MATHEMATICAL TRIPOS Part II PAPER 4 Before you begin read ...

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16 31B Asymptotic Methods The stationary Schrödinger equation in one dimension has the form ǫ 2 d2 ψ = −(E − V (x)) ψ, dx2 where ǫ can be assumed to be small. Using the Liouville–Green method, show that two approximate solutions in a region where V (x) < E are 1 ψ(x) ∼ {± (E − V (x)) 1/4 exp i ∫ x } (E − V (x ′ )) 1/2 dx ′ , ǫ where c is suitably chosen. Without deriving connection formulae in detail, describe how one obtains the condition ∫ 1 b ( (E − V (x ′ )) 1/2 dx ′ = n + 1 ) π (∗) ǫ 2 a for the approximate energies E of bound states in a smooth potential well. appropriate values of a, b and n. c State the Estimate the range of n for which (∗) gives a good approximation to the true bound state energies in the cases (i) V (x) = |x|, (ii) V (x) = x 2 + λx 6 with λ small and positive, (iii) V (x) = x 2 − λx 6 with λ small and positive. Part II, Paper 4
17 32A Principles of Quantum Mechanics Setting = 1, the raising and lowering operators J ± = J 1 ± iJ 2 for angular momentum satisfy [J 3 , J ± ] = ±J ± , J ± |j m〉 = √ (j ∓ m)(j ± m + 1) |j m ± 1〉 , where J 3 |j m〉 = m|j m〉. Find the matrix representation S ± for J ± in the basis {|1 1〉, |1 0〉, |1 − 1〉} of j = 1 states. Hence, calculate the matrix representation S of J. Suppose that the angular momentum of the state v = |1 m〉 is measured in the direction n = (0, sin θ, cos θ) to be +1. Find the components of v, expressing each component by a single term consisting of products of powers of sin(θ/2) and cos(θ/2) multiplied by constants. Suppose that two measurements of a total angular momentum 1 system are made. The first is made in the third direction with value +1, and the second measurement is subsequently immediately made in direction n. What is the probability that the second measurement is also +1 Part II, Paper 4 [TURN OVER
Page 1 and 2: MATHEMATICAL TRIPOS Part II Friday,
Page 3 and 4: 4G Coding and Cryptography Describe
Page 5 and 6: 5 10E Cosmology The number density
Page 7 and 8: 7 13K Statistical Modelling Let (X
Page 9 and 10: 9 17F Graph Theory (a) Show that ev
Page 11 and 12: 11 22G Linear Analysis Let X be a B
Page 13 and 14: 13 26K Applied Probability (a) Defi
Page 15: 15 30B Partial Differential Equatio
Page 19 and 20: 19 34C Statistical Physics Non-rela
Page 21: 21 38D Waves The shallow-water equa

MATHEMATICAL TRIPOS Part II PAPER 4 Before you begin read ...

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