Wave functions and quantization workshop CH112 Workshop 9, Spring 2003 http://quantum.bu.edu/notes/quantummechanics/wavefunctionsandquantizationworkshop.pdf Last updated Friday, October 31, 2003 16:33:01 Copyright 2003 Dan Dill (dan@bu.edu) Department of Chemistry, Boston University, Boston MA 02215 This workshop is based on four quantum concepts. First, in the quantum world, a "particle" is represented by a wave function, y. Second, the value of the squared modulus of the wave function at each point in space,» yhxl» 2, is proportional to the probability that the particle is at that position; this is known as the Born interpretation of the meaning of y. Third, since the total probability of finding a particle somewhere is 1, the wave function satisfies - +» yhxl» 2 x = 1; this is known as the normalization condition. Fourth, wave functions are determined by the Schrödinger equation, curvature of y at x -kinetic energy at x µyat x The goal of the workshop is to use the Schrödinger equation to develop a qualitative understanding of all of the key features of wave functions and energies of quantum systems: ä that in regions of high kinetic energy wave functions oscillate rapidly and so have small wavelengths; ä that in regions of low kinetic energy that wave functions oscillate slowly and so have large wavelengths; ä that in regions of negative (!) kinetic energy, instead of oscillating, wave function diverge away from zero; ä and, finally, that the observed (quantized) energies, are just those energies for which the wave function is prevented from diverging to infinity. We will be able to do all of this just by making simple sketches of the form of the wave function required by the Schrödinger equation. It is easy to make these sketches, once we learn what the rules are, and these sketches allow us to accurately predict what wave functions look like without doing any calculations at all. Kinetic energy from potential energy and total energy The structure of atoms and molecules are determined by the balances of forces between electrons and nuclei. These forces are due primarily to Coulomb attractions and repulsions. A convenient way to represent the effect of these forces is in terms of how the potential energy changes as the positions of the electrons and nuclei change. For example, here is the potential energy experienced by an electron with orbital momentum quantum number = 2 (a d electron), in He +.
2 Wave functions and quantization EêRy 0.6 0.4 0.2 10 20 30 40 rêa 0-0.2-0.4-0.6 Potential energy experienced by an electron with orbital momentum quantum number = 2 (a d electron), in He +. Length is in units of the Bohr radius, a 0, and energy is in units of the Rydberg, Ry = 13.6 ev, the ionization energy of the hydrogen atom. The horizontal line is the total energy, -0.2 Ry. In this expression one unit of energy corresponds to one Rydberg of energy, and one unit of length corresponds to one Bohr radius. To find the kinetic energy at each position, we need to specify the total energy. This is represented by the horizontal line in the figure.
Wave functions and quantization 3 1. Make a graph of the kinetic energy of the electron in the figure. Recall that the total energy is the sum of the potential energy and the kinetic energy. Your graph should show both positive a negative values for the kinetic energy. Positive values occur in so-called classically allowed regions. Negative values occur in so-call classically forbidden regions. 2. Interpret your graph in terms the speed, v, of the electron, using the expression for kinetic energy mv 2 ê 2. Where is the speed v greatest? Where is v = 0? How does the speed differ in allowed and forbidden regions? 3. Do you suppose the probability of a particle being in a forbidden region can be different from 0? 4. The boundary between an allowed and a forbidden region is called a classical turning point. Why? Properties of wave functions in allowed regions 5. Assume potential energy is constant with a value of 0 and total energy is constant with a value of +1, in arbitrary units. Sketch the corresponding potential energy curve and 6. For the potential energy and total energy in the previous problem, assume the values of the wave function yh0l = 1 and yh0.1l = 1.1. Use the Schrödinger equation to make a 7. Assume potential energy is constant with a value of 0 and total energy is constant with a value of +4, in arbitrary units. Sketch the corresponding potential energy curve and 8. For the potential energy and total energy in the previous problem, assume the values of 9. What conclusions can you make comparing your wave functions for the two different potential curves and total energies? Are your results consistent with the de Broglie relation, l =h ê p, between wavelength and linear momentum? 10. Assume potential energy varies as -10 + x and that the total energy is 0, in arbitrary units. Sketch the corresponding potential energy curve and indicate the total energy by a horizontal line. 11. For the potential energy and total energy in the previous problem, assume the values of qualitative sketch of the wave function by predicting its value at x = 0.2, 0.3,, 7.9, 8.0. 12. What conclusions can you make comparing your wave function to those for the two different constant potential energies? Are your results consistent with the de Broglie relation, l =h ê p, between wavelength and linear momentum? Properties of wave functions in forbidden regions 13. Assume potential energy is constant with a value of 0 and total energy is constant with a value of -1, in arbitrary units. Sketch the corresponding potential energy curve and
4 Wave functions and quantization 14. For the potential energy and total energy in the previous problem, assume the values of 15. For the potential energy and total energy in the previous problem, assume the values of the wave function yh0l =-1.0 and yh0.1l =-.9. Use the Schrödinger equation to make a 16. Assume potential energy is constant with a value of 0 and total energy is constant with a value of -4, in arbitrary units. Sketch the corresponding potential energy curve and 17. For the potential energy and total energy in the previous problem, assume the values of 18. For the potential energy and total energy in the previous problem, assume the values of the wave function yh0l =-1.0 and yh0.1l =-.9. Use the Schrödinger equation to make a 19. What conclusions can you make comparing your wave functions for the two different total energies? Properties of wave functions across turning points 20. Assume that potential energy is 0 for 0 x 5, and 5 for x > 5, in arbitrary units, and assume that total energy is constant with a value of +1, in arbitrary units. Sketch the corresponding potential curve and 21. For the potential energy and total energy in the previous problem, assume the values of the wave function yh0l = 1 and yh0.1l = 1.1. Use the Schrödinger equation to make a qualitative sketch of the wave function by predicting its value at x = 0.2, 0.3,, 7.9, 8.0. 22. Assume that potential energy is the same as in the previous problem but that the total energy is constant with a value of -4, in arbitrary units. Sketch the corresponding potential curve and 23. For the potential energy and total energy in the previous problem, assume the values of the wave function yh0l = 1 and yh0.1l = 1.1. Use the Schrödinger equation to make a qualitative sketch of the wave function by predicting its value at x = 0.2, 0.3,, 7.9, 8.0. 24. What conclusions can you make comparing your wave functions for the two different total energies? Normalization and quantization 25. Assume that potential energy is +1000 for x < 0, 0 for 0 x 5, and 5 for x > 5, in arbitrary units, and assume that total energy is constant with a value of +1, in arbitrary units. Sketch the corresponding potential curve and indicate the total energy by a horizontal line. 26. For kinetic energy variation given in the previous question, assume the values of the wave function yh0l = 0 and yh0.1l = 0.1; also assume the value of the wave function at x = 5 is yh5l = 0.2 and that y =0 at two additional points between x = 0 and x = 5. Use
Wave functions and quantization 5 the Schrödinger equation to make a qualitative sketch of the wave function by predicting its value at x = 0.2, 0.3,. 27. Is your wave function normalizable? If so, why? If not, why not? 28. How would your wave function sketch change if the total energy was increased by 0.1 unit? 29. How would your wave function sketch change if the total energy was decreased by 0.1 unit? 30. Is it possible, by adjusting the total energy, to make you wave function normalizable? If so why? If not, why not? 31. Do you suppose the probability of a particle being in a forbidden region can be different from 0? Sketching wave functions with the Schrödinger Shooter The Schrödinger Shooter is a computer program that automates sketching of wave functions. The program allows you to select different kinetic energy variations, by specifying potential energy variation and the total energy. Then by varying the total energy you can find wave functions that are normalizable. We will later learn that their energies are just those that account for observed spectra of particles confined by the potential energy variation.
6 Wave functions and quantization 32. Find the lowest energy of a square well. Use File > New potential function > Square well, and then the default values width, L = 3, and height 64. How many loops (half-oscillations) does the lowest energy wave function have? 33. Find the second and third lowest energies of the square well in the previous question. How many loops (half-oscillations) do the second and third lowest energy wave functions have? 34. Discover how the energy depends on the number of loops, j, in the wave function. Test your result by predicting the value of the fourth lowest energy of the square well in the previous question. Is your prediction confirmed exactly? If not, why not? 35. How does the lowest energy change when the width, L, of the well is increased. Does the result make sense in terms of the de Broglie relation, p = h ê l, between wavelength and linear momentum? 36. Discover the proportionality between well width, L, and the lowest energy value. Test your result by predicting the values of the second and third lowest energies when the width of the well is changed from 3 to 6 to 12. Is your prediction confirmed exactly? If not, why not? 37. Combine the dependence on number of loops, j, and width of the well, L, into a single expression for the energy of a square well in terms of j and L. Test your result by predicting the value of the fourth lowest energy of a well of width L = 10. Is your prediction confirmed exactly? If not, why not? 38. Compare the wave functions in the forbidden region for j = 3 and L = 1, 3, and 6. What prediction can you make? For which case do the energy values agree most closely with your expression in the previous part. Does this result make sense? If so, why. If not, why not? 39. Do you suppose the probability of a particle being in a forbidden region can be different from 0? Homework: Schrödinger Shooter exploration of one-electron atoms The potential energy of an electron bound to a nucleus of charge +Ze is VHrL =-ÅÅÅÅÅÅÅÅÅÅ 2 Z + r 2 H + 1L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ r where length, r, in units of the Bohr radius, a 0 = 0.529 Þ, and potential energy is in units of the Rydberg, Ry = 13.6 ev, the ionization energy of the hydrogen atom. H1. Find the lowest four energies for atomic number Z = 1, 2 and 3, and orbital momentum quantum number = 0, 1 and 2. Use File > New potential function > Square well, select the values of Z and, and then find the energies. H2. Make a table of your results, and then, with pencil and paper only (it is essential that you do this by hand, and that you not use any calculational aids), discover the dependence of the energy on the number of loops in the wave function, the atomic number, and the orbital momentum quantum number. H3. Check your proposed dependence of the energy on the number of loops in the wave function, the atomic number, and the orbital momentum quantum number by predicting the four lowest energies for Z = 4 (Be 3+, and = 2, and for Z = 3 (Li 2+ ) and = 3, and then checking your predictions with the Shooter.