THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE PHYS3034 PAPER 1 (Senior Physics – Three subjects) SEMESTER 1, 2021 TIME ALLOWED: 3 HOURS READING TIME: 10 MINUTES INSTRUCTIONS: There are three sections in this paper, each worth 45 marks: Section A: QUANTUM PHYSICS Section B: STATISTICAL MECHANICS Section C: COMPUTATIONAL PHYSICS All questions have the value shown. You should attempt all three (3) subjects. SEMESTER 1, 2021 Page 2 of 9 DATA speed of light: c = 3.0× 108ms 1 fundamental charge: e = 1.6× 10 19C electron mass: me = 9.1× 10 31 kg atomic mass unit: u = 1.66× 10 27 kg Avogadro’s number: NA = 6.022× 1023 Boltzmann constant: kB = 1.38× 10 23 JK 1 electron volt: eV = 1.6× 10 19 J gravitational constant: G = 6.67× 10 11Nm2 kg 2 Planck’s constant: h = 6.626× 10 34 J s (Planck’s constant)/2pi: ~ = 1.055× 10 34 J s Permeability of free space: μ0 = 4pi × 10 7 N A 2 Permittivity of free space: 0 = 8.854× 10 12 C2 N 1 m 2 Stefan-Boltzmann constant: σ = 5.67× 10 8 Wm 2kg 4 Gas constant: R = 8.315× 10 11 J/mol·K SEMESTER 1, 2021 Page 3 of 9 SECTION A: QUANTUM PHYSICS 1. The quantum states of the hydrogen atom were presented in the lecture. (a) List the quantum numbers, which describe hydrogen, and briefly explain their phys- ical significance. (b) Give an expression for the degeneracy of the energy levels of hydrogen. (c) Briefly describe the fine structure splitting in hydrogen atom. (d) Briefly describe the hyperfine splitting in hydrogen atom. (e) In muonic hydrogen the electron is replaced by a muon, which has the same charge as the electron, but a mass mμc2 = 106MeV. (i) Calculate the energy of the ground state of muonic hydrogen. (ii) Calculate the Bohr radius for the muonic atom. (iii) Compare your result to the hydrogen atom. What does it imply for the Lamb shift in muonic hydrogen (25 marks) 2. The state of a helium atom can be represented as |ψ〉 = |ψspatial〉|ψspin〉 (a) What are the symmetry requirements for the three ket states in the above equation Justify your answer. (b) Why can the states of the Helium atom be classified as either singlet or triplet states Justify your answer. (c) Consider a Helium atom with the electron configuration 1s2s. Which of the possible states has the lowest energy Briefly explain your reasoning. (d) The hydrogen nucleus has a spin 1/2 but the deuterium nucleus has spin 1. Why is the hydrogen atom a suitable candidate for Bose Einstein condensation, but the deuterium atom is not Justify your answer. (20 marks) SEMESTER 1, 2021 Page 4 of 9 SECTION B: STATISTICAL MECHANICS 1. Explain briefly (in about 50 words for each) what is meant by each of the following. (a) The density of states (b) The grand partition function (c) The ultraviolet catastrophe (d) Hawking radiation of a black hole (8 marks) 2. Planck solved the “UV catastrophe” by summing over quantized energy states with wave- length λ for a blackbody with temperature T . The emitted spectrum is known as the Planck function such that P (λ, T ) = 2hc2 λ5 1 ehc/λkT 1 (1) (a) Derive the Planck function in terms of frequency ν (c = λν) using the relation P (ν, T ) = P (λ, T ) ∣∣∣∣dλdν ∣∣∣∣ (2) (b) In the figure above (left), P (λ, T ) has a maximum brightness that peaks at a chang- ing wavelength as a function of temperature, λ1T = c1, known as Wien’s First Law. If you plot the same spectrum using P (ν, T ), it looks similar in shape except that the peak frequency ν2 corresponds to a wavelength λ2 that obeys a different form, λ2T = c2, known as Wien’s Second Law. (c1 and c2 are different constants.) Comment on why there are two versions of Wien’s Law for a fixed temperature T . SEMESTER 1, 2021 Page 5 of 9 (c) In the figure above (right), the two stars Acrux and Gacrux in the Southern Cross are bluer and redder than the Sun, respectively. The surface temperatures of the stars are as follows: 24000 K (Acrux), 5800 K (Sun), 3600 K (Gacrux). (i) Determine the peak wavelengths for all three stars (in microns) using both Wien’s Laws where c1 = 2.8 × 10 3 and c2 = 5.0 × 10 3 in SI units (six numbers). (ii) Determine the ratio of the total energies emitted by both Southern Cross stars when compared to the Sun (two numbers). (d) Give an example other than stars where we can witness Wien’s Law in action in the natural world, i.e. where cooler objects are red and hotter objects are blue. (13 marks) 3. Consider a single harmonic oscillator with allowed energies 0, hν, 2hν, · · · (a) Evaluate the partition function Z for this simple system; express the answer as an expansion and in its most compact form. (b) Write down an expression for the average energy of this simple system at a temper- ature T . (c) Derive an expression for the total energy of this system if it has N identical oscil- lators at a temperature T . (d) Derive an expression for the heat capacity of the system in (c). (12 marks) 4. Study the figure above and answer the following questions. (a) What system is being described here SEMESTER 1, 2021 Page 6 of 9 (b) What is the quantity F (c) What is the quantity μ and why is it sometimes different from F (d) What must happen to the system for the slope through = F to become shallower or steeper Explain. (12 marks) SEMESTER 1, 2021 Page 7 of 9 SECTION C: COMPUTATIONAL PHYSICS 1. This question concerns numerical errors and approximations. (a) Briefly but carefully explain what is meant by: (i) truncation error; (ii) numerical instability. (b) Consider the non-dimensional dynamical equations for a 1-D linear oscillator: dx dt = v, dv dt = x. (3) (i) Show that the total energy, E = 1 2 (x2 + v2) , is a conserved quantity for the oscillator. (ii) Show that one step of Euler’s method with time step τ for this system is x2 = x1 + τv1, v2 = v1 τx1 . (iii) Show that in one step of Euler’s method, the energy increases according to E2 = (1 + τ 2)E1. (4) (iv) Extrapolate Equation (4) to derive an expression for how the radius in x–v space evolves using Euler’s method. Use your result to sketch the trajectory of a numerical solution to Equations (3) in the x–v plane. (15 marks) SEMESTER 1, 2021 Page 8 of 9 2. Consider the (non-dimensional) 1-D linear advection equation: a t = c a x , with c > 0 a constant. (5) (a) Show that a(x, t) = f(x ct) is a solution to Equation (5), with initial condition a(x, 0) = f(x). (b) Sketch the solution to Equation (5) as a function of x at times t = 0 and t > 0, for the initial condition a(x, 0) = exp ( 1 2 x2 ) . (c) Consider a numerical solution of Equation (5) using the scheme an+1j = a n j + 1 2 g ( anj+1 anj 1 ) , where g = cτ/h. Is this scheme useful for solving the problem at hand Justify your answer carefully, using equations as needed. (d) Consider a numerical solution of Equation (5) using the scheme an+1j = 1 2 ( anj 1 + a n j+1 ) + 1 2 g ( anj+1 anj 1 ) , where g = cτ/h. Is this scheme useful for solving the problem at hand Justify your answer carefully, using equations as needed. (15 marks) SEMESTER 1, 2021 Page 9 of 9 3. The 2-D diffusion equation is T t = κ ( 2T x2 + 2T y2 ) where κ > 0 is constant. (6) (a) Derive the Forward Time, Centered Space (FTCS) scheme for solving the 2-D dif- fusion equation: T n+1j l = T n j l + κτ h2 ( T nj 1 l + T n j+1 l + T n j l 1 + T n j l+1 4T nj l ) . (7) (b) By applying von Neumann stability analysis to Equation (7), with trial solutions of the form T nj l = A n(eikxjh + eikylh), derive the stability condition for the method: κτ h2 ≤ 1 4 . (8) (c) Consider the non-dimensional 2-D Laplace equation, 2φ x2 + 2φ y2 = 0 , (9) where φ(x, t) is a scaled electric potential, and x and y are positions in units of a chosen length scale. A student wants to solve the equation numerically with given boundary conditions on a square grid with L points in x and L points in y. The student chooses to use the Jacobi relaxation method, which may be written: φn+1j l = 1 4 ( φnj+1 l + φ n j 1 l + φ n j l 1 + φ n j l+1 ) . (10) (i) Carefully explain the Jacobi relaxation method. Your explanation should in- clude a derivation of Equation (10) and an account of how this equation is applied to the Laplace problem at hand. (ii) Briefly explain how the computational time taken to obtain an accurate numer- ical solution to the given Laplace problem using the Jacobi method will scale with L. (15 marks) THERE ARE NOMORE QUESTIONS.
