Getting to the Schr ödinger equation

1. Getting to the Schrödinger equation Announcements: • Learning Assistant program informational session will be held today at 6pm in UMC 235. Fliers are available if you are interested. • Next homework assignment will be available by tomorrow • I will be giving a public talk about physics at the Large Hadron Collider (LHC) at 2pm on Saturday in G1B30. If you are interested in particle physics, you may find it interesting. Erwin Schrödinger (1887 – 1961) Physics 2170 – Spring 2009

2. Where we go from here We will finish up classical waves Classical waves obey the wave equation: Then we will go back to matter waves which obey a different wave equation called the time dependent Schrödinger equation: On Friday we will derive the time independent Schrödinger equation: Physics 2170 – Spring 2009

3. Solving the standard wave equation • Guess the functional form(s) of the solution • Plug into differential equation to check for correctness, find any constraints on constants • Need as many independent functions as there are derivatives. • Apply all boundary conditions (more constraints on constants) The standard wave equation is Generic prescription for solving differential equations in physics: Physics 2170 – Spring 2009

4. Step 2: Check solution and find constraints Claim that is a solution to Time to check the solution and see what constraints we have LHS: RHS: Setting LHS = RHS: This works as long as We normally write this as so this constraint just means or Physics 2170 – Spring 2009

5. Constructing general solution from independent functions Since the wave equation has two derivatives, there must be two independent functional forms. y t=0 x The general solution is Can also be written as and we have the constraint that We have finished steps 1, 2, & 3 of solving the differential equation. Last step is applying boundary conditions. This is the part that actually depends on the details of the problem. Physics 2170 – Spring 2009

6. L 0 Boundary conditions for guitar string Wave equation Functional form: Guitar string is fixed at x=0 and x=L. Boundary conditions are that y(x,t)=0 at x=0 and x=L. Requiring y=0 when x=0 means which is This only works if B=0. So this means Physics 2170 – Spring 2009

7. Clicker question 1 Set frequency to DA Boundary conditions require y(x,t)=0 at x=0 & x=L. We found for y(x,t)=0 at x=0 we need B=0 so our solution is . By evaluating y(x,t) at x=L, derive the possible values for k. • k can have any value • /(2L), /L, 3/(2L), 2/L … • /L • /L, 2/L, 3/L, 4/L … • 2L, 2L/2, 2L/3, 2L/4, …. n=1 n=2 n=3 To have y(x,t) = 0 at x = L we need This means that we need This is true for kL = np. That is, So the boundary conditions quantize k. This also quantizes w because of the other constraint we have: Physics 2170 – Spring 2009

8. Summary of our wave equation solution 1. Found the general solution to the wave equation or 2. Put solution into wave equation to get constraint 3. Have two independent functional forms for two derivatives n=1 4. Applied boundary conditions for guitar string. y(x,t) = 0 at x=0 and x=L. Found that B=0 and k=np/L. n=2 n=3 Our final result: with and Physics 2170 – Spring 2009

9. Standing waves Standing wave Standing wave constructed from two traveling waves moving in opposite directions Physics 2170 – Spring 2009

10. Examples of standing waves For standing waves on violin string, only certain values of k and w are allowed due to boundary conditions (location of nodes). Same is true for electromagnetic waves in a microwave oven: We also get only certain waves for electrons in an atom. We will find that this is due to boundary conditions applied to solutions of Schrödinger equation. Physics 2170 – Spring 2009

11. Clicker question 2 Set frequency to DA Case I: no fixed ends y For which of the three cases do you expect to have only certain frequencies and wavelengths allowed? That is, in which cases will the allowed frequencies be quantized? x Case II: one fixed end y x • Case I • Case II • Case III • More than one case Case III: two fixed ends y x After applying the 1st boundary condition we found B=0 but we did not have quantization. After the 2nd boundary condition we found k=np/L. This is the quantization. Physics 2170 – Spring 2009

12. Boundary conditions cause the quantization Free electron Electron bound in atom E Boundary Conditions  standing waves No Boundary Conditions  traveling waves Only certain energies allowed Quantized energies Any energy allowed Physics 2170 – Spring 2009

13. Getting to Schrödinger’s wave equation Works for light (photons), why doesn’t it work for electrons? Physics 2170 – Spring 2009

14. Clicker question 3 Set frequency to DA The equation E = hc/l is… • true for photons and electrons • true for photons but not electrons • true for electrons but not photons • not true for either electrons or photons works for photons and electrons works for photons and electrons only works for massless particles (photons) Physics 2170 – Spring 2009

15. Getting to Schrödinger’s wave equation Works for light (photons), why doesn’t it work for electrons? We found that solutions to this equation are or with the constraint which can be written Multiplying by ħ we get which is just But we know that E=pc only works for massless particles so this equation can’t work for electrons. Physics 2170 – Spring 2009

16. Getting to Schrödinger’s wave equation doesn’t work for electrons. What does? or Note that each derivative of x gives us a k (momentum) while each derivative of t gives us an w (energy). Equal numbers of derivatives result in For massive particles we need So we need two derivatives of x for p2 but only one derivative of w for K. If we add in potential energy as well we get the Schrödinger equation… Physics 2170 – Spring 2009

17. The Schrödinger equation The Schrodinger equation for a matter wave in one dimension (x,t): Total energy Kinetic energy Potential energy = + This is the time dependent Schrödinger equation (discussed in 7.11) and is also the most general form. Physics 2170 – Spring 2009

18. The Schrödinger equation applied to a plane wave A plane matter wave can be written as If we plug this into the Schrödinger equation, what do we get? E=K+U seems to make sense Physics 2170 – Spring 2009