1. Review of waves T = period = time of one cycle  = 2  f = angular frequency = number of radians per second t Waves in time: f = 1/T =  /2  = frequency = number of cycles per second A = wavelength = length of one cycle k = 2  / = wave number = number of radians per meter x Waves in space:

2. Sinusoidal waves A traveling wave changes in both space and time: Using k and  gives us the simplest equation so we will generally use them instead of, T, and f. Note we can write the energy of a photon as and the momentum of any particle as and for more than 1 dimension: For a massive particles can also write kinetic energy as sines and cosines or complex exponentials

3. Plane waves Different k’s correspond to different energies since Which is the highest energy wave? a) b) c) d)

4. Plane waves Each of these waves is made up of a single k and . How far do they extend in space? 1) 2) 3) 4) a)Wave 1 extends farthest b)Wave 4 extends farthest c)All the waves extend forever in space There is no term that damps them in space.

5. Plane Waves vs. Wave Packets Plane wave: Wave packet: Matter waves are more like wave packets than plane waves. Mathematically, we obtain wave packets by adding up plane waves. Method of adding up sine waves to obtain an arbitrary function (like a wave packet) is called Fourier Analysis.

6. Superposition If  1 (x,t) and  2 (x,t) are both solutions to the wave equation then the sum  1 (x,t) +  2 (x,t) is also a solution. Actually, any linear combination A  1 (x,t) + B  2 (x,t) is a solution. This is the superposition principle. This seems pretty straightforward but is actually an important result We are going to make a wave packet out of a superposition of plane waves

7. Superposition

8. Addition of two waves Two waves each of form come together and interfere. Construct them anyway you wish to get the maximum interference. What is the maximum intensity that is observed in the interference pattern? a)0 b)E 2 c)2E 2 d)4E 2 One wave: I 1 = E 2 =E 1 2 +E 1 2 +2E 1 *E 2 Two waves: I 2 = (E 1 +E 2 ) 2 Alternatively, just add to get 2E and then square.

9. Two slit interference Standard two slit interference with light causes waves to interfere The electric field E(x) of the wave as a function of position. It can be positive or negative. The intensity is proportional to the square of the field and is ≥ 0. This is what we see! The intensity also tells us the probability that a photon will land in a particular x region. E(x)

10. Same thing with electrons Electrons have a wavelength and so two slit interference has the same effect as for light The probability of any given electron hitting near x is determined by is called the wave function. It contains everything we know about the electron but only gives probabilities about where the electron is located

11. Addition of two waves Two waves each of form come together and interfere. What is the average intensity that is observed in the interference pattern? a)0 b)E 2 c)2E 2 d)4E 2 This is the average of the max (4E 2 ) and the min intensity (0).

12. Addition of two waves What happened to the average energy (intensity) of the sum of the two interfering waves when they are in constructive interference? a)It is the same as the average energy of one of the individual waves. b)It is the equal to twice the average energy of one of the individual waves. c)It is equal to four times the average energy of one of the individual waves. Average energy of one individual wave is 0.5*E 2

13. Addition of two waves Two waves each of form come together and interfere constructively. Each has average intensity (energy) of 0.5E 2. The sum of them has four times this average intensity, or 2E 2. Where did the extra intensity come from?