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Quantum Mechanics in

Quantum Mechanics in. 3-D. Modifying the Schr ö dinger Equation. The real universe has three space dimensions: x , y , and z What needs to change about this formula? Wave function needs to be more complicated The momentum p is now a vector. Time Independent 3D Schr ö dinger.

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Quantum Mechanics in

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  1. Quantum Mechanics in 3-D Modifying the Schrödinger Equation • The real universe has three space dimensions: x, y, and z • What needs to change about this formula? • Wave function needs to be more complicated • The momentum p is now a vector

  2. Time Independent 3D Schrödinger If the potential is independent of time • We have reduced our wave function from four variables to three (good) • But it’s still a partial differential equation (bad) • Need to find tricks to make this problem solvable • The interpretation of the wave function is pretty much the same • The amplitude squared is the probability density • To find probability in a region, integrate • Have to normalize like before

  3. Separation of Variables ??? none x only y only z only • We need to solve this equation – try separation of variables? • Substitute it in, divide by wave function • This method rarely works, because naturally occurring problems are rarely set up in Cartesian coordinates • Two problems we will solve: • Free particle • Particle in a 3D box

  4. Free particle in 3D none x only y only z only • First term is pure function of x, but nothing else has x in it • Therefore, this term must be independent of x • Same argument applies to the other two terms • We can easily solve all these equations

  5. Particle in a 3D box Lz • We get same differential equations as before: • This time it is more useful to get real solutions: • Must vanish at x = 0 and x = Lx • Same type of solutions for y and z • Need to normalize wave functions Ly Lx

  6. Spherical Coordinates • Very few problems have “Cartesian symmetry” • Look at hydrogen-like atom • Many problems have spherical symmetry • Independent of angles z r sin • Switch from Cartesian to Spherical Coordinates • r is the distance from the origin to the point •  is the angle compared to the z-axis •  is the angle of the projected “shadow”compared to the x-axis r cos  r y  r sincos r sinsin x Note that  = 0   = 2 In math notation,  and  are swapped

  7. Derivatives in Spherical Coordinates • WARNING: This is nasty! • We need to write derivatives in terms of the new coordinates • Think of x, y, and z as functions of r, ,  and use the chain rule Work, work, . . . Let’s rewrite Schrödinger’s equation in spherical coordinates now

  8. Schrödinger’s Eq. in Spherical Coords. Assume potential depends only on r, and call the mass  • Change to spherical coordinates on the left • Multiply result by r2 • Now we can try separation of variables,this time in new coordinates • Substitute in • Divide by the wave function, and bring first term on left over to the right

  9. Schrödinger’s Eq. in Spherical Coords. (2) Left side is independent of r Right side is independent of  and  Both sides must be constant – call them -L2 Multiply first equation by –Y Multiply second equation by R/2r2

  10. The Problem Broken in Two • No dependence on V(r) • Can be solved for all spherically symmetric problems • No more partial derivatives! • Looks like 1D-Schrödinger • The L2 term looks like anaddition to the potential • The effective potential is just this term • Very similar to how classical mechanics solves this problem

  11. Solving the angle equation • Strategy: • Guess some solutions • Rotate them and find some more

  12. Solving the angle equation (2) • Will any l work? • We want Y to befinite • l  0 • We want it to be continuous • l is an integer • l= 0,1, 2, 3, … Note that  = 0   = 2 • Rotate them and find some more • Example: 180º rotation around x - axis • Work, work, until you find as many as you can • Normalize them (more on this later)

  13. Spherical Harmonics • The functions you get this way are called spherical harmonics • They arise in any problem with spherical symmetry • The angular solutions are always the same • Look them up, don’t calculate them • What do l and m really mean? • Consider angular momentum operator • z-angular momentum is m • Total angular momentum squared is 2(l2+l)

  14. Sample Problems An electron in a spherically symmetric potential has a total angular momentum squared of 62. If we measure the angular momentum around the z –axis, what are the possible outcomes? An electron in a spherically symmetric potential has a total angular momentum quantum number l < 3. How many possible pairs (l,m) are there?

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