Physics 443: Quantum Mechanics I
Midterm Exam II: 12:30–13:45 Thursday 29 March 2018 in SA124a for seventy-five (75) minutes.
This is a closed-book examination. No calculators are permitted.
Answer fully five (5) of the eight (8) questions. Cross out solutions you do not want marked,
or else only your first five (5) answers are marked if you complete more than five (5).
All questions have equal value. Answers must be written in the test booklets provided.
1. What is the double-slit experiment, how was it first realized in the laboratory, and how to corpuscularity and undu-
larity manifest in this experiment?
2. Explain the meaning of the Heisenberg-Born-Jordan matrix elements, the role of frequency in these elements, and
the nature and meaning of the eigenvalues if the matrix is hermitian.
3. Describe how a matrix is diagonalized and the roles of orthogonal matrices for real-valued matrices and unitary
matrices for complex-valued matrices; also explain the relation between the eigenvalues and eigenvectors and the
nature of degeneracy.
4. Prove that two nondegenerate matrices share eigenvectors if they commute with each other.
5. Write the Hamiltonian for the quantized simple harmonic oscillator in terms of momentum and position operators;
then write the ladder and number operators, from which you then need to explain the energy spectrum.
6. Write the Hamiltonian for the quantized anharmonic oscillator, show that the Hamiltonian is stationary in time, and
explain how the anharmonic term describes slightly irregular spacing of spectral lines.
7. Explain the projection operator and its role in obtaining measurement outcomes along with the probability for such
outcomes.
8. Explain why the momentum eigenstates of a free particle are stationary states.
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