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lecture_15 - CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4...

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6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well 6.6 Simple Harmonic Oscillator 6.7 Barriers and Tunneling CHAPTER 6 Quantum Mechanics II Quantum Mechanics II I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that?” because you will get “down the drain” into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. - Richard Feynman

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Quantum Mechanics The fundamental difference between classical (or Newtonian) mechanics (CM) and quantum mechanics (QM) lies in what they describe. In QM the kind of certainty about the future characteristics of CM is impossible because the initial state of a particle cannot be established with sufficient accuracy (Uncertainty Principle). The quantities whose relationships QM explores are probabilities . For example, Bohr theory asserts that the radius of the electron’s orbit in a ground-state hydrogen atom is always exactly a 0 = 5.3 × 10 -11 m. QM states that a 0 = 5.3 × 10 -11 m is the most probable radius.
Wave Function The quantity with which QM is concerned is the wave function Ψ of a body. While itself has no physical interpretation, the square of its absolute magnitude | | 2 evaluated at a particular place at a particular time is proportional to the probability of finding the body there at that time. The problem of QM is to determine Ψ for a body when its freedom of motion is limited by the action of external forces. Wave functions are usually complex, =A + iB . The probability density | | 2 = * = A 2 + B 2 , always a positive real quantity. Normalization: Ψ -∞ +∞ 2 dV = 1

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Well-Behaved Wave Functions The wave function Ψ must be continuous and single-valued everywhere. The partial derivatives / x, / y, / z must be continuous and single-valued everywhere. The wave function must be normalizable, which means that must go to 0 as x →± ∝ , y , z in order that over all space be a finite constant. Ψ 2 dV Probability : for a particle restricted to motion in the x direction, the probability of finding it between x 1 and x 2 is given by P x 1 x 2 = Ψ x 1 x 2 2 dx
Partial Derivatives Suppose we have a function f(x,y) of two variables, x and y, and we want to know how f varies with only one of them, say x. f

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This note was uploaded on 04/15/2008 for the course PHY 361 taught by Professor Alarcon during the Spring '08 term at ASU.

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lecture_15 - CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4...

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