Wavefunctionsallowed inquantummechanicsY(x)P(x)no1.Single-valued;nob)+a)2.Continuous;P(x)3.Nowhereinfinite(exception:Dirac delta-function);4.Piecewise continuousfirstP(x.yPoxvederivatives;5. SSquare-integrable(exceptionplane wave)vesY(0)(0)1, 3, 5 required by Bornh)00g)interpretation;2 and 4 required by kinetic(x)P(x)nooperator (2nd derivative of )yesi
Wave functions allowed in quantum mechanics 7 1. Single-valued; 2. Continuous; 3. Nowhere infinite (exception: Dirac delta-function); 4. Piecewise continuous first derivatives; 5. Square-integrable (exception: plane wave) 1, 3, 5 required by Born interpretation; 2 and 4 required by kinetic operator (2nd derivative of Ψ)
Kineticenergyvs.potentialenergyIV-O(c)(a(b)(d)(a)SinceV=O,E=T.ForhigherT,ismorewiggly,which meansthat入isshorter. (Since is periodic fora free particle,入is defined.)(b) As Vincreases from left to right, becomes less wiggly. (c)-(d) is most wigglywhereVislowestandTisgreatest
Kinetic energy vs. potential energy 8 (a) Since V = 0, E = T . For higher T , ψ is more wiggly, which means that λ is shorter. (Since ψ is periodic for a free particle, λ is defined.) (b) As V increases from left to right, ψ becomes less wiggly. (c)–(d) ψ is most wiggly where V is lowest and T is greatest
Quantization,energylevelsanddegeneracy:ananalogyof stable statesofa chaireaEsnormalpositionnon the sideThedegreeofdegeneracyequalstwoinclinedE1onthesupportEoDifferent degrees of degeneracy of energy levels lead to densityof states (DOS);UsingkasxaxisleadstobandstructureArranging x axis according to reaction leads to reaction pathway9
Quantization, energy levels and degeneracy: an analogy of stable states of a chair on the support Potential energy of the chair inclined The degree of degeneracy equals two on the side normal position 9 • Different degrees of degeneracy of energy levels lead to density of states (DOS); • Using k as x axis leads to band structure • Arranging x axis according to reaction leads to reaction pathway