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237module6

# 237module6 - 2 y 2 V x where V x = ‰ ≤ x ≤ L ∞...

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Quantum Mechanics I - Module 6 Discussion Questions 1. What is degeneracy? 2. What is meant by the statement “energies add and wavefunctions multi- ply?” Problems 1. Consider a particle of mass m in N dimensions with H = 1 2 m N X i =1 p 2 i + V ( x i ) where V ( x i ) = 0 for all x i for which 0 x i L and V ( x i ) = otherwise. (a) What are the energy eigenfunctions and their eigenvalues? (b) What are the ten lowest eigenvalues? (Assume N 5.) (c) How many distinct states are there for each of these ten energies? 2. Consider the two dimensional system H = 1 2 m ( p 2 x + p 2 y ) + 1 2
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Unformatted text preview: 2 y 2 + V ( x ) where V ( x ) = ‰ , ≤ x ≤ L ∞ , otherwise . (a) What are the energy levels of the system described by H ? (b) Imagine that the parameters of the system are adjusted (frequently called tuning) so that the ﬁrst excited state is degenerate. What are the eigenvalues in this case? (c) Show that the tuning described in part (b) implies that only a ﬁnite number of states are nondegenerate. Give their energies as multiples of ~ ω . 1...
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