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Modern Physics Homework #5 1. Compare the angular momentum of a ground-state electron in the Bohr model of the hydrogen atom with its value in quantum theory. 2. How much more likely is the electron in a ground-state hydrogen atom to be at the distance ao from the nucleus than at the distance ao /2? Than at the distance 2 ao? 3. A 2s electron in a hydrogen atom is more likely than a 2p electron to be closer to the nucleus than r = ao (that is between r = 0 and r = ao). Verify this by calculating the relevant probabilities? 4. Use energy considerations to show that the greatest distance a 1-s electron can be from the nucleus for the hydrogen atom is 2 ao. Find the probability of finding the 1-s electron at a distance greater than 2 ao according to quantum mechanics. 5. Calculate the location at which the radial probability density is a maximum for the 2-s state of the hydrogen atom. Then calculate the expectation value of the radial coordinate in this state. Which answer if either is consistent with the Bohr model prediction. 6. Can there be solutions with E < 0 for the time-independent Schrodinger equation for a system with zero potential? Explain your reasoning. 7. A particle of total energy 9 Vo is incident from the –x axis on a potential given by x 0 8 Vo V 0 0 x a 5 Vo x a Find the probability that the particle will be transmitted on through to the positive side. 8. For a particle in a box, show that the fractional difference in the energy between adjacent eigenvalues is ΔE n 2 n 1 2 En n and use this formula to discuss the classical limit of the system. 9. Prove that wave function, A 1 2 u e 2 u 2 /2 where u 4 km x is a solution to the simple harmonic oscillator and derive its associated energy eigenvalue.