Final Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall.

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1 Final Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall.

2 Chapter 38 Quantum Mechanics

3 Units of Chapter Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 38-3 The Heisenberg Uncertainty Principle 38-4 Philosophic Implications; Probability versus Determinism 38-5 The Schrödinger Equation in One Dimension Time-Independent Form 38-6 Time-Dependent Schrödinger Equation

4 Units of Chapter Free Particles; Plane Waves and Wave Packets 38-8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) 38-9 Finite Potential Well Tunneling Through a Barrier

5 The Wave-Particle Duality p E c = h 2π k ω c If waves can behave like particle, then particles can behave like waves λ = h p De Broglie wavelength

$diffraction$

6 The Wave Nature of Matter Photon diffraction Electron diffraction

7 Double-Slit Experiment with Electrons Figure 38.4

8 38.3 The Heisenberg Uncertainty Principle Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of its momentum to the electron.

9 38.3 The Heisenberg Uncertainty The combination of uncertainties gives: which is called the Principle (Δx) (Δp x ) h 2π Heisenberg uncertainty principle. It tells us that the position (x) and momentum (p) cannot simultaneously be measured with precision.

11 Summary of Chapter 38 Part I In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2 represents the probability of finding particle in a given location. The wave function Ψ satisfies the Time-Dependent Schrödinger Equation 2 2m x 2 + V(x) Ψ(x,t) = i t Ψ(x,t) 2

12 One-Dimensional Wave Equation 2 x 2 D(x,t) = 1 v 2 2 t 2 D(x,t) One-Dimensional Schrödinger Equation 2 2m 2 x + V(x) 2 Ψ(x,t) = i t Ψ(x,t)

13 38.6 Time-Dependent Schrödinger Equation Writing Ψ(x,t) =ψ (x) f (t) we find that the Schrödinger equation can be separated, that is, its time- and space-dependent parts can be solved for separately.

14 Time-Dependent Schrödinger Equation 2 2m 2 x + V(x) 2 Ψ(x,t) = i t Ψ(x,t) Separation of the Time and Space Dependencies Ψ(x,t) =ψ (x) f (t) 2 1 2m ψ (x) i 1 f (t) d 2 ψ (x) + V(x) = E 2 dx d dt f (t) = E

16 38.6 Time-Dependent Schrödinger Equation The time dependence can be found easily; it is: i 1 f (t) d dt f (t) = E f (t) = e i E t This function has an absolute value of 1; it does not affect the probability density in space.

17 Time-Dependent Schrödinger Equation 2 2m 2 x + V(x) 2 Ψ(x,t) = i t Ψ(x,t) Time-Independent Schrödinger Equation 2 2m d 2 ψ (x) + V(x)ψ (x) = Eψ (x) 2 dx d dt f (t) = 1 Ef (t) i Ψ(x,t) = ψ (x)exp[ i E t]

18 38.5 The Schrödinger Equation in One Dimension Time-Independent Form The Schrödinger equation cannot be derived, just as Newton s laws cannot. However, we know that it must describe a traveling wave, and that energy must be conserved. Therefore, the wave function will take the form: where ψ(x) = Asinkx + Bcoskx p = h 2π k = k k = p

19 38.7 Free Particles; Plane Waves and Wave Packets Free particle: no force, so V = 0. The Schrödinger equation becomes: 2 2m 2 ψ (x) = E ψ (x) 2 x which can be written d 2 2mE ψ (x) + ψ (x) = 0 2 dx 2

20 38.7 Free Particles; Plane Waves and Wave Packets Free particle: no force, so V = 0. d 2 2mE ψ (x) + ψ (x) = 0 2 dx 2 The solution to this equation is ψ(x) = Asinkx + Bcoskx with k = 2mE 2

21 38.7 Free Particles; Plane Waves and Wave Packets Example 38-4: Free electron. An electron with energy E = 6.3 ev is in free space (where V = 0). Find (a) the wavelength λ (in nm) and (b) the wave function for the electron (assuming B = 0).

22 38.7 Free Particles; Plane Waves and Wave Packets The solution for a free particle is a plane wave, as shown in part (a) of the figure; more realistic is a wave packet, as shown in part (b). The wave packet h a s b o t h a r a n g e o f momenta and a finite uncertainty in width.

23 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) One of the few potentials where the Schrödinger equation can be solved exactly is the infinitely deep square well. As is shown, this potential is zero from the origin to a distance, and is infinite elsewhere.

24 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) The solution for the region between the walls is that of a free particle: Requiring that ψ = 0 at x = 0 and x = gives B = 0 and k = nπ/. This means that the energy is limited to the values: E n = n 2 h 2 ; n = 1, 2, 3, ml

25 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) The wave function for each of the quantum states is: ψ(x) = Asink n x ; k n = n π l The constant A is determined by the condition + + ψ(x) 2 dx = 1; Asink n x 2 dx = A 2 sink n x 2 dx = 1; l 0 ψ n = 2 l sink nx

26 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) These plots show for several values of n the energy levels, wave function, and probability distribution. energy levels wave function probability distribution

27 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-5: Electron in an infinite potential well. (a) Calculate the three lowest energy levels for an electron trapped in an infinitely deep square well potential of width = 0.1 nm (about the diameter of a hydrogen atom in its ground state). (b) If a photon were emitted when the electron jumps from the n = 2 state to the n = 1 state, what would its wavelength be?

28 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-6: Calculating a normalization constant. Show that the normalization constant A for all wave functions describing a particle in an infinite potential well of width has a value of: A = 2/ l.

29 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-7: Probability near center of rigid box. An electron is in an infinitely deep square well potential of width = 1.0 x m. If the electron is in the ground state, what is the probability of finding it in a region dx = 1.0 x m of width at the center of the well (at x = 0.5 x m)?

30 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-8: Probability of e - in ¼ of box. Determine the probability of finding an electron in the left quarter of a rigid box i.e., between one wall at x = 0 and position x = /4. Assume the electron is in the ground state.

31 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-9: Most likely and average positions. Two quantities that we often want to know are the most likely position of the particle and the average position of the particle. Consider the electron in the box of width = 1.00 x m in the first excited state n = 2. (a) What is its most likely position? (b) What is its average position?

32 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) ψ (x) 2 = 2 l sin2 2 π l x x = l 0 x ψ (x) 2 dx

33 38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box) Example 38-10: A confined bacterium. A tiny bacterium with a mass of about kg is confined between two rigid walls 0.1 mm apart. (a) Estimate its minimum speed. (b) If instead, its speed is about 1 mm in 100 s, estimate the quantum number of its state.

34 38.9 Finite Potential Well A finite potential well has a potential of zero between x = 0 and x =, but outside that range the potential is a constant U 0.

35 38.9 Finite Potential Well The potential outside the well is no longer zero; it falls off exponentially. The wave function in the well is different than that outside it; we require that both the wave function and its first derivative be equal at the position of each wall (so it is continuous and smooth), and that the wave function be normalized.

36 38.9 Finite Potential Well ψ I = Ce Gx ψ II = Asin kx + Bcos kx ψ III = De Gx x = 0 x = l We require that both the wave function and its first derivative be equal at the position of each wall (so it is continuous and smooth), and that the wave function be normalized.

37 38.9 Finite Potential Well These graphs show the wave functions and probability distributions for the first three energy states. Figure 38-13a goes here. Figure 38-13b goes here.

38 Simulations

39 38.10 Tunneling Through a Barrier Since the wave function does not go to zero immediately upon encountering a finite barrier, there is some probability of finding the particle represented by the wave function on the other side of the barrier. This is called tunneling. Figure goes here.

40 38.10 Tunneling Through a Barrier The probability that a particle tunnels through a barrier can be expressed as a transmission coefficient, T, and a reflection coefficient, R (where T + R = 1). If T is small, The smaller E is with respect to U 0, the smaller the probability that the particle will tunnel through the barrier.

41 38.10 Tunneling Through a Barrier Example 38-11: Barrier penetration. A 50-eV electron approaches a square barrier 70 ev high and (a) 1.0 nm thick, (b) 0.10 nm thick. What is the probability that the electron will tunnel through?

42 38.10 Tunneling Through a Barrier Alpha decay is a tunneling process; this is why alpha decay lifetimes are so variable. Scanning tunneling microscopes image the surface of a material by moving so as to keep the tunneling current constant. In doing so, they map an image of the surface. Figure goes here.

43 38.10 Tunneling Through a Barrier The 7.1 nm "quantum corral" shown above was created and imaged with an ultra-high vacuum cryogenic STM. Each sharp peak in the circle is an iron atom resting on atomically flat copper.

44 Macro scale image of an etched tungsten scanning tunneling microscopy (STM) tip. Uniform geometry contributes to a more stable tip.

45 Atomic Force Microscope

46 Atomic Force Microscope The enzyme-dna complex can be imaged using AFM. It allows to resolve the different stages of the restriction reaction: binding of the MTase and the motor subunits, grabbing of the DNA by the motors, formation of the initial DNA loop, and the enzyme translocation.

Final Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall.

Final Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall. Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2