Calculate the probability of an electron in the 2s state of the hydrogen atom being inside the region of the proton (radius ≈ 1.2 x 10-15 m). Repeat for a 2p electron.
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Calculate the probability of an electron in the 2s state of the hydrogen atom being inside the region of the proton (radius ≈ 1.2 x 10-15 m). Repeat for a 2p electron.
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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, (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,…Taking the n=3 states as a representative example, explain the relationship between the complexity of hydrogen’s standing waves in the radial direction and their complexity in the angular direction at a given value of n. What relationship would this be considered a direct relationship or inverse relationship?
- The expectation value,Calculate the average orbital radius of a 3d electron in the hydrogen atom. Compare with the Bohr radius for a n 3 electron. (a) What is the probability of a 3d electron in the hydrogen atom being at a greater radius than the n 3 Bohr electron?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?An electron in a hydrogen atom failing from an excited state (n=7) to a relaxed state has the same wavelength as an electron moving at a speed of 7281 m/s. Determine the relaxed orbit that this electron relaxed to.The energies in a 2D particle-in-a-box are given by h² 8mL 2 in which the box is a square enclosure with Lx = Ly = L, and nx, ny = 1, 2, 3,... . (a) If the particle is an electron and L = 300 pm (assume three significant figures), find the value of the lowest energy level in units of 10-18 J (that is, if the energy is 5.00 × 10-18 J, you would report it as "5.00"). E n, n (n₂ ² + n₂²) y x ya 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.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.An electron is in a three-dimensional box. The xx- and zz-sides of the box have the same length, but the yy-side has a different length. The two lowest energy levels are 2.18 eVeV and 3.47 eVeV, and the degeneracy of each of these levels (including the degeneracy due to the electron spin) is two. What is the length LY for side of the box? What are the lengths LXLX, LZLZ for sides of the box? What is the energy for the next higher energy state? What are the quantum numbers for the next higher energy state? What is the degeneracy (including the spin degeneracy) for the next higher energy state?