Complete the following table for these simple quantum mechanical models. You don't need to memorize the...

  

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Complete the following table for these simple quantum mechanical models. You don't need to memorize the equations but you should become familiar with them. Being familiar with these simple models will help as we build up more complicated models in module 8. Each model will require you to fill in and/or annotate 5 rows of the table according to the directions in bold under the "model" column. Model Variables (Give definitions) Hamiltonian (Identify terms as kinetic or potential energy) Energy Levels Wavefunctions Quantum numbers (Give possible values) Energy Level Diagram (Sketch in approximate energy levels) Physical Systems Particle-in-a-box (1D) Harmonic Oscillator X: x: m: m or : L: k: h d =- + V(x) 2mdx h d 1 = 2dx +kx 0; 0xL V(x)=;otherwise nh En 8mL 4n(x)= =sin nx L v = hw (v + 2) = hv (v + 2) E hw|v+ V = NH, (a/2x)e a= 1/2 (k\ ax 2 (Give examples where these models could be applied) Variables Model m: R: I: (Give definitions) Dimensions Hamiltonian (Identify terms as kinetic or potential energy) Particle-on-a-sphere/Rigid Rotor Particle-on-a-ring m or : R: h d 2mRdo Energy Levels Emi = m 21 Wavefunctions Quantum numbers (Give possible values) Energy Level Diagram (Sketch in approximate energy levels) Physical Systems (Give examples where these models could be applied) 1 4m (4) = -eim 2 I: 1 = 1 sin 000 , h 2m a -A 1 a sin 0- + 2 sin 0042 l(l+1)h E = 21 1,m, (0, 0) = (0, 4) Complete the following table for these simple quantum mechanical models. You don't need to memorize the equations but you should become familiar with them. Being familiar with these simple models will help as we build up more complicated models in module 8. Each model will require you to fill in and/or annotate 5 rows of the table according to the directions in bold under the "model" column. Model Variables (Give definitions) Hamiltonian (Identify terms as kinetic or potential energy) Energy Levels Wavefunctions Quantum numbers (Give possible values) Energy Level Diagram (Sketch in approximate energy levels) Physical Systems Particle-in-a-box (1D) Harmonic Oscillator X: x: m: m or : L: k: h d =- + V(x) 2mdx h d 1 = 2dx +kx 0; 0xL V(x)=;otherwise nh En 8mL 4n(x)= =sin nx L v = hw (v + 2) = hv (v + 2) E hw|v+ V = NH, (a/2x)e a= 1/2 (k\ ax 2 (Give examples where these models could be applied) Variables Model m: R: I: (Give definitions) Dimensions Hamiltonian (Identify terms as kinetic or potential energy) Particle-on-a-sphere/Rigid Rotor Particle-on-a-ring m or : R: h d 2mRdo Energy Levels Emi = m 21 Wavefunctions Quantum numbers (Give possible values) Energy Level Diagram (Sketch in approximate energy levels) Physical Systems (Give examples where these models could be applied) 1 4m (4) = -eim 2 I: 1 = 1 sin 000 , h 2m a -A 1 a sin 0- + 2 sin 0042 l(l+1)h E = 21 1,m, (0, 0) = (0, 4)