Skip to content
Quantum Mechanics

Diagnostic Problems

Diagnostic problems are short. Their purpose is not grading; it is routing. Work quickly, then follow the repair links for any item that is shaky.

For a longer prerequisite quiz, use Self-Diagnostic Quiz. For a full readiness map, use Diagnostic Checklist.

Quick Diagnostic Set

Section titled “Quick Diagnostic Set”
  1. QM-PROB-D001, normalization. Normalize v=(1,i)∈C2v=(1,i)\in\mathbb C^2.
Check

The norm is 2\sqrt2, so the normalized vector is (1,i)/2(1,i)/\sqrt2. Review Inner Products and Finite-Dimensional Hilbert Spaces if this is shaky.

  1. QM-PROB-D002, Born rule. If ∣ψ⟩=(∣0⟩+∣1⟩)/2\lvert\psi\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2, what is the probability of measuring ∣0⟩\lvert0\rangle in the computational basis?
Check

The probability is ∣1/2∣2=1/2\lvert1/\sqrt2\rvert^2=1/2. Review Born Rule.

  1. QM-PROB-D003, boundary conditions. A function on 0<x<L0<x<L satisfies f(0)=f(L)=0f(0)=f(L)=0 and f′′+k2f=0f''+k^2f=0. What nonzero sine solutions are allowed?
Check

The allowed wave numbers are k=nπ/Lk=n\pi/L for n=1,2,…n=1,2,\ldots. Review Boundary Conditions and Infinite Square Well.

  1. QM-PROB-D004, commutator. If [X,P]=iℏI[X,P]=i\hbar I, what sign appears in [P,X][P,X]?
Check

[P,X]=−[X,P]=−iℏI[P,X]=-[X,P]=-i\hbar I. Review Commutators and Canonical Commutation Relations.

  1. QM-PROB-D005, tensor-product dimension. If dim⁡HA=2\dim\mathcal H_A=2 and dim⁡HB=3\dim\mathcal H_B=3, what is dim⁡(HA⊗HB)\dim(\mathcal H_A\otimes\mathcal H_B)?
Check

The dimension is 2⋅3=62\cdot3=6. Review Tensor Product Exercises.

  1. QM-PROB-D006, density matrix. Is ρ=I/2\rho=I/2 a valid one-qubit density matrix?
Check

Yes. It is Hermitian, positive, and has trace one. It is mixed because Tr⁡ρ2=1/2\operatorname{Tr}\rho^2=1/2. Review Density Operators.

  1. QM-PROB-D007, scattering. Why is transmission usually a current ratio rather than just an amplitude-squared ratio?
Check

Current includes velocity or wave-number factors. If incident and transmitted wave numbers differ, amplitudes alone do not give the probability flux ratio. Review Reflection and Transmission Coefficients.

How to Interpret Results

Section titled “How to Interpret Results”

If two or more items feel shaky, pause before entering a graduate route. If only one item is shaky, follow its repair link and then continue with Undergraduate Problem Map or Problems by Method.

References

Section titled “References”
  • D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
  • R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.