Chapter #14 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Oxygen is the second most abundant element in the human body (by number) and the most abundant by mass. However, MRI detection of oxygen atoms is not practical. Why? Get solution

2. Electrons in hydrogen have a much larger magnetic moment (both orbital and spin) than the magnetic moment of the proton. Why then are MRI machines not tuned, for example, to the resonant frequency of the spin magnetic moment of the electron rather than the proton? Get solution

3. Consider an MRI machine with a 1.5 T static field. How far from the resonant frequency would one have to be in order for the spin-flip probability to decrease from its maximum value to each of the following?(a) one-half of the maximum value(b) zero Get solution

4. The random thermal energy of each molecule in the human body is roughly E ≈ kT, where k is Boltzmann’s constant (k = 1.38 × 10−23 J K−1) and T is the temperature. Compare this thermal energy to the potential energy experienced by a proton in a 1.5 T MRI machine. What does the answer say about the efficiency with which protons will align into lower energy states in such a magnetic field? Get solution

5. (a) Verify that the matrix corresponding to UNOT produces the correct output when applied to the | ↓ 〉 and | ↑〉 states.(b) Compute the matrix corresponding to ..., and calculate the result when it is applied to | ↓ 〉 and | ↑〉. What logical operation does ... correspond to? Get solution

6. (a) Show explicitly that the classical XOR gate cannot be inverted.(b) Calculate the matrix corresponding to the inverse of the quantum XOR gate Get solution

7. Show that .... Get solution

8. Apply UXOR to the mixed state (...) (| ↓ ↓〉 + | ↑ ↑〉) Explain what the result means. Get solution