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Advanced university · Physics

Advanced university Physics practice

15 questions on statistical mechanics, analytical mechanics, relativity, quantum mechanics. The answer key is a separate download.

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  1. A canonical system has a nondegenerate ground level at 0 and an excited level of degeneracy 10. If Δ/(kBT)=2.3, calculate Z. Round the final result to 4 decimal places if needed.

  2. Using {f,g}=f_q g_p−f_p g_q, calculate {10q², 4p²} at q=3, p=0. Treat variables as dimensionless. Round the final result to 4 decimal places if needed.

  3. Using {f,g}=f_q g_p−f_p g_q, calculate {2q², 10p²} at q=3, p=3. Treat variables as dimensionless. Round the final result to 4 decimal places if needed.

  4. Using {f,g}=f_q g_p−f_p g_q, calculate {1q², 9p²} at q=6, p=-6. Treat variables as dimensionless. Round the final result to 4 decimal places if needed.

  5. Two events have cΔt=36 m and Δx=18 m, with no y or z separation. Using s²=(cΔt)²−(Δx)², calculate s². Round the final result to 4 decimal places if needed. Answer in m².

  6. A nondegenerate unperturbed state has component probabilities 0.3 and 0.7 in a basis where V=diag(-7,-20) meV. Calculate its first-order energy correction. Round the final result to 4 decimal places if needed. Answer in meV.

  7. A normalized state has probabilities 0.5 and 0.5 for two orthogonal eigenstates of A with eigenvalues -6 and 26. Calculate ⟨A⟩. Round the final result to 4 decimal places if needed.

  8. For H(q,p)=p²/(2×9) + 16q²/2 in SI units, find q̇ at p=-69 kg·m/s. Round the final result to 4 decimal places if needed. Answer in m/s.

  9. In a microcanonical model, the multiplicity increases by a factor 218. Find ΔS/kB. Round the final result to 4 decimal places if needed.

  10. A canonical system has a nondegenerate ground level at 0 and an excited level of degeneracy 6. If Δ/(kBT)=1, calculate Z. Round the final result to 4 decimal places if needed.

  11. An observable returns -1 with probability 0.9, and 10 otherwise. Calculate its quantum measurement variance. Round the final result to 4 decimal places if needed.

  12. In a microcanonical model, the multiplicity increases by a factor 193. Find ΔS/kB. Round the final result to 4 decimal places if needed.

  13. Use SI-compatible constants in L=4ẋ²/2 − 1x⁴. Find ẍ when x=-4 m. Round the final result to 4 decimal places if needed. Answer in m/s².

  14. An observable returns -8 with probability 0.3, and 4 otherwise. Calculate its quantum measurement variance. Round the final result to 4 decimal places if needed.

  15. For a two-level canonical model with ground degeneracy 1, excited degeneracy 6 and Δ/(kBT)=2.2, find the total probability of being in the excited level. Round the final result to 4 decimal places if needed.