CH301 Chapter 12 Notes

CH301 Chapter 12 Notes - CH301 Chapter 12 Notes: Part 2:...

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CH301 Chapter 12 Notes: Part 2: Solution to the Schrödinger Equation: The Particle in a 1-Dimensional Box Ψ n(x) = (2/L)1/2 sin(n Π x/L) This equation summarizes a set of n sinusoidal curves: Ψ 1 , Ψ2 , Ψ3 , Ψ4 , Ψ5……… Allowed energies of a particle in a 1-D box: Each solution has its own specific energy: E 1 , E 2 , E 3 , E 4 , E 5…………. n is known as a QUANTUM NUMBER. In this case, n is a positive, nonzero integer. Any other value of n would not work - it would not be a solution to the Schrödinger equation! What happens to E for any value of n if we let m or L increase? Figure 12.14 shows the energies, wavefunctions and square of the wavefunctions for the first three solutions (n=1, n=2, n=3). In class we'll discuss the third one in detail. Interpreting the solution for a particle in a 1-D box (Figure 12.14): Each wavefunction is equal to zero at x = 0 and x = L (as required by the boundary conditions) Note the energy level ‘ladder’ -- QUANTIZED energies. Sketch the plots of to Ψ3 and Ψ32 here and add the comments/notes: Remember Ψ2 gives the idea of WHERE the particle is likely to be! How does this compare to what you would expect for a macroscopic object? What do I need to know? Be able to sketch and interpret any of the first 8 or so solutions: e.g., Ψ3, or Ψ22 …. Where are the NODES. What is a Node? Where will you be most/least likely to find the particle? What does Ψ2 mean? Energy needed to jump from one level to another: (Figure 12.14) We can calculate the energy to go from Ψn 1 to level Ψn 2 : where n 1 and n 2 are two different values of n. We see that the gaps themselves, ΔE, are also inversely proportional to L (actually L2) and m. Example: A ball mass 0.5g in a 1D box 2m long: What energy is needed to go from n=4 to n=7 ? Zero Point Energy: Quantum Weirdness What about n=0? Ψ 0 does NOT exist! The smallest allowed value of n is 1. The energy of a particle in a 1-D box can never be zero! To calculate ZPE use n=1 and your values of m and L. Example: Assume we confine all of these in 1-D boxes 100cm long: - an electron - a marble - a bowling ball Which do you think has the smallest ZERO POINT ENERGY? Δ E = (n22-n12) h2 8mL2 En = n2h2
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What do I need to know? Be able to sketch the energy level diagram and interpret it: (what happens as we change n,L) Be able to do calculations of values of E for a particular wavefunction given n,m,L Be able to calculate ΔE for a particular wavefunction given two values of n, and m,L Be able to calculate the ZPE if given m,L (obviously you know what n is!!! ) Also the sort of qualitative examples we’ve done. Always remember that you should get very tiny numbers if your object is macroscopic. Why? The Schrodinger Equation for the H atom
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This note was uploaded on 09/08/2009 for the course CH 301 taught by Professor Fakhreddine/lyon during the Spring '07 term at University of Texas.

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CH301 Chapter 12 Notes - CH301 Chapter 12 Notes: Part 2:...

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