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5. A free particle has the following wave function at t = 0: V(x,0) = Ne-a|x| = [Ne-ª* x>0 Near x < 0 where N and a are real, positive constants. a) Normalize this wave function. b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function is symmetric around a = 0 what do you expect (x) to be?). The following integral could be useful: 1. me -bx dx = m! fm+1

5. A free particle has the following wave function at t = 0: V(x,0) = Ne-a|x| = [Ne-ª* x>0 Near x < 0 where N and a are real, positive constants. a) Normalize this wave function. b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function is symmetric around a = 0 what do you expect (x) to be?). The following integral could be useful: 1. me -bx dx = m! fm+1

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5. A free particle has the following wave function at t = 0: V(x, 0) = Ne- -a|x| = So 0 Ne-ax Neax where N and a are real, positive constants. a) Normalize this wave function. b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function is symmetric around x = 0 what do you expect (x) to be?). The following integral could be useful: -bx dx xme = m! fm+1 x > 0 x < 0 c) What is the expectation value of the momentum, (p)? d) We can represent this wave function at t = 0 as a superposition (linear combination) of complex exponentials: (x,0) = Ne-a|x| =[ A(k)etkx dk -∞ Determine the amplitude function A(k). e) Find an expression for (x, t). You can leave your answer in the form of an integral over k but make sure all dependence on k is made explicit. Optional: this choice of wavefunction V(x, 0) is really just for practice with calculating some expectation values, normal- ization, etc, but it's actually a BAD choice for a wavefunction. Why is it a bad choice? Why might it fail at being a realistic wavefuction?
Transcribed Image Text:5. A free particle has the following wave function at t = 0: V(x, 0) = Ne- -a|x| = So 0 Ne-ax Neax where N and a are real, positive constants. a) Normalize this wave function. b) Determine Ax by finding (x) and (x²). You can simplify the work by using symmetry (for example, if the wave function is symmetric around x = 0 what do you expect (x) to be?). The following integral could be useful: -bx dx xme = m! fm+1 x > 0 x < 0 c) What is the expectation value of the momentum, (p)? d) We can represent this wave function at t = 0 as a superposition (linear combination) of complex exponentials: (x,0) = Ne-a|x| =[ A(k)etkx dk -∞ Determine the amplitude function A(k). e) Find an expression for (x, t). You can leave your answer in the form of an integral over k but make sure all dependence on k is made explicit. Optional: this choice of wavefunction V(x, 0) is really just for practice with calculating some expectation values, normal- ization, etc, but it's actually a BAD choice for a wavefunction. Why is it a bad choice? Why might it fail at being a realistic wavefuction?
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