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237module4 - well potential always has one bound eigenvalue...

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Quantum Mechanics I - Module 4 Getting Started 1. Here are some integrals that you may have already encountered. These appear frequently in quantum mechanics and you should know how to evaluate them. Try to integrate the following (if you don’t know the “trick” to integrating these, your TA will give you a hint): (a) R + -∞ e - x 2 dx (b) R + -∞ xe - x 2 dx (c) R + -∞ x 2 e - x 2 dx 2. How would you express the orthogonality of the two normalized state | n i and | n 0 i ? Be sure to use Dirac notation! 3. Look at the chart on page 226 of your textbook. For each of the eight cases, describe (either in words or with a picture) the form of the wavefunction in each region. Also, write the general form of the wavefunction (as an equation!) in each region. Discussion Questions 1. Show from a qualitative argument that a one-dimensional finite square
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Unformatted text preview: well potential always has one bound eigenvalue, no matter how shallow the binding region. What would the eigenfunction look like if the binding region were very shallow? 2. Why do the lowest eigenvalues and eigenfunctions of an infinite square well provide the best approximation to the corresponding eigenvalues and eigenfunctions of a finite square well? 3. If the eigenfunctions of a potential have definite parities, the one of lowest energy always has even parity. Explain why. Enrichment Problem A particle of mass m in an infinite square well has a wavefunction at t = 0 proportional to: sin 3 πx 2 L cos πx 2 L . 1. What is ψ ( x,t ) for t > 0? 2. What are the expectation values of x and p , including time dependence? 1...
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