Getting to the Schrödinger equation

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1 Getting to the Schrödinger equation Announcements: Homework #6 is available to be picked up. Erwin Schrödinger ( ) Physics 2170 Fall

2 Uncertainty Example from Monday.. An engineer is dropping marbles from a ladder of height H with sophisticated equipment and trying to hit a crack on the floor everytime. Despite his great care, show that she will miss the crack by an average distance of ( /m) 1/ 2 ( H /g) 1/ 4 where g is the acceleration due to gravity. H = ½ a t 2 t = 2H g ΔE Δt ħ/2 ΔΕ ħ/(2δt) ΔΕ =(Δp) 2 /2m Δp = 2mΔE Δx Δp ħ/2 Δx 2Δp = 2 1 2mΔE = 2 Δt m = 1 2 m 1/ 2 ( Δt) 1/ 2 = 1 2 m 1/ 2 H g 1/ 4 Physics 2170 Fall

3 Wave Function Comments Wave Function is complex means can have imaginary and real components doesn t have to have both. The fact that wave functions are complex makes it obvious we should not attribute to wave functions a physical existence (ie water waves). Shouldn t try to answer exactly what is waving and what is it waving in! We shall find that wave functions contain all the information which the uncertainty principle allows us to know about the associated particle. Physics 2170 Fall

4 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 Wednesday we will derive the time independent Schrödinger equation: Physics 2170 Fall

5 Reading quiz 1 Set frequency to AD Which of the following is a true statement about standing waves? A. Standing waves have points called antinodes which are motionless B. Standing waves have points called nodes which are motionless C. Standing waves can be constructed from two traveling waves moving in opposite directions D. A and C are both true E. B and C are both true Physics 2170 Fall

6 Reading quiz 1 Set frequency to AD Which of the following is a true statement about standing waves? A. Standing waves have points called antinodes which are motionless B. Standing waves have points called nodes which are motionless C. Standing waves can be constructed from two traveling waves moving in opposite directions D. A and C are both true E. B and C are both true Antinodes move the most while nodes do not move at all. Physics 2170 Fall

7 E Example Wave Equations You Have Seen Electromagnetic waves: Vibrations on a string: y x x c = speed of light Solutions: E(x,t) Magnitude is non-spatial: = Strength of Electric field v=speed of wave Solutions: y(x,t) Magnitude is spatial: = Vertical displacement of String Physics 2170 Fall

8 Solving the standard wave equation The standard wave equation is Generic prescription for solving differential equations in physics: 1. Guess the functional form(s) of the solution 2. Plug into differential equation to check for correctness, find any constraints on constants 3. Need as many independent functions as there are derivatives. 4. Apply all boundary conditions (more constraints on constants) Physics 2170 Fall

9 Clicker question 2 The standard wave equation is Set frequency to AD Step 1: Guessing the functional form of the solution Which of the following function forms is a possible solution to this differential equation? A. B. C. D. E. More than one of the above Physics 2170 Fall

10 Clicker question 2 The standard wave equation is Set frequency to AD Step 1: Guessing the functional form of the solution Which of the following function forms is a possible solution to this differential equation? A. The A. form leads to B. or Different functions on C. the left side and right D. side. This is incorrect. E. More than one of the above The B. form leads to which doesn t work. The C. form leads to which doesn t work. Physics 2170 Fall

11 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 Fall

12 Constructing general solution from independent functions Since the wave equation has two derivatives, there must be two independent functional forms. 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. t=0 Last step is applying boundary conditions. This is the part that actually depends on the details of the problem. y x Physics 2170 Fall

13 Boundary conditions for guitar string Wave equation Functional form: 0 L 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 Fall

14 Clicker question 3 Set frequency to AD 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. A. k can have any value B. π/(2l), π/l, 3π/(2L), 2π/L C. π/l D. π/l, 2π/L, 3π/L, 4π/L E. 2L, 2L/2, 2L/3, 2L/4,. Physics 2170 Fall

15 Clicker question 3 Set frequency to AD 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. A. k can have any value n=1 B. π/(2l), π/l, 3π/(2L), 2π/L C. π/l n=2 D. π/l, 2π/L, 3π/L, 4π/L E. 2L, 2L/2, 2L/3, 2L/4,. n=3 To have y(x,t) = 0 at x = L we need This means that we need This is true for kl = nπ. That is, So the boundary conditions quantize k. This also quantizes ω because of the other constraint we have: Physics 2170 Fall

Summary of our wave equation solution 1. Found the general solution to the wave equation y = Asin(kx)cos(ωt) + Bcos(kx)sin(ωt) or 2. Put solution into wave equation to get constraint 3.

16 Summary of our wave equation solution 1. Found the general solution to the wave equation y = Asin(kx)cos(ωt) + Bcos(kx)sin(ωt) 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=nπ/l. Our final result: y = Asin(kx)cos(ωt) n=2 n=3 with and Physics 2170 Fall

17 Standing waves Standing wave Standing wave constructed from two traveling waves moving in opposite directions Physics 2170 Fall

Examples of standing waves For standing waves on violin string, only certain values of k and ω are allowed due to boundary conditions (location of nodes).

18 Examples of standing waves For standing waves on violin string, only certain values of k and ω 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 Fall

19 Clicker question 4 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? A. Case I B. Case II C. Case III D. More than one case Set frequency to AD Case I: no fixed ends Case II: one fixed end Case III: two fixed ends y y y x x x Physics 2170 Fall

20 Clicker question 4 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? A. Case I B. Case II C. Case III D. More than one case Set frequency to AD Case I: no fixed ends Case II: one fixed end Case III: two fixed ends y y y x x x After applying the 1 st boundary condition we found B=0 but we did not have quantization. After the 2 nd boundary condition we found k=nπ/l. This is the quantization. Physics 2170 Fall

21 Boundary conditions cause the quantization Electron bound in atom Free electron E Boundary Conditions standing waves Only certain energies allowed Quantized energies No Boundary Conditions traveling waves Any energy allowed Physics 2170 Fall

22 Getting to Schrödinger s wave equation Works for light (photons), why doesn t it work for electrons? Physics 2170 Fall

23 Clicker question 4 Set frequency to AD The equation E = hc/λ is A. true for photons and electrons B. true for photons but not electrons C. true for electrons but not photons D. not true for either electrons or photons Physics 2170 Fall

24 Clicker question 4 Set frequency to AD The equation E = hc/λ is A. true for photons and electrons B. true for photons but not electrons C. true for electrons but not photons D. 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 Fall

25 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 ω (energy). Equal numbers of derivatives result in For massive particles we need So we need two derivatives of x for p 2 but only one derivative of t for K. If we add in potential energy as well we get the Schrödinger equation Physics 2170 Fall

Professor Jasper Halekas Van Allen 70 MWF 12:30-1:20 Lecture

Professor Jasper Halekas Van Allen 70 MWF 1:30-1:0 Lecture Back on regular schedule for the next two weeks There *will* be labs and homeworks due this week and next EM Waves (light/photons) Amplitude E