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The “classically forbidden regions” are where … a. a particle’s total energy is less than its kinetic energy b. a particle’s total energy is greater than its kinetic energy c. a particle’s total energy is less than its potential energy d. a particle’s total energy is greater than its potential energy Answer is c. Today’s class: • 1 more thing in 1-d Quantum harmonic oscillator • Back to 3-d • Pauli exclusion principle One last example for the 1D Schrödinger equation In past classes we did: • Free particle: V=0 everywhere • Particle in rigid box: V=0 inside, V=∞ outside • Particle in non-rigid box: V=0 ins., 0<V<∞ outside Next: • Harmonic oscillator: V(x) x2 One last 1D example: The harmonic oscillator Classical harmonic oscillator: Mass ‘m’ experiences restoring force F = -kx (x: displacement from equilibrium point). In QM: don’t want to deal with forces. What can we do? Can derive corresponding PE function: x F = -kx 1 2 U ( x) F ( x' )dx' kx 2 0 x Why is this important? Vibration in molecules!! (Used for molecular detection) x 1 2 U ( x) kx 2 U(x) Potential close to the ground-state looks very similar to x2 x (Å) Real time molecular detection Real time molecular detection Simple harmonic oscillator All we have to do is to solve the Schrödinger equation with a parabolic potential: 2 d 2 ( x ) 1 2 kx ( x ) E ( x ) 2 2m dx 2 (… or a lengthy calculation...) (“Hermite polynomials”) Energy levels Simple harmonic oscillator (cont.) Probability density (“Where is the particle most likely to be found”) Harmonic oscillator Iodine spectrum Absorption (%) Atmospheric greenhouse effect Notice the CO2 absorption is saturated. The details of the interactions are key to understanding the effects of more CO2 in the atmosphere. That’s all about the quantum harmonic oscillator Up next: Brief review of energy eigenstates and superposition states. Review: “Particle in rigid 3D box” 2 2 2 2 2 2 2 ( x, y, z ) V ( x, y, z ) ( x, y, z ) E ( x, y, z ) 2m x y z V(x,y,z) = 0 inside; ∞ outside. Separation of variables approach: (x,y,z) = X(x)Y(y)Z(z) Solutions: X ( x) A sin k x x , c a 2 2 2 k n , Ex n , n 1,2,3.... 2 a 2ma and similar for Y(y) and Z(z). And the total energy is: Or for a cube (a=b=c): E Ex E y Ez E E0 ( nx2 n 2y nz2 ) , E0 2 2 2ma 2 b Degeneracy Sometimes, there are several solutions with the exact same energy. Such solutions are called ‘degenerate’. E = E0(nx2+ny2+nz2) Degeneracy of 1 means “non-degenerate” Pauli exclusion principle. • Particles come in two types (there are basically only two possibilities) – This is a consequence of relativity and quantum together. The two types are Fermions and Bosons (we’ll come back to this) Enrico Fermi Satyendra Nath Bose Wolfgang Pauli Pauli exclusion principle. • No two identical fermions can be put into the same quantum state. They are, in a way, antisocial. Wolfgang Pauli E1 Stay out E0 Pauli exclusion principle. • You can put them in another energy state (takes more energy) • You can instead change their internal quantum numbers (if they have them). • Electrons do have 1 internal number that can be +1/2 or -1/2 so 2 of them can get into a state. Wolfgang Pauli E1 Stay out E0 Electron Spin. • The internal number that can be switched in electrons is called spin. • It is an angular momentum that can only 1 1 take on two quantized states + ℏ or − ℏ 2 2 • Two electrons can occupy any given state (one with “up spin” and one with “down spin” • This is different for different particles (bosons love to party) • Usually we use arrows E0 not tiny faces. Imagine a 3D cubic box of sides L x L x L. What is the degeneracy and how many electrons can be put into the a state with energy equal to the first excited energy? 2 2 2 E E0 ( n x n y n z ) Degeneracy of ground state Number of electrons you can put in there. L a) b) c) d) 1, 2 1,1 3, 6 3, 3 L L 1st excited state: 2,1,1 1,2,1 1,1,2 : all same E2 = 6 E0 So 3 degeneracy. And each “state” can take +1/2 and -1/2 electrons so 6 total What is the minimum energy required to put 4 electrons into the 3-d rigid box? We are given 4 electrons with zero energy to start with. a) b) c) d) e) 12 E0 39 E0 18 E0 0 E0 24 E0 2 in the ground state and 2 in the next state.