Who I am: Dan Dessau Dessau@Colorado Prof. Shepard will be back on Wed.

1. Who I am: Dan Dessau Dessau@Colorado.edu Prof. Shepard will be back on Wed.

2. In 2D: Chapter 8. The 3D Schrodinger Equation In 1D: In 3D: Instead of doing a lengthy mathematical proof we will see from various examples that this indeed seems to be the correct generalization of the 1D equation.

3. Simplest case: 3D box, infinite wall strength V(x,y,z) = 0 inside, = infinite outside. Use separation of variables y(x,y,z) = X(x)Y(y)Z(z)

4. y(x,y,z) = X(x)Y(y)Z(z) plug in and see what happens! Divide both sides by XYZ

5. The left side is a constant with respect to x: A) True B) False  The right side is independent from x!  left side must be independent from x as well!!

6. If we call this const. '-kx2' we can write: X"(x) = - kx2 X(x)  This is the Schrödinger equation for a particle in a one-dimensional rigid box!! We already know the solutions for this equation:

7. Repeat for Y and Z: And:

8. or: with: And the total energy is: Now, remember: y(x,y,z) = X(x)Y(y)Z(z) Done! • What is the energy of a particle in the 3D box? • Ex • Ey • Ez • Ex = Ey = Ez • Ex + Ey + Ez

9. 2D box: Square of the wave function for nx=ny=1 y Probability to find electron at a certain location in the box. Percent relative to maximum |Y(y)|2 |X(x)|2 x y(x,y,z) = X(x)Y(y)Z(z)

10. 2D box: Square of the wave function of selected excited states 100% 0% ny nx

11. Degeneracy Sometimes, there are several solutions with the exact same energy. Such solutions are called ‘degenerate’.

12. The ground state energy of the 2d box of size L x L is 2E0, where E0 = p2ħ2/2mL2 is the ground state energy of a 1d box of size L. y L E=E0(nx2+ny2) x L What is the energy of the 1st excited state of this 2D box? • 3Eo • 4Eo • 5Eo • 8Eo

13. L x L The ground state energy of the 2d box of size L x L is 2E0, where E0 = p2ħ2/2mL2 is the ground state energy of a 1d box of size L. E=E0(nx2+ny2) What is the energy of the 1st excited state of this 2D box? • 3Eo • 4Eo • 5Eo • 8Eo nx=1, ny=2 or nx=2 ny=1

14. L L L • Imagine a 3d cubic box of sides L x L x L. What is the degeneracy of the ground state and the first excited state? • Gnd, 1st • 1 , 1 • 3, 1 • 1, 3 • 3, 3

15. L L L • Imagine a 3d cubic box of sides L x L x L. What is the degeneracy of the ground state and the first excited state? • Gnd, 1st • 1 , 1 • 3, 1 • 1, 3 • 3, 3 Ground state = 1,1,1 1st excited state: 2,1,1 1,2,1 1,1,2