Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

no text concepts found

Transcript

Recap I Lecture 40 © Matthias Liepe, 2012 Recap II Today: • More quantum mechanics • Heisenberg’s uncertainty principle • 1-D infinite square well • Finite square well, tunneling… Werner Heisenberg Infinite 1-D Square Well: Wave functions and Quantized Energy n x n ( x) A sin , for 0 x L. L h2 n 2 E1 , n 1, 2, 3, . En n 2 8mL 2 | 1(x |2) 11.1 0.9 11(x ( x)) 0.7 0.5 0.8 Energy 0.3 0.3 -0.2 0.1 L1 0 -0.1 L1 0 -0.7 2 | 1(x 21.1|) -1.2 2 (x ( x)) 0.9 0.8 0.7 16E1 0.5 0.3 -0.2 0.3 L1 0 0.1 -0.1 -0.7 -1.2 L1 0 2 | 1(x 31.1|) 33(x ( x)) 0.8 E1 0.7 0.5 L1 0 0.3 0.1 -0.7 -0.1 -1.2 Position x 9E1 4E1 0.9 0.3 -0.2 25E1 L1 0 Position x Electron (de Broglie Waves) in an Infinite 1-D Square Well Which set of energy levels corresponds to the larger value of well size L? A. B. C. Set 1 Set 2 The spacing of energy levels does not depend on L Finite 1-D square well: For an electron in a potential well of finite depth we must solve the time-independent Schrödinger equation with appropriate boundary conditions to get the wave functions. 10 10 8 8 6 6 4 4 2 2 0 -2 -1 0 En (x) V(x) -0.5 0 x/L 0.5 1 -2 -1 En (x) V(x) -0.5 0 x/L 0.5 1 Quantum Tunneling Particle can “tunnel” through a barrier that it classically could not surmount Example: Scanning Tunneling Microscope (STM) Example: Scanning Tunneling Microscope (STM) Spin • Electrons (and many other particles) have an intrinsic property (like mass & charge) called spin angular momentum, S, (or just called spin). • The component of S measured along any axis is quantized. • For electrons: S ss 1 h 2 , • Have spin where s = 1/2 is the spin quantum number of an electron • Spin component along any axis can only have one the h following two values: S z ms 2 with spin magnetic quantum number ms = ½ or ½. The Pauli Exclusion Principle (Wolfgang Pauli, 1925): No two electrons confined to the same trap can have the same set of values for their quantum numbers (n, ms …). In a given trap, only two electrons, at most, can occupy a state with the same (single-particle) wave function (solution of the Schrödinger equation). One must have ms ½ and the other must have ms ½.