties of Wave function • Wave function respects the symmetry of the system • For example if the system is symmetric around zero • x-x • then wave function is either symmetric or antisymmetric around zero: Lecture XVI

  4. Count knots 0 knots in the box symmetric 1 knot in the box antisymmetric 2 knots in the box Symmetric N-th state: (N-1) knots in the box N-odd – symmetric N-even - antisymmetric Lecture XVI

  5. Particle in a finite potential well U0 I II III x 0 L • Particle mass m in a finite potential well: • U(x)=0, if 0<x<L, • U(x)=U0, if x<0-or-x>L • Boundary conditions: Lecture XVI

  6. Particle in a finite potential well U0 I II III x 0 L • Inside the box (region II) • Possible solutions: sin(kx) and cos(kx) Lecture XVI

  7. Particle in a finite potential well U0 I II III x 0 L • Outside the box (regions I and III) • Possible solutions: exp(Gx) and exp(-Gx) Lecture XVI

  8. Wave functions 2 knots in the box Symmetric 1 knot in the box antisymmetric 0 knots in the box symmetric Lecture XVI

  9. Probability to find particle at x • Particle can be found outside the box!!! • E=U0+KE • KE must be positive • KE=E-U0, but U0>E • Energy not conserved?! • Fine print: Heisenberg uncertainty principle • Time spent outside the box is less than h/2p divided by energy misbalance, then energy non-conservation is “virtual”=undetectable Lecture XVI

  10. Probability to find particle at x Consider electron mc2=0.5 MeV with U0=2eV, E=1eV How much time does it spend outside the box? Characteristic depth of penetration x0: exp(-Gx)=exp(-x/x0) Lecture XVI

  11. You can go through the wall!!! • It’s called tunneling effect • Probability of tunneling P=|y|2=exp(-2GL), L-width of the barrier • Transmission coefficient T~P=exp(-2GL) Lecture XVI

  12. Problem 39-34 • A 1.0 mA current of 1.0 MeV protons strike 2.0 MeV high barrier of 2.0x10-13m thick. Estimate the current beyond the barrier. p Lecture XVI

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