Quantum Mechanics

Quantum Mechanics - Quantum Mechanics and Atomic Structure...

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1 Chem 6BH: Quantum Mechanics Quantum Mechanics and Atomic Structure Chem 6BH: Quantum Mechanics Wave-Particle Duality Energy (electromagnetic radiation) behaves like waves (diffraction, wavelength, frequency) but also possesses particle behavour (photons in photoelectric effect). Invoke wave- particle duality of EMR to explain. If energy can exhibit both particle and wave natures, what about matter? deBroglie: “Matter Waves” Large items have very small wavelengths Electrons are small enough however to exhibit measurable wave properties ! = h m e v = h p
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2 Chem 6BH: Quantum Mechanics Wave-Particle Duality: Problems Classical mechanics: particles have trajectories and can precisely be described according to location ( x ) and linear momentum ( p ) Quantum mechanics: can’t think like this: waves are spread out as they oscillate Duality denies the ability of knowing the trajectory of particles ( complementarity ) Heisenberg Uncertainty principle: Important for wave equations ( )( ) ( )( ) ( )( ) p p 4 or p 2 x m x h x ! " " = " " # " " # ! Chem 6BH: Quantum Mechanics Wavefunctions ( Ψ ’s) How do we revise the mathematical description of matter considering duality exists? Schr ödinger : replace precise trajectory of a particle by a wavefunction ( Ψ ): a function that varies in value with position. Born : physical interpretation? Let Ψ 2 = probability density ( Ψ 2 )(volume) = probability Ψ (or Ψ 2 ) = 0 represents a node Schr ödinger Equation ( ) ( ) ( ) ( ) 2 2 2 d V E or E 2 d x x x x H m x ! ! ! ! ! + = = !
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3 Chem 6BH: Quantum Mechanics Schrödinger Equation The SWE relates the second derivative of Ψ to the value of Ψ at each point; impossible to solve exactly (except simple cases) One simple solution: particle in a box Solution of SWE using this yields: and, since Δ E = h ν and Δ E = E n+1 - E n ( ) 2 sin x x n x x n x x L L ! " # $ = % & ( L = length of box n = 1,2,3… x = distance between 0 and L E n = n 2 h 2 8 mL 2 ( ) 2 2 2 1 8 e n h hc h m L ! " + = = Since n is an integer, E is quantized with discrete values of E Chem 6BH: Quantum Mechanics SWE: Particle in a Rectangular Box What if the box is 3-dimensional? Solution of SWE using this yields: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 sin , , x y z x y z n n n n n n n n x x L L x y z x y z ! " " " " " # $ = % & ( = ) ) E n x n y n z = h 2 8 m n x 2 L x 2 + n y 2 L y 2 + n z 2 L z 2 ! " # # $ % & & E n x n y n z = h 2 8 mL 2 n x 2 + n y 2 + n z 2 ! " $ %
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4 Chem 6BH: Quantum Mechanics Standing Waves Chem 6BH: Quantum Mechanics Zero Point Energy and Wavefunction Shape ZPE: An implication arising out of the solution of the SWE for energy: a particle can not have zero energy (n 0) and therefore can never be perfectly still In line with uncertainty principle; purely a QM phenomenon---too small to be important for big particles Finally: Shape of the wavefunction: using Ψ 2 we get probability density in the box; nodes are seen as constrictions
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