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Chapter 34 Quantum Mechanics

Chapter 34 Quantum Mechanics. 1927 Solvay Conference. Wave function. Wavelength of de Broglie wave:. “Displacement” of wave → wave function Ψ. Energy density of EM wave. Number density of photon:. Probability of finding photon:. Normalization condition:. 3. Probability wave.

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Chapter 34 Quantum Mechanics

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  1. Chapter 34 Quantum Mechanics

  2. 1927 Solvay Conference

  3. Wave function Wavelength of de Broglie wave: “Displacement” of wave → wave function Ψ Energy density of EM wave Number density of photon: Probability of finding photon: Normalization condition: 3

  4. Probability wave |Ψ|2dVat a certain point represents the probability of finding the particle within volume dV about the given position and time. 4

  5. *Coherent & de-coherence de-coherence Interference between coherent wave functions If we can measure which slit the electron passes Measurement disturbs the state of particle! Superposition state & Schrödinger cat 5

  6. Uncertainty principle Measurement disturbs the state of particle Observe an electron by a photon: Always an uncertainty in position or momentum The uncertainty principle 6

  7. Heisenberg uncertainty principle W. K. Heisenberg Nobel 1932 The position and the momentum of a particle can not be precisely determined simultaneously. 1) It is a natural result of duality 2) A criterion of classical and quantum physics 3) The uncertainty is inherent rather than caused by the limitation of instruments or methods 7

  8. Other forms of uncertainty principle Inherent principles in quantum mechanics Microscopic particles will not stay at rest! Zero point energy (at 0K) & Dirac sea 8

  9. A quantum criterion Example1: Calculate the uncertainty in position for (a) a 150-g baseball and (b) an electron both measured with 50 m/s to a precision of 0.3%. Solution: (a) It’s too small, so we can ignore the limitation (b) QM is necessary! 108m/s to precision of 0.3% ? 9

  10. Film Electron beam Single slit Diffraction of electron Thinking: An electron beam (each with p) falls on a single slit, show that the central bright fringe satisfies the uncertainty relation 10

  11. Probability vs determinism Probability is inherent in nature, not a limitation on our abilities to calculate or to measure Copenhagen interpretation of quantum mechanics “God does not play dice.” ——A. Einstein “Anyone who is not shocked by quantum theory has not understood it” —— N. Bohr 11

  12. Schrödinger equation (1) An equation to determine the wave function Ψ A free particle (E, p) moves along x axis can be also treated as a wave: Wave function in general form (complex value) : 12

  13. Schrödinger equation (2) Nonrelativistic free particle: in general case, considering the potential energy 13

  14. Schrödinger equation (3) 3D time-dependent S-equation: E. Schrödinger Nobel 1933 → Hamilton operator Basic, can’t be derived, check by experiments 14

  15. Schrödinger equation (4) Time-independent case: Time-independent Schrödinger equation: 1) E → energy 2) Stationary state / eigenstate 15

  16. Solution of S-equation Solve the S-equation to analyze quantum systems 1) Each solution represents a stationary state 2) The system may be in a superposition state 3) The wave function of system should be continuous, finite and normalized. 16

  17. Infinitely deep square well potential Particle in an infinitely deep square well potential or “rigid box” Trapped in the well, can’t escape Classical case: Free motion until collision Equal probability at any point 17

  18. Solution by S-equation (1) Can’t escape, so 0 < x < L, U(x) = 0, so: Or: General solution is 18

  19. Solution by S-equation (2) ψ must be continuous, so Quantized! The wave function: 19

  20. Quantum properties (1) • The energy of particle is quantized n: quantum number of state 2) The minimum energy is not zero! Zero point energy Microscopic particles will not stay at rest! 20

  21. Quantum properties (2) 3) Figure of wave function & probability distribution Like a standing wave Notice: The probability is not constant! 21

  22. Quantum properties (3) 4) de Broglie wavelength? In accord with the figure of wave function! 5) Uncertainty principle: 22

  23. Electron in the well Example2: Calculate the energy of ground state and first exited state for an electron trapped in an IDSWP of width L = 0.1nm. Solution: The ground state (n=1) has energy First exited state: λ of photon if jumping from n = 2 to n = 1 ? 23

  24. Probability in well Example3: In the state of (a) Where does the particle have maximum probability densities? (b) What is the probability to find the particle in region 0 < x < L/4 ? Solution: (a) So: 24

  25. (b) What is the probability to find the particle in region 0 < x < L/4 ? 25

  26. Probability for other ψ Homework: A particle trapped in a special well has a wave function , and C is a constant. What is the probability to find the particle in region 0 < x < L/3 ? 26

  27. Finite potential well If the well potential is not infinitely deep? Particle can get out even if E < U0 ! 27

  28. U U0 2 3 1 x L 0 Tunneling through a barrier Particle can also penetrate a thin barrier ( E < U0 ) This effect is called “quantum tunneling” Tunneling probability: 28

  29. 238 U  Th + He 234 4 *Applications of tunneling 1) Radioactive decay 2) Scanning Tunneling Microscope (STM) Invented 1981, Nobel 1986 29

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