Show that the radial probability density function is P(r)= 4π r2R(r2)dr
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Show that the radial probability density function is
P(r)= 4π r2R(r2)dr
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Solved in 3 steps with 2 images
- Show that the uncertainty principle can be expressed in the form ∆L ∆θ ≥ h/2, where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined?Show that the radial wave function R21 for n = 2 and ℓ = 1 is normalized.Show that the hydrogen wave function Ψ211 is normalized
- Find the directions in space where the angular probability density for the l = 2, ml = 0 electron in hydrogen has its maxima and minima.An electron is trapped in a region between two infinitely high energy barriers. In the region between the barriers the potential energy of the electron is zero. The normalized wave function of the electron in the region between the walls is ψ(x) = Asin(bx), where A=0.5nm1/2 and b=1.18nm-1. What is the probability to find the electron between x = 0.99nm and x = 1.01nm.The expectation value, (r), for a hydrogen atom in the 3d₂2 orbital can be written ∞ 4 r = [² dr r² (2) * e-2r/3a0 (r) = = 8 (3⁹.5) a ³ where the integrals over 0 and have already been evaluated and included in this expression. (a) Starting with the integral shown above, define x = r/ao and use this to simplify this integral by expressing it as an integration over x. Be careful and don't forget about the volume element and the integration limits. (b) From part (a), you should now have an integral over x and this variable essentially represents the radial distance of the electron in units of do. Determine (r) for this state. You may find the tabulated integral given above (before problem 3) helpful. (c) The angular part of the 3d₂2 orbital is given by the spherical harmonic, Y₂ (0,4): 5 -√√ 16π Y₂ (0,0) = (3 cos² 0 - 1) Using the standard limits of these variables (0 ≤ 0≤ л; 0≤ ≤ 2π), determine the angles at which this function has nodes. (d) Describe the nodal surfaces for the orbital,…
- A particle is in a three-dimensional cubical box that has side length L. For the state nX = 3, nY = 2, and nZ = 1, for what planes (in addition to the walls of the box) is the probability distribution function zero?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?