More on waves + uncertainty principle

Transcription

1 More on waves + uncertainty principle ** No class Fri. Oct. 18! The 2 nd midterm will be Nov. 2 in same place as last midterm, namely Humanities at 7:30pm. Welcome on Columbus day Christopher Columbus arrived Oct. 12, 1492! - Federal offices are shutdown, but how would we tell? Announcements: Electron wave function of C 60 Physics 2170 Fall

2 Comple Numbers Physics 2170 Fall

3 Why Intensity not amplitude? Physics 2170 Fall

4 The wave function The wave function does not tell you the path of an object like a normal traveling wave function. ψ() The wave function (squared) gives the probability of finding the object at a given. Physics 2170 Fall L Some details about the wave function: Wave function is really a function of the 3D position, ψ(,y,z) but many of the problems are 1D so we will start with that: ψ(). Wave functions are comple valued (have real and imaginary parts) and are never directly observable. The probability density is observable and is L ψ*() is comple conjugate: replace i by i in ψ().

5 Getting probabilities from wave functions ψ() Wave function ψ(): L L Probability density For an infinitesimal distance δ the probability is To find the probability over a finite distance we need to integrate. L a b L Probability of finding electron between a and b is the area under the curve. Probability of electron being between a and b is Physics 2170 Fall

6 Normalization Probability density = Probability of electron being between a and b is L a b L What if this calculation gives a number larger than 1? Doesn t make any sense! A properly normalized wave function obeys the normalization constraint: This is simply a statement that the electron is located somewhere! That is, the probability of finding the electron somewhere is 100%. Physics 2170 Fall

7 Clicker question 1 Set frequency to DA If ψ() is a properly normalized wave function, what is the value of A as shown in the figure? A A. A = 1 B. A = 1 / L C. A = 1 / 2L D. E. L L Physics 2170 Fall

8 Clicker question 1 If ψ() is a properly normalized wave function, what is the value of A as shown in the figure? Set frequency to DA A. A = 1 B. A = 1 / L C. A = 1 / 2L D. E. L L The probability is 0 for >L so we only need to consider the region between L and L. The electron must be somewhere in that region. In that region we know so applying the normalization condition we get: A so A = 1 / 2L Notice it has units of 1/Length Physics 2170 Fall

9 Working with probabilities Probability density of grades: 0.06 What is Probability of getting between ? P() What is the probability of getting a score between 38-40? 45 Probability = P()*d = 0.07*2.5 = (~16/100 students) Physics 2170 Fall

10 Summary of wave function The wave function ψ() is just the spatial part of the wave function. The general wave function is also a function of time and is Ψ(,t). is the full wave function gives the space and time dependent probability of detecting the particle Electron double slit eperiment: Intensity is magnitude of wave function Large Magnitude ( Ψ )= probability of detecting electron here is high Small Magnitude ( Ψ )= probability of detecting electron here is low Physics 2170 Fall

11 Clicker question 2 ψ() = 0 for <0 = Ae -/L for 0 ψ() A Set frequency to DA What must be the value of A to have a properly normalized wave function? A. B. C. D. E. 0 0 L Physics 2170 Fall

12 Clicker question 2 ψ() = 0 for <0 = Ae -/L for 0 ψ() A Set frequency to DA What must be the value of A to have a properly normalized wave function? A. B. C. D. E. 0 0 L Physics 2170 Fall

13 Review of waves Waves in time: A T = period = time of one cycle t f = 1/T = ω/2π = frequency = number of cycles per second ω = 2πf = angular frequency = number of radians per second Waves in space: λ = wavelength = length of one cycle k = 2π/λ = wave number = number of radians per meter Physics 2170 Fall

14 More on sinusoidal waves A traveling wave changes in both space and time: Using k and ω gives us the simplest equation so we will generally use them instead of λ, 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 Physics 2170 Fall

15 Plane waves Plane waves etend infinitely in space and time. Can be made up of sine and cosine waves or comple eponentials Same thing by Euler s theorem: Different k s correspond to different energies since Physics 2170 Fall

16 Superposition If Ψ 1 (,t) and Ψ 2 (,t) are both solutions to the wave equation then the sum Ψ 1 (,t) + Ψ 2 (,t) is also a solution. Actually, any linear combination AΨ 1 (,t) + BΨ 2 (,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 Physics 2170 Fall

17 Superposition Physics 2170 Fall