MATH 325

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PAPER CODE MATH 325 EXAMINER: Prof T Teubner DEPARTMENT: Mathematical Sciences NOVEMBER 2025 MARKED HOMEWORK Quantum Mechanics Time allowed: One week INSTRUCTIONS TO CANDIDATES: This paper contains 4 questions. Candidates should attempt all questions. Full marks will be awarded for complete solutions to all questions. Online submission on Canvas by Monday, 17.11.2025, 5pm. By submitting solutions to this assessment you affirm that you have read and understood the Academic Integrity Policy detailed in Ap- pendix L of the Code of Practice on Assessment, https://www.liverpool.ac.uk/media/livacuk/tqsd/code-of- practice-on-assessment/appendix L cop assess.pdf and have successfully passed the Academic Integrity Tutorial and Quiz. The marks achieved on this assessment remain provisional un- til they are ratified by the Board of Examiners. Paper Code MATH 325 Page 1 of 4 CONTINUED 1. A particle moves on the positive x-axis and is described, at some moment in time, by the wave function ψ(x) = Ae x sinx , with A a real and positive constant. (i) Determine A such that ψ is correctly normalised. [2 marks] (ii) Calculate the average position and momentum of the particle. Deter- mine the position of the global maximum of the probability density. Briefly discuss this in comparison to the particle’s average position. [6 marks] (iii) State Heisenberg’s uncertainty principle and show that it is satisfied for the particle with the wave function ψ(x). [6 marks] Hint: You may use the integrals ∫∞ 0 e 2x sin2 x dx = ∫∞ 0 xe 2x sin2 x dx = 1 8 ,∫∞ 0 x2e 2x sin2 x dx = 5 32 , ∫∞ 0 e 2x sinx cosx dx = 1 8 , ∫∞ 0 e 2x cos2 x dx = 3 8 . 2. A particles of mass m and energy E moves on the x-axis under the influence of the potential V given by V (x) = ∞ for x ≤ 0 , 0 for 0 < x < L , V1 for x ≥ L , where V1 and L are real and positive constants. (i) By solving the (time independent) Schro¨dinger equation in the three regions, find the general form of bound state solutions for this potential. There is no need to determine the normalisation constants. [4 marks] (ii) Now impose the required continuity conditions and show that the al- lowed eigenenergies must satisfy the quantisation condition tan √ 2mEL hˉ = √ E V1 E . [6 marks] (iii) Deduce the condition which V1 must fulfill for there to be at least one bound state. [4 marks] (iv) By referring to general symmetry properties of energy eigenfunctions (as discussed in the lectures) and the form of the potential in this problem, explain why the solutions are neither symmetric nor antisymmetric w.r.t. x = L/2. [3 marks] Paper Code MATH 325 Page 2 of 4 CONTINUED 3. A beam of particles of mass m and energy E moves in the positive direction along the x-axis and interacts with the potential step V (x) = { V1 for x < 0 , 0 for 0 ≤ x V1. (i) Solve the Schro¨dinger equation for this setup in the regions x 0. By imposing the required continuity conditions, determine the normalisation constants in both regions in terms of the normalisation of the incoming wave and hence give the wave function in both regions. [6 marks] (ii) Using your results from part (i), derive the reflection and transmission coefficients, R and T , defined as the ratios between the probability fluxes of the reflected and the incoming, and the transmitted and the incoming particles, respectively. Briefly discuss your results in comparison with the same setup in classical mechanics. [6 marks] (iii) A scattering experiment with this setup but unknown V1 uses incoming particles with known energy E > V1. If the measurements give the result R = 1/4, derive the value of V1 in terms of E. [3 marks] Paper Code MATH 325 Page 3 of 4 CONTINUED 4. The Hamiltonian H for the Simple Harmonic Oscillator is H = ( a a + 12 ) hˉω , where ω is a real, positive constant and a and a are the annihilation and creation operators, respectively. (i) The orthonormal energy eigenstates |ψn (n ≥ 0) can be obtained from the (normalised) ground state |ψ0 by |ψn = 1√ n! ( a )n |ψ0 . Using this relation, the commutator of the simple harmonic oscillator, [a, a ] = 1, and a|ψ0 = 0, verify that ψ1|ψ1 = 1 . Show that ψ3|ψ1 = 0 . [4 marks] (ii) Now consider a system in the normalised state |ψ , where |ψ = A ( 3 |ψ0 i √ 5 |ψ2 + √ 6 |ψ4 ) and A > 0 is a real constant. Determine A. Using the number operator N = a a, with N |ψn = n|ψn , calculate the energy expectation value for the state |ψ . [4 marks] (iii) The operators a and a can be defined in terms of the position and momentum operators, x and p , by a = α√ 2 ( 1 mω p ix ) and a = α√ 2 ( 1 mω p + ix ) , where α2 = mω/hˉ. First express x 2 in terms of a and a , then use the relations derived in the lectures, aψn = √ n ψn 1 and a ψn = √ n+ 1 ψn+1 , to calculate ψ5|x 2|ψ7 and ψ6|x 2|ψ6 . [5 marks] Paper Code MATH 325 Page 4 of 4 END

MATH 325最先出现在KJESSAY历史案例。