UPSC Combined Geo-Scientist, Mains · 2020 · Mains Paper II
Q1(l). Probabilities of quantum states
Question
ψ=(√3/3)φ₁+(2/3)φ₂+(√2/3)φ₃, where the φᵢ are orthonormal. Find the probabilities and the numbers expected in each state for an ensemble of 800 systems.
Answer in brief
P₁=1/3, P₂=4/9, P₃=2/9. Expected counts are 266.67, 355.56 and 177.78, respectively.
Key points
- Born' s rule gives Pᵢ=|cᵢ|². Squaring the three coefficients gives 3/9, 4/9 and 2/9, which sum to one.
- Mean counts are 800Pᵢ: 800/3, 3200/9 and 1600/9. These expectations need not be integers.
- Actual measured counts are integers, sum to 800 and fluctuate statistically.
The full model answer
As written for the answer booklet: the diagram carries the working, the lines stay short. Read here only; it is not offered as a download.
CoversProbabilities of quantum statesQuantum mechanics
All parts of Question 1
- 1(a) Critical-point inflection5 marks
- 1(b) Boyle and inversion temperatures5 marks
- 1(c) Accuracy at high diffraction angle5 marks
- 1(d) Composition change and non-expansion work5 marks
- 1(e) Van't Hoff plots5 marks
- 1(f) Potassium sulfate activities5 marks
- 1(g) Reaction order and rate constant5 marks
- 1(h) Catalytic rate enhancement5 marks
- 1(i) Electrochemical potential of zinc ions5 marks
- 1(j) Open-circuit emf and equilibrium5 marks
- 1(k) Hermitian operators5 marks
- 1(l) Probabilities of quantum states5 marks
- 1(m) Hydrogen rotational spectra5 marks
- 1(n) Wavelength and the einstein5 marks
- 1(o) Uranyl-sensitised decomposition5 marks
- 1(p) HI photolysis yield5 marks