22. Using the normalization condition, show that the constant A has the value (ma/ħn)/4 for the one- dimensional simple harmonic oscillator in its ground state.
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- For an infinite potential well of length L, determine the difference in probability that a particle might be found between x = 0.25L and x = 0.75L between the n = 3 state and the n = 5 states.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 -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
- 2. Consider a particle of mass m in an infinite square well, 0(≤ x ≤a). At the time t = 0 the particle is in the ground (n = 1). Then at t> 0 a weak time-dependent external potential is turned on: H' = Axe T(x, t) = Ae-iwt e-(mw/ħ).x² which is a solution to Schrödinger's equation. Determine the potential V(x) that is consistent with this wave function, Note: You do not have to normalize V since Schrödinger's equation is linear.An electron is in an infinite potential well of width 364 pm, and is in the normalised superposition state Ψ=cos(θ) ψ5-sin(θ) i ψ8. If the value of θ is -1.03 radians, what is the expectation value of energy, in eV, of the electron?