Modifying the Schrödinger Equation 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

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

Separation of Variables We need to solve this equation – try separation of variables? Substitute it in, divide by wave function none ??? x only y only z only 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

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

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

In math notation,  and  are swapped 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

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

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

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

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 an addition to the potential The effective potential is just this term Very similar to how classical mechanics solves this problem

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

Solving the angle equation (2) Will any l work? We want Y to be finite 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)

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)

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?