Quantum Physics (Normal): Assignment 1 Lecturer: Dr. Michael Schmidt ([email protected]) Submit the scanned PDF version of the handwritten assignment or text-based PDF through Turnitin by 11:59pm Friday 19 May 2017. You are expected to show all working out for each question. 1. The first excited state of a particle in an infinite square well of width L are 15 Points ϕ2(x) =r2 L sin2πx L  (1) (a) Show that ϕ2(x) is correctly normalised. (b) Why it is standard practice to give wave functions in normalised form? (c) Show that ϕ2(x) is not a momentum eigenstate. 2. Semi-classical approximation [~ 1] to α-decay. 30 Points (a) Using the ansatz ψ(x) = A(x)eiφ(x), show that the real and imaginary part of the timeindependent Schr¨odinger equation with potential V (x) leads to Aφ02 −iAφ00−2iA0φ0−A00 = 2m(E −V (x)) ~2 A . (2) (b) Find solutions to real an imaginary part of Eq. (2). Hint: Neglect the term A00 to find an approximate solution to the real part. Show that ψ(x) =X ± C±p p(x) e±i ~Rx dyp(y) with p(x) = 2m(E −V (x)) . (3) Argue why this can be considered as semi-classical limit. (c) To better understand the semi-classical limit, consider the first two terms in Eq. (2). For ~ 1, the first term dominates over the second. Rewrite |φ0|2 ~|φ00| in terms of a condition for the de Broglie wavelength λ ≡ h/p and briefly discuss its physical significance. (d) Apply Eq. (3) to a simple model of α-decay, the spontaneous emission of a α-particle (2 protons and 2 neutrons) by a radioactive nucleus. We model the nucleus by a potential well inside the nucleus (with Z + 2 protons) and Coulomb potential outside the nucleus. V (r) =(−V0 r < L 2eZe 4π0r = 2Zα r r ≥ L [α = e2 4π0 ] (4) The α-particle is not in a bound state and can be considered freely moving in the nucleus. It tunnels through the classically forbidden region of the potential. The transmission probability is given by the ratio of the probability densities at the exit and entry point of the tunnelling. i. Show that the transition probability is given by T ' e−2γ with γ = 1 ~RR L |p(x)|dx Hint: Assume that the barrier is infinitely broad (compared to the size of the nucleus). ii. Derive the probability that the α-particle escapes the nucleus. Hint: Use the substitution r = Rsin2 u to do the integral for the transition probability. The escape probability is the product of transition probability and frequency of collisions of the α-particle with the wall. iii. Give an expression for the half-life time. 1 3. In the lecture we solved the harmonic oscillator using the ladder operator method. Using the commutation relation of the momentum operator ˆ p and the position operator ˆ x, show the commutation relation of the ladder operators ˆ a and ˆ a†, [ˆ a,ˆ a†] = 1. 5 Points 4. Consider a charged particle (e.g. electron) of mass m and charge q < 0 in a magnetic field B pointing in the negative z-direction. The Hamiltonian ˆ H with vector potential ~ A is given by 35 Points ˆ H = 1 2mp~− q c ~ A2 ~ A = B 2  −y x 0  . (5)(a) What motion do you expect in classical physics? (b) Write down the Hamiltonian and replace p~ →−i~~ ∇ and show that it is separable in a sum of two Hamiltonians ˆ Hxy and ˆ Hz describing the motion in the x−y plane and along the z-axis, respectively. What does it imply for the wave function? (c) Show that [ ˆ H, ˆ pz] = 0. What does it mean physically? (d) Show that the Hamiltonian ˆ Hxy can be written in the form of the Hamiltonian of the harmonic oscillator, ˆ Hho = ~ωa†a + 1 2
.i. Introduce raising and lowering operators a† and a, a =r c 2q~Bˆ Πx + iˆ Πy (6) with ˆ Πx ≡ ˆ px − q c ˆ Ax and ˆ Πy ≡ ˆ py − q c ˆ Ay. Show that they satisfy the same commutation relation as the ones for the harmonic oscillator, i.e. [ˆ a,ˆ a†] = 1. ii. Rewrite the Hamiltonian and determine the frequency ω. iii. Using your knowledge about the harmonic oscillator, what can you say about the energy levels of an electron in the magnetic field? iv. Use the ladder termination condition to show that ψ(x,y) = f(x+iy)exp−qB 4~cx2 + y2
is the ground state for an arbitrary function f(x + iy). (bonus) Take f(z) = Nnzn with a normalisation constant Nn, which is chosen such that the wave function is properly normalised. Rewrite the solution using polar coordinates z = reiϕ and determine the average radius squared, hr2i. Discuss the implications of your result for an electron which is confined to a two-dimensional disc perpendicular to the magnetic field. Hint: Use the definition of the Γ function Γ(t) ≡R∞ 0 xt−1e−xdx to do the integral. 5. Consider a diatomic molecule with a moment of inertia about its centre of mass of I and a natural vibration frequency of ν0. 15 Points (a) Write down expressions for the (i) rotational energy, and (ii) vibrational energy of the molecule, indicating the possible values for relevant quantum numbers. (b) Using the selection rules show that the vibrational-rotational absorption spectrum consists of two groups of lines whose frequencies are given by hν = hν0 ± ~2 I l, l = 1,2,3,... . (7) (c) Using the result from (b) show that the spacing (in frequency) between the lines in both groups is equal to ∆ν = ~ 2πI Revised on May 4, 2017 Dr. Michael Schmidt 2