125. An attractive square well potential is 55 represented by -V for r a The scattering due to this potential in low energy limit is proportional to nth power of a. Here n is (1) 2 (2) 4 (3) 5 (4) 6
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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 -16. Consider the wave function Mx) = (alT)" exp(-ax12) Calculate (x) for n = 1, 2. Can you quickly write down the result for (")?QUESTION 7 Use the Schrödinger equation to calculate the energy of a 1-dimensional particle-in-a-box system in which the normalized wave function is 4' = e sin(6x). The box boundaries are at x=0 and x=r/3. The potential energy is zero when 0 < x <- and o outside of these boundaries. 18h? m h2 8m h2 36n2m none are correct
- QUESTION 6 Consider a 1-dimensional particle-in-a-box system. How long is the box in radians if the wave function is Y =sin(8x) ? 4 4л none are correct T/2 O O O22 A particle is confined to the one-dimensional infinite poten- tial well of Fig. 39-2. If the particle is in its ground state, what is its probability of detection between (a) x = 0 and x = 0.30L. (b) x = 0.70L and x = L, and (c) x = 0.30L and x = 0.70L? U(x) Fig 39-2a 4. 00, -Vo, V(z) = 16a 0, Use the WKB approximation to determine the minimum value that V must have in order for this potential to allow for a bound state.