GRE Physics Domain 5: Quantum Mechanics Welcome to your GRE Physics Domain 5: Quantum Mechanics 1. GRE Physics: Quantum Mechanics A particle in a one-dimensional box of width \( L \) is in a state described by the wave function \( \psi(x) = A \sin(\frac{2\pi x}{L}) \). What is the energy of the particle? A. \( \frac{h^2}{8mL^2} \) B. \( \frac{h^2}{4mL^2} \) C. \( \frac{h^2}{2mL^2} \) D. \( \frac{h^2}{mL^2} \) None 2. GRE Physics: Quantum Mechanics What is the uncertainty in position (\( \Delta x \)) for a particle if the uncertainty in its momentum (\( \Delta p \)) is known to be \( \frac{\hbar}{2} \)? A. \( \frac{\hbar}{2} \) B. \( \hbar \) C. \( 2\hbar \) D. \( \frac{\hbar}{\sqrt{2}} \) None 3. GRE Physics: Quantum Mechanics Which of the following represents the correct commutation relation between position (\( \hat{x} \)) and momentum (\( \hat{p} \)) operators? A. \( [\hat{x}, \hat{p}] = i\hbar \) B. \( [\hat{x}, \hat{p}] = \hbar \) C. \( [\hat{x}, \hat{p}] = -i\hbar \) D. \( [\hat{x}, \hat{p}] = 0 \) None 4. GRE Physics: Quantum Mechanics For a particle in a three-dimensional spherical box, the energy eigenfunctions are labeled by quantum numbers \( n, l, \) and \( m \). What is the degeneracy of the energy level characterized by quantum number \( n \)? A. \( n \) B. \( n^2 \) C. \( 2n + 1 \) D. \( 2n^2 \) None 5. GRE Physics: Quantum Mechanics What is the probability of finding a particle in the ground state in a one-dimensional box of length \( L \) between \( \frac{L}{4} \) and \( \frac{3L}{4} \)? A. 0.5 B. 0.25 C. 0.75 D. 1 None 6. GRE Physics: Quantum Mechanics Which principle states that no two fermions can occupy the same quantum state? A. Heisenberg's uncertainty principle B. Pauli exclusion principle C. Hund's rule D. Aufbau principle None 7. GRE Physics: Quantum Mechanics In quantum mechanics, what does the operator \( \hat{H} \) represent? A. Momentum B. Position C. Hamiltonian D. Spin None 8. GRE Physics: Quantum Mechanics If a quantum system is described by the normalized wave function \( \psi(x) \), what is the probability of finding the particle in the interval \( a \leq x \leq b \)? A. \( \int_a^b \psi(x) dx \) B. \( \int_a^b |\psi(x)|^2 dx \) C. \( |\int_a^b \psi(x) dx|^2 \) D. \( \int_a^b |\psi(x)| dx \) None 9. GRE Physics: Quantum Mechanics What is the spin quantum number for an electron? A. \( 1/2 \) B. \( 1 \) C. \( 3/2 \) D. \( 2 \) None 10. GRE Physics: Quantum Mechanics In the context of quantum mechanics, what does the term 'orthonormality' refer to? A. The condition where operators commute B. The conservation of probability C. The property of wave functions being orthogonal and normalized D. The linear superposition of states None 1 out of 10 Time is Up! Time's up
