Determine the integral | P(r) dr for the radial probability density for the ground state of the hydrogen atom P(r) = re-2rla O-1 O 0.5
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- (1) Find the average orbital radius for the electron in the 3p state of hydrogen. Compare your answer with the radius of the Bohr orbit for n=3. (2) What is the probability that this electron is outside the radius given by the Bohr model?The expectation value,Calculate the number of angles that L can make with the z-axis for an l=3 electron.(d) The following orbital belongs to the 3d subshell of the Hydrogen atom: r Y(r, 0, 0) = A(Z) θ, φ) 2 r e 3ao sin² (0) e²i зао where A and ao are constants. Using the operator for the z-component of orbital angular momentum (L₂ = -ih d/do) determine the m, for this particular orbital. (e) Consider the wavefunction, r r Y(r,0,0) = A-e 2do cos(0) do (i) Identify the radial part of this orbital function and the number of radial nodes. (ii) Identify the angular part of the orbital function and the number of angular nodes. Z (iii) Using this information and the L₂ = -ih d/do operator obtain the n, 1, and, m quantum numbers and identify the orbital.Suppose you measure the angular momentum in the z-direction L, for an /= 2 hydrogen atom in the state | > 2 > |0 > +i/ |2 >. The eigenvalues of %3D V10 10 Lz are – 2h, -ħ, 0, ħ, 2ħfor the eigenvectors | – 2 >, |– 1>, |0 >, |1 >, |2 >, respectively. What is AL,? V31 10 7 19 25(a) How many angles can L make with the z -axis for an l = 2 electron? (b) Calculate the value of the smallest angle.The wave function for the ground state of hydrogen is given by 100(0,0) = Ae¯¯r/ª Find the constant A that will normalize this wave func- tion over all space.The wave function for hydrogen in the 1s state may be expressed as Psi(r) = Ae−r/a0, where A = 1/sqrt(pi*a03) Determine the probability for locating the electron between r = 0 and r = a0.In a particular state of the hydrogen atom, the angle between the angular momentum vector L →and the z-axis is u = 26.6°. If this is the smallest angle for this particular value of the orbital quantum number l, what is l?