Physics 486 Midterm Exam #1 Spring 2018 Thursday February 22, 9:30 am 10:50 am

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1 Physics 486 Midterm Exam #1 Spring 18 Thursday February, 9: am 1:5 am This is a closed book exam. No use of calculators or any other electronic devices is allowed. Work the problems only in your answer booklets only. The exam questions will not be collected at the end, so anything you write on these question pages will NOT be graded You have 8 minutes to work the problems. At the beginning of the exam: 1) Write your name and netid on your answer booklet(s). ) Turn your cell phone off. ) Put away all calculators, phones, computers, notes, and books. During the exam: 1) Show your work and/or reasoning. Answers with no work or explanation get no points. But... ) Don t write long essays explaining your reasoning. We only need to see enough work to confirm that you understand what you re doing and are not just guessing. (If you are guessing, explain that, then verify your guess explicitly.) A good annotated sketch is often the best explanation of all! ) All question parts on this exam are independent: you can get full points on any part even if your answers to all the other parts are incorrect. You should attempt all the question parts! If you get stuck, move on to the next one and come back later. The worst thing you can do is stall on one question and not get to others whose solution may be very simple. 4) Partial credit will be given for incorrect answers if the work is understandable and some of it is correct. IMPORTANT: If you think you ve made a mistake but can t find it, explain what you think is wrong you may well get partial credit for noticing your error! 5) It is fine to leave answers as radicals or irreducible fractions (e.g. 1 or 5/7), but you will lose points for not simplifying answers to an irreducible form (e.g. 4(x y ) / ( 9x Phys 486 Midterm #1 Spring 18 9y) is unacceptable.) When you re done with the exam: Academic Integrity: Turn in EVERYTHING : answer booklet and question pages The giving of assistance to or receiving of assistance from another person, or the use of unauthorized materials during University Examinations can be grounds for disciplinary action, up to and including expulsion from the University. Please be aware that prior to or during an examination, the instructional staff may wish to rearrange the student seating. Such action does not mean that anyone is suspected of inappropriate behavior.

2 Phys 486 Midterm #1 Spring 18

3 Problem 1 : Always Real Phys 486 Midterm #1 Spring 18 Any complex function ψ(x) can be expressed as the sum of a real part and an imaginary part: ψ (x) = f (x) + i g(x) where f (x) and g(x) are both real. Show that the expectation value of momentum, p, for any complex wavefunction ψ (x) = f (x) + i g(x) is a real number. Hint: You will need to use one integration by parts. Problem : Bohr Atom In Bohr s original model of the atom, electrons of charge e and mass m move in circular orbits of radius rn around a heavy, stationary nucleus of charge +Ze. Since the circular motion of the electrons is periodic in their azimuthal angle φ, the allowed values of the radius rn can be determined by applying the quantization rule p φ dφ = nh where generalized momentum pφ = angular momentum L = mvr. Using this quantization rule for angular momentum, and some very elementary classical mechanics expressions, calculate the allowed radii rn for the electron in terms of Z, e, m, n and physical/numerical constants. Problem : Infinite Well Consider a particle in this infinite potential well, which is symmetric around x = : for π V(x) = < x < π. elsewhere Determine the ground-state wavefunction, i.e. the wavefunction of lowest energy ψ1(x). You do NOT need to determine the ground-state energy OR the time-dependent part of the wavefunction... just find ψ1(x).

4 Problem 4 : A Minimalist Wave Packet Phys 486 Midterm #1 Spring 18 A free particle in a region with V = is approximately described by this wavefunction at t = : Ψ(x,) =ψ (x) = A( e i11x + e i1 x + e i1 x ) where A is a known constant. Work in units where the particle s mass m =.5 and the constant = 1. (a) Write down the time-dependent wavefunction Ψ(x, t) that describes how the starting wavefunction Ψ(x, ) given above will evolve with time. (b) At time t =, the probability distribution Ψ*Ψ of this wavefunction has a pretty sharp peak at x =. With what velocity does this peak move? Please give the approximate speed of the peak and the direction in which it moves (+x or x direction). Problem 5 : A Wavefunction to Play With At time t =, a particle is in a state with wavefunction ψ (x) = A e x +i5x where A is a constant of unspecified value. Hints for dealing with absolute values: It is always safest to deal with the two cases x > and x < separately. Alternatively, you may sometimes find it efficient to use the sign function sgn(x) to express both cases with one expression; it is defined as sgn(x) +1 for x > and 1 for x <. (It is undefined at x =.) (a) Find the standard deviation σx of the particle s position at time t =. (b) Find the mean momentum p of the particle at time t =. (c) What can you conclude about the value of the potential V(x) at x =? Hint: making a quick sketch of ψ(x) would be extremely helpful!

5 v v v v = df (x 1,..., x n ) = i=1 n i=1 ( v ˆr i ) ˆr i f x i i dl path = d l du du da = l u l v du dv dv = l u l v l du dvdw w Conceptual version: d l u l u du d l path = d l u d A = d l u d l v dv = (d l u d l v ) d l w Taylor f (x) = n= (1 + x) n 1+ nx sin x x cos x 1 x tan x x e x 1 + x f (n) (x ) (x x ) n n! 1 st order approx for x 1 : sin 1 x x cos 1 x π x tan 1 x x ln(1+ x) x sin cos tan 1 1 [ ] [ ] [ ] sin a sinb = 1 cos(a b) cos(a + b) cos a cosb = 1 cos(a + b) + cos(a b) sin a cosb = 1 sin(a + b) + sin(a b) Complex Numbers e iθ = cosθ + isinθ imag y z = x + iy = re iθ r x real z* x iy = re iθ z z * z = r θ cos = 1+ cosθ sin θ = 1 cosθ sin(a + b) = sin a cosb + cos asinb cos(a + b) = cos a cosb sin asinb Integral Table x x a = 1 a a cos 1 x sin φ dφ = cos φ dφ = π sin φ dφ = φ sin(φ) 4 cos φ dφ = φ + sin(φ) 4 sin θ dθ cos n θ sinθ dθ = cosn+1 θ n + 1 = cos θ cosθ cos θ dθ = sinθ sin θ ( ) x ± a = ln x + x ± a a x = x sin 1 a (a ± x) = 1 a ± x a + x = 1 x a tan 1 a a x = 1 a ln a + x a x (a ± x ) = x / a a ± x x a ± x = ± a ± x x (a ± x) = a a ± x + ln( a ± x) x a ± x = ± 1 ln ( a ± x ) x ( a ± x ) = 1 / a ± x ln(ax) = x ln(ax) x ln(ax) x = 1 [ ln(ax) ] a x = x a x + a x tan 1 a x x a x = x a x + a x tan 1 a x x ± a = x x ± a ± a ln x + x ± a (x acosθ) sinθ dθ (x + a ax cosθ) = 1 a x cosθ / x x + a ax cosθ

6 Old Quantum Theory (19 195) E = hf =!ω p = h λ =!k Quantization : Bohr E = nh Rules Wave Mechanics for a probability distribution P(x) : mean x = x max P(x) x, variance σ x x x x min Wilson- Sommerfeld! one period p q dq = n q h ( ) = x x, σ x standard deviation probability P(x,t) = Ψ(x,t) = Ψ* Ψ Schrödinger Equation! Ψ m x + VΨ = i! Ψ t operators: Ê = i! t, ˆp =! i x, ˆx = x expectation value Q of physical observable Q(x, p) : Q = Ψ* Heisenberg uncertainty principle : σ p σ x! / ˆQ x, i! x Ψ wavefunction boundary conditions : a. Wavefunctions are always continuous. b. Wavefunctions have continuous derivatives, except at points where V = c. Wavefunctions are zero in any region where V = ±. Miscellaneous Math Gaussian probability distribution: P(x; x,σ ) = 1 ( σ e x x ) / σ Sinusoidal π sin (aφ) cos (aφ) dφ = π sin(a) 4a π sin(nφ) sin(mφ) cos(nφ)cos(mφ) dφ = δ π nm sin(nφ) cos(mφ)dφ = Fourier f (x) = 1 A(k) e ikx dk where A(k) = 1 f (x)e ikx 1 e ik1x e ikx = δ k 1 k if k 1 = k ( ) = if k 1 k Exponential e ax = e ax a x e ax = e ax ( ax +1) a x e ax = e ax ( a x + ax + ) a Gaussian e ax bx = π b a e4a x e ax bx = π b b e 4a x e ax bx = a / b ( 4a )e π 4a 5/ a + b Classical Mechanics security blanket " L( q i,!q i,t) = T U Lagrange EOM: L q i = d dt H!q i ( L /!q i ) L equals T+U when! r a =! r a (q i ) dh / dt = L / t L!q i Generalized momentum p i L!q i, force Q i L q i Hamilton s EOM: H q i = dp i dt Special Relativity: E = (pc) + (mc ), H = dq i p i dt Common : F Forces grav = Gm m 1, F r elec = q q 1 4πε r, F = mv cf r γ = 1 1 (v / c), E = γ mc, p = γ mv, v = pc E