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\Px/ the uncertainty principle he

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7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -
Transcribed Image Text:7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -
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