10:50 4 of 6 SPA5319 (2023) SECTION B Answer ALL questions in Section B [Expect to use no more than three pages of A4 for each Section-B question.] Question B1 This question involves a finite potential barrier with V everywhere else, as displayed below: = Vo in the region 0 < x < L and V = 0 V = Vo (1) (2) V=0 x=0 x=L a) Write down and solve the time-independent Schrödinger equation in regions (1), (2) and (3) of this system, for particles with energy E > Vo. In each region, identify which parts of your solutions correspond to particles moving to the left and right. [6 marks] b) For a beam of particles with energy E > V that is incident from the right, identify the parts of the solutions you derived in part a) that correspond to incident, reflected and transmitted particles. Identify which part of your solutions can be set to zero in this case, and explain why this can be done. [4 marks] c) Without performing the calculations explicitly, outline the different steps you would perform in order to show that the transmission coefficient for this scenario is given by 4q²k² T = (k² − q²)² sin²(qL) +4q²k² ' where k² = 2mE/ħ² and q² = 2m(E - Vo)/h². [7 marks] d) Explain how you can use the expression for T from part c) of this question to determine the transmission coefficient for the case in which the particles have E < V. Write down the transmission coefficient in this case, and compare it to the corresponding expression you would obtain from using the Gamow factor formalism to calculate the rate of tunneling. Explain how these two different results are related, and why they differ from each other. SPA5319 (2023) Question B2 The wavefunction of an electron in a hydrogen atom is written as follows: 1 r e -r/200 cos 0, 3/2 Αναπαρ ao 4Teoh²/mee2 0.0529 nm is the Bohr radius. where ao = a) Identify the value of the quantum numbers n, I and m₁ for this electron. [8 marks] Page 5 [6 marks] b) State the energy level of this electron in eV, and determine the amount of energy that would be required to promote it one energy level higher. qmplus.qmul.ac.uk - - Private 10:50 where k² = 2mE/h² and q² = 2m(E - Vo)/h². 5 of 6 [7 marks] ou can use the expression for T from part c) of this question to determine the transmission coefficient for the case in which the particles have E < V. Write down the transmission coefficient in this case, and compare it to the corresponding expression you would obtain from using the Gamow factor formalism to calculate the rate of tunneling. Explain how these two different results are related, and why they differ from each other. SPA5319 (2023) Question B2 The wavefunction of an electron in a hydrogen atom is written as follows: 1 r 3/2 Αναπα ao e -r/200 cose, where ao = 4πεoh²/mee² ~ 0.0529 nm is the Bohr radius. a) Identify the value of the quantum numbers n, I and m₁ for this electron. [8 marks] Page 5 [6 marks] b) State the energy level of this electron in eV, and determine the amount of energy that would be required to promote it one energy level higher. [Hint: The groundstate energy of the hydrogen atom is -13.6 eV.] d) [5 marks] Write down an expression for the probability density associated with this particle, and sketch a diagram to illustrate the dependence of this probability density on the angular coordinates and o. Identify whether the particle is in an s or p orbital. [6 marks] Write down an expression for the probability of finding this particle at a distance r from the central nucleus. Find all values of r at which this probability is at an extremum, and use your results to identify the value of r at which you would be most likely to find this particle if you observed its position. End of Paper - An Appendix of 1 page follows [8 marks] Turn over Page 6 qmplus.qmul.ac.uk - Private SPA5319 (2023)