© Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Physics 322 Final Exam Study Guide Date of Exam: Tuesday, May 12, 2:00 pm - 4:00 pm Rules: 1. Closed Book and Closed Note. A formula sheet will be provided. 2. You will be allowed the entire 2 ½ hours for the exam, although I will be aiming for it to be 90 minutes in length. 3. You will be allowed a calculator (although I will avoid problems where it will be helpful). 4. If you are stuck because you can’t remember a formula or an indefinite integral, I will assure you that you can ask me… I am not big on formula memorization. 5. Combat Arms is not to be played until after the final exam is over. Topics on this Exam: This final exam is cumulative, but focused on more recent material. As a warning, the problems that you all had difficulty with on previous exams and homework might may a comeback in some form. In addition to material covered in previous exams, you can expect the problems on the final exam to cover the concepts in Chapter 7, Chapter 8 (Sections 1 through 4), and Chapter 11 (Sections 1 through 3, 5, and 6) of Harris’ Modern Physics textbook and problem sets 9 through 12. BIG HINTS: • • • Understand the answers to the problem sets you did – some problems on the exam will be very similar to them. Understand the examples in the textbook. By now you have noticed that on the midterms, I avoid problems with particularly nasty integrals/derivatives, focusing instead on mathematically simpler problems that test your conceptual knowledge and ability to apply those concepts. This same approach holds for the problems I will be writing for the final exam. Chapter 7) Quantum Mechanics in Three Dimensions and the Hydrogen Atom 1. The Schrödinger Equation in 3 Dimensions a. Know the general changes that have to go on in order to get our previously 1 dimensional Schrödinger equations to work in three dimensions. I don’t mean memorizing the newer equations, I mean understanding just what needed to be modified in moving to 3 dimensions. 2. The 3D Infinite Well a. Understand how the technique of separation of variables is used to determine the allowed wave functions and states for a 3 dimensional –– Page 1 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. infinite well. b. Know under what situations a degeneracy in allowed energy states occurs and more importantly what it means. c. Why are there three quantum numbers instead of just one for a 3-D infinite well? What does this mean physically? d. When will degenerate energy levels split? How might it affect the allowed photon wavelengths due to electron transitions between different energy states? 3. Energy Quantization and Spectral Lines in Hydrogen a. Know the basic characteristics of the hydrogen spectra and why it strongly suggests that the allowed energy levels of the electron in the hydrogen atom are quantized. b. Given the expression for the energy levels of the hydrogen atom, be able to compute the wavelength of a photon emitted/absorbed by an electron to accomplish a transition between any two energy levels? 4. The Schrödinger Equation for a Central Force a. Understand the spherical coordinate system used by Harris (in addition to the radial coordinate r, it takes θ as the “polar angle” which measures angle from the “north pole” of the coordinate system and ϕ as the azimuthal angle which measures angle about the z-axis from the x-axis). b. Understand what is meant by the separability of the wave function into three wave functions in each coordinate: ψ ( r,θ , φ ) = R ( r ) Θ (θ ) Φ (φ ) c. Examine “A Closer Look” in p. 247 and understand how the separability of the wave function allows use to find three equations: the radial, the polar, and the azimuthal equations which when solved let us solve for R ( r ) , Θ (θ ) , and Φ (φ ) respectively. You don’t need to memorize the equations, I just want you to understand what they represent. 5. Angular Behavior in a Central Force a. Understand how the solution to the azimuthal equation Φ (φ ) is obtained and why the solution must be sinusoidal in nature (HINT: Smoothness and boundary conditions). b. Know what the solutions to the azimuthal equation Φ (φ ) look like (such as those shown in Figure 7.9 on p. 249). c. Why does he azimuthal equation Φ (φ ) lead to the quantization of the zcomponent of the orbital angular momentum, Lz in terms of (magnetic) quantum number mℓ ? –– Page 2 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. d. Understand conceptually how we arrive at the quantization of the magnitude of the orbital angular momentum |L| via the solution of the polar equation. e. Understand how the orbital angular momentum |L| and the orbital quantum number ℓ are related. f. Operationally, what limits does the orbital quantum number ℓ place on the allowed values of (magnetic) quantum number mℓ and why does this physically make sense? g. How do the quantum numbers ℓ and mℓ relate to the orbital motion of the electron (as illustrated in Figures 7.13 and 7.14)? 6. The Hydrogen Atom a. Know conceptually how the radial solutions, as shown in Table 7.4, are obtained (even if actually solving for them is beyond the scope of this course). b. How does the principal quantum number n restrict the values on the quantum numbers ℓ and mℓ ? c. Know why as the principal quantum number n increases there is an increasing degeneracy in the energy levels of the hydrogen atom (HINT: Examine Table 7.5). d. Know why it is we can normalize the radial and angular components of the hydrogen atom wave equation independently. e. Annoyingly, you should probably memorize the order of the letters used in spectroscopic notation to indicate the various ℓ states (NOTE: The order is s, p, d, f, g, h,...) 7. Radial Probability a. Understand what is represented by the radial probability in equation 7-38 (a probability per unit radial distance). b. Know the difference between most likely location, most likely radius, and average radius (aka the expectation value of the radius) and how to compute those three values. A few of you really blew this on problem set 10... 8. The Hydrogen-like Atom a. Basically, know how you have to modify the hydrogen atom model (it’s only the potential that changes) when dealing with hydrogen-like atoms which are atoms with Z protons and one electron. –– Page 3 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Chapter 8) Spin and Atomic Physics [Sections 1 to 4] 1. Evidence of Angular Momentum Quantization: A New Property a. Know how the magnetic dipole moment is related to angular momentum. b. Know how the Stern-Gerlach apparatus was designed to separate a beam of particles fired through it by their magnetic dipole moment. Specifically how does it manage to exert a force (as opposed to a torque) on the dipoles. c. How did firing ground state hydrogen atoms through the Stern-Gerlach apparatus reveal the existence of an intrinsic angular momentum (called spin) in the electron? d. Why does the spin of the proton in the hydrogen atom not play a huge role in the Stern-Gerlach experiments? HINT: Look at the proper relativistic quantum mechanical expression for the magnetic dipole moment, equation 8-7. e. What is meant by the statement “the electron is a spin-½ particle”? f. What is different about the dimensionless number s used to quantize spin and other quantum numbers? 2. Identical Particles a. Understand what is meant by “identical particles”. b. Understand the setup of the separable “Two Particles in a Box” problem (detailed in “A Closer Look” in p. 301) used to solve for the expression of two-particle wave function for two particles in an infinite one-dimensional well. c. Why must identical particles be indistinguishable and why does that mean we must have a “symmetric” probability density? d. What do the symmetric and antisymmetric two-particle wave functions represent? e. What are fermions? What are bosons? 3. The Exclusion Principle a. Know what is meant by the (Pauli) Exclusion Principle. b. Why does the Exclusion Principle apply to fermions but not bosons? 4. Multi-electron Atoms and the Periodic Table a. Know the rule of thumb for the order of filling electron shells and subshells, the “ n + ℓ ” rule. b. Know what electronic configuration notation indicates (e.g. – 1s2 or 2s22p63s23p1). c. What makes the periodic table of the elements periodic? d. Which electrons (in any given atom) are the ones that govern the chemistry of atoms? –– Page 4 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Chapter 11) Nuclear Physics 1. Basic Structure a. Know the basic definitions of: nucleon, atomic number (Z), mass number (A), and isotopes. b. What is the relationship between radius of a nucleus and the number of nucleons and HOW was this determined? 2. Binding a. What is the “strong nuclear force” and what do we know about it? b. What role does the fact the strong nuclear force only has a range of ~1 fm have on the nature of internucleon bonds (compared to say electrostatic forces). c. What is an atomic mass unit (u)? I don’t mean just knowing to look up unit conversion, I mean knowing that 1 u = 1 gm/mol. d. What is meant by “binding energy” and why is it we can say a nucleus with a high binding energy has a very low energy? e. What role do the following play in determining the binding energy of the nucleus? The strong internucleon attraction. Coulomb repulsion. The exclusion principle. f. Explain why the binding energy is really just the energy equivalent of “the mass of the parts” minus “the mass of the whole”? How can there be a difference between the mass of the parts that make up the whole and the whole nucleus? g. What is the curve of stability (Figure 11.13) and why does it exist? h. Why do we care about the binding energy per nucleon as a function of Z? i. Understand how to use the data in Appendix I if it is given to you! 3. Nuclear Models a. What was the motivation for the liquid drop model of the nucleus? b. Know why the various terms (the volume term, the surface term, the Coulomb term, and the asymmetry term) in the liquid drop model relate to physically and why. c. How were the terms in the semi-empirical binding energy formula determined? d. Why did we also (briefly) consider a second model for the nucleus of the atom called the shell model? What does the shell model allow us to explain that the liquid drop model doesn’t? e. What are magic numbers? How is their explanation under the shell model similar to the concept of “noble” gases on the periodic table? NOTE: Don’t memorize the numbers, learn the concept. –– Page 5 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. 4. Radioactivity a. What is meant by the term “radioactive decay”? b. What drives every form of radioactive decay? c. What is Q? (I mean conceptually) What values of Q are associated with radioactive decay processes? d. What is a parent nucleus and what is a daughter nucleus? e. What is α decay? What is β– decay? What is β+ decay? What is electron capture? What is γ decay? f. How did observations of the kinetic energies of β– particles in β– decay result in the discovery/identification of the neutrino? g. Understand why Qβ − = m parent − mdaughter c 2 if those are atomic masses? h. Understand why Qβ + ( = (m ) parent ) − mdaughter − 2me c 2 if those are atomic masses? i. Understand why QEC = m parent − mdaughter c 2 if those are atomic masses? ( ) j. What is spontaneous fission and how is it distinct from other forms of radioactive decay? k. What happens in the nucleus (in terms of changes in Z and N) during α decay? during β– decay? during β+ decay? during electron capture? during γ decay? during spontaneous fission? 5. Radioactive Decay a. Understand how the idea that the number of nuclei decaying per unit time dN (aka the decay rate) being proportional to the number of nuclei ( ∝N ) dt leads to an exponential decline in the number of nuclei over time. b. What is the definition of decay constant? half-life? How are the two related? c. Be able to compute how much of an original radioactive isotope is available after n half-lives have passed. d. How does radioactive dating work? What information do we need to have to make it work? –– Page 6 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Physical Constants for Final Exam (First Draft) 1eV = 1.602 × 10 −19 J e = 1.602 × 10 −19 C c = 3.00 × 10 8 m s h = 6.626 × 10 −34 J i s h != = 1.055 × 10 −34 J i s 2π J kB = 1.38 × 10 −23 K 1 2 = 8.99 × 10 9 NC·m2 4π ε0 me = 9.1094 × 10 −31 kg m p = 1.6726 × 10 −27 kg mn = 1.6749 × 10 −27 kg 1u = 1.6605 × 10 −27 kg = 931.5 mH = 1.007825u MeV c2 mn = 1.008665u m p = 1.007276u me = 0.000549u –– Page 7 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Equations for Final Exam (First Draft) Special Relativity (Repeats from Midterm #1) Implications of Einstein’s Postulates 1 γv = v2 1− 2 c Δt = γ v Δt 0 L L= 0 γv Energy and Momentum Lorentz Transformations Velocity Transformations x ' = γ v ( x − vt ) x = γ v ( x '+ vt ') ⎛ v ⎞ t ' = γ v ⎜ − 2 x + t⎟ ⎝ c ⎠ ⎛ v ⎞ t = γ v ⎜ + 2 x '+ t '⎟ ⎝ c ⎠ 1 γu = 1− u2 c2 KE = (γ u − 1) mc 2 ! ! p = γ u mu E = γ u mc 2 Einternal = mc 2 ux ' = E 2 = p2c2 + m 2c4 uy uz ux − v , uy ' = , uz ' = ux v ux v uv 1− 2 1− 2 1 − x2 c c c Doppler Shift fobs = fsource 1− v2 c2 v 1 + cosθ c Waves as Particles and Particles as Waves (Repeats from Midterm #1) Properties of Light Waves/Particles λf = c hc E = hf = = pc λ Properties of Matter Waves/Particles E = hf = !ω p= h = !k λ probability density ∝ Ψ ( x,t ) 2 Photoelectric Effect / Compton Effect KEmax = hf − φ h λ′ − λ = (1 − cosθ ) me c Free-Particle Schrödinger Equation (1-D) ∂Ψ ( x,t ) −! ∂ 2 Ψ ( x,t ) = i! 2 2m ∂x ∂t Heisenberg Uncertainty Principle ! ΔxΔpx ≥ 2 ! ΔEΔt ≥ 2 Bohr Atom mvr = n! 4π ε0 ) ! 2 2 ( r= n E=− –– Page 8 of 11 –– me2 me4 2 ( 4π ε0 ) 2 1 1 = −13.6eV 2 2 n ! n 2 © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Quantum Mechanics: 1-D Bound States (Repeats from Midterm #2) Time-Dependent Schrödinger Equation − ∂Ψ ( x,t ) ! ∂ Ψ ( x,t ) + U ( x ) Ψ ( x,t ) = i! 2 2m ∂x ∂t 2 2 Ψ ( x,t ) dx = 1 ∫ 2 all space Time-Independent Schrödinger Equation Ψ ( x,t ) = ψ ( x )φ ( t ) where φ ( t ) = e E −i t ! ! 2 d 2ψ ( x ) + U ( x )ψ ( x ) = Eψ ( x ) 2m dx 2 Infinite Well - Particle in a Box − ⎧⎪ 0, 0<x<L U ( x) = ⎨ ⎩⎪ ∞, x < 0 or x > L ⎧ 2 ⎛ nπ x ⎞ ⎪ sin ⎜ , 0<x<L ψ ( x) = ⎨ L ⎝ L ⎟⎠ ⎪ 0, x < 0 or x > L ⎩ n 2π 2 ! 2 En = 2mL2 General Solutions 2m ( E − U 0 ) !2 ψ ( x ) = Ae+ikx + De−ikx k= 2m (U 0 − E ) !2 ψ ( x ) = Ce+α x + De−α x α= Simple Harmonic Oscillator Potential ⎛ En = ⎜ n + ⎝ 1⎞ ⎟ !ω 0 where n = 0,1, 2, 3,... and ω 0 = 2⎠ κ m Operators xˆ = x ∂ ∂x ∂ Eˆ = i! ∂t pˆ = −i! Q= ∫ ˆ ( x,t ) dx Ψ * ( x,t ) QΨ all space ΔQ = Q 2 − Q 2 Quantum Mechanics: Unbound States (Repeats from Midterm #2) Potential Steps for Stationary States 2 2 ψ inc = ψ ref + ψ trans 2 ψ k T = trans 2 trans ψ inc kinc R= For E > U 0 : T= ( ( R= ( 4 E ( E − U0 ) E + E − U0 E − E − U0 E + E − U0 Quantum Tunneling – Wide-Barriers 2 ψ ref 2 ψ inc 2 kref kinc 2m (U 0 − E ) L = αL = ≫1 δ ! E ⎛ E ⎞ −2 δL T ≅ 16 ⎜ 1 − ⎟ e U0 ⎝ U0 ⎠ Potential Barriers for Stationary States ) ) ) 2 Quantum Tunneling – For E < U 0 : E ⎛ E⎞ 1− ⎟ ⎜ U0 ⎝ U0 ⎠ 2m (U 0 − E ) L ⎤ E ⎛ E⎞ ⎥ + 4 ⎜ 1− ⎟ ! U0 ⎝ U0 ⎠ ⎥⎦ 4 2 2 T= ⎡ sinh 2 ⎢ ⎢⎣ R = 1− T –– Page 9 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Quantum Mechanics: 3-D Bound States (Repeats from Midterm #2) Time-Dependent Schrödinger Equation ∂2 ∂2 ∂2 ∇2 = 2 + 2 + 2 ∂x ∂y ∂z !2 2 " " " " ∇ Ψ ( r,t ) + U ( r ) Ψ ( r,t ) = i!Ψ ( r,t ) 2m " 2 ∫ Ψ ( r,t ) dV = 1 − Time-Independent Schrödinger Equation !2 2 " " " " − ∇ ψ ( r ) + U ( r )ψ ( r ) = Eψ ( r ) 2m " 2 ∫ ψ ( r ) dV = 1 all space all space 3-D Infinite Well ⎛ 2⎞ ψ nx ,ny ,nz ( x, y, z ) = ⎜ ⎟ ⎝ L⎠ Enx ,ny ,nz = 3/2 ⎛ n π x ⎞ ⎛ n yπ y ⎞ ⎛ n z π z ⎞ sin ⎜ x ⎟ sin ⎜ ⎟ sin ⎝ Lx ⎠ ⎝ Ly ⎠ ⎜⎝ Lz ⎟⎠ 2 π 2 ! 2 ⎛ nx2 ny nz2 ⎞ + + 2m ⎜⎝ L2x L2y L2z ⎟⎠ Quantum Mechanics: Central Forces and The Hydrogen Atom Quantized Quantities in Hydrogen Atom En = − me 4 1 2 2 2 ( 4π ε0 ) ! 2 n where n = 1, 2, 3,... L = ℓ ( ℓ + 1)! where ℓ = 0,1, 2, 3,..., n − 1 L z = mℓ ! where mℓ = 0,1, 2, 3,..., ℓ Normalization Hydrogen Atom π 2π 4π ε0 ) ! 2 ( 2 2 2 ∫ R ( r ) r dr ∫θ = 0 ∫φ = 0 Θ (θ ) sinθ dθ dφ = 1 a0 = me2 = 0.0529nm r=0 ∞ or split up ∞ ∫ R 2 ( r ) r 2 dr = 1 r=0 π ∫ ∫ 2π θ =0 φ=0 Radial Probability P ( r ) = r 2 R2 ( r ) Θ 2 (θ ) sin θ dθ dφ = 1 –– Page 10 of 11 –– © Copyright 2015 Juan E. Cabanela. All Rights Reserved. Duplication only permitted for students in my Physics 322 course. Quantum Mechanics: Spin and Atomic Physics Magnetic Dipole Moment !!" e " µL = − L 2m " " " τ =µ×B " " " " F = −∇ − µ ·B ( # Spin S = s ( s + 1)! "# " q # µs = g S 2m Sz = ms ! where ms = −s, −s + 1,..., s − 1, s ) Identical Particles for B varying only in z direction ψ S ( x1 , x2 ) ≡ ψ n ( x1 )ψ n ' ( x2 ) + ψ n ' ( x1 )ψ n ( x2 ) " ∂B ∂B e F = µ z z zˆ = − mℓ % z zˆ ∂z 2m ∂z [Symmetric] ψ A ( x1 , x2 ) ≡ ψ n ( x1 )ψ n ' ( x2 ) − ψ n ' ( x1 )ψ n ( x2 ) [Antisymmetric] Nuclear Physics Size Radioactivity ( r = A × R0 where R0 = 1.2fm ) Qβ + BE = ZmH + Nmn − M A X c 2 Z ( = (m = (m QEC Liquid Drop Model − mdaughter − 2me c 2 parent − mdaughter c 2 Radioactive Decay Z ( Z − 1) (N − Z) 2/3 BE = c! − c3 − c4 1A − c 2A 1/3 ! A $% "$#A$% "$# Volume Surface Term BEMeV = 15.8A − 17.8A Coulomb Term 2/3 Asymmetry Term Z ( Z − 1) (N − Z) − 0.71 − 23.7 1/3 A A 2 –– Page 11 of 11 –– ) parent 2 Term ) Qβ − = m parent − mdaughter c 2 Binding Energy (Experimental) ( ) Q = mi − m f c 2 1/3 dN = − λ N = −R dt N = N 0 e− λt N0 = N 0 e− λT1/2 2 ln 2 λ= T1/2 )
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