The expectation value of the z-co-ordinate,, in the ground state of th hydrogen atom (wave function: 100(r) normalization constant and ao is the Bohr radius), is Ae-Tlao, where 'A' is th 0.25 ao 0.5 ao O ao
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Q: For what energy levels in the hydrogen atom will we not find l = 2 states?
A: To determine:For what energy levels in the hydrogen atom will we not find l = 2 states?
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- Show by direct substitution that the wave function corresponding to n = 1, l = 0, ml = 0 is a solution of the attached equation corresponding to the ground-state energy of hydrogen.In the subshell e = 3, (a) what is the greatest (most positive) me value, (b) how many states are available with the greatest mn, value, and (c) what is the total number of states in the subshell? (a) Number Units (b) Number Units (c) Number UnitsThe 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,…
- The total probability of finding an electron in the hydrogen atom is related to the integral ∫ r2 e-2r/ao dr Where r is the distance of the electron from the nucleus and ao is the Bohr radius. Evaluate thisintegral.The radial probability density of a hydrogen wavefunction in the 1s state is given by P(r) = |4rr2 (R13 (r))²| and the radial wavefunction R1s (r) = a0 , where ao is 3/2 the Bohr radius. Using the standard integral x"e - ka dx n! calculate the standard deviation in the radial position from the nucleus for the 1s state in the Hydrogen atom. Give your answer in units of the Bohr radius ao.(a) What is the magnitude of the orbital angular momentum in a state with e = 2? (b) What is the magnitude of its largest projection on an imposed axis? (a) Number 2.50998008 Units J.s (b) Number 2.11 Units J.s
- If we neglect interaction between electrons, the ground state energy of the helium atom is E =2 z2((- e2)/(2ao)) = -108.848eV (Z=2). The true (measured) value is – 79.006eV.Calculate the interaction energy e2/r12 supposing that both electrons are in the 1s state and r12 that the spin wave function is anti-symmetric. What E is the ground state energy?(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.Hydrogen gas can be placed inside a strong magnetic field B=12T. The energy of 1s electron in hydrogen atom is 13.6 eV ( 1eV= 1.6*10 J ). a) What is a wavelength of radiation corresponding to a transition between 2p and 1s levels when magnetic field is zero? b) What is a magnetic moment of the atom with its electron initially in s state and in p state? c) What is the wavelength change for the transition from p- to s- if magnetic field is turned on?