the probability of an electron in the ground state of the hydrogen atom being at a distance r from the center of the atom is P1s(r) = 4r 2 a 3 0 e −2r/a0 r = a0, is the value of r that has the greatest probability. Prove this
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the probability of an electron in the ground state of the
hydrogen atom being at a distance r from the center of the atom is
P1s(r) =
4r
2
a
3
0
e
−2r/a0
r = a0, is the value of r that has the greatest probability. Prove this
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- The quantum-mechanical treatment of the hydrogen atom gives an expression for the wave function ψ, , of the 1s orbital:where ris the distance from the nucleus and a₀ is 52.92 pm. The electron probability density is the probability of finding the elec-tron in a tiny volume at distance rfrom the nucleus and is pro-portional to ψ² . The radial probability distribution is the total probability of finding the electron at all points at distance rfromthe nucleus and is proportional to 4πr² ψ² . Calculate the values(to three significant figures) of ψ, ψ² , and 4πr2² ψ² to fill in the fol-lowing table, and sketch plots of these quantities versus r.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 ground-state wave function for the electron in a hydro- gen atom is 1 1,(7) = VTa where r is the radial coordinate of the electron and a, is the Bohr radius. (a) Show that the wave function as given is normalized. (b) Find the probability of locating the electron between r, = a,/2 and r, = 3a,/2.
- 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?Needs Complete typed solution with 100 % accuracy.a 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 in a hydrogen atom is approximated by a one-dimensional infinite square well potential. The normalised wavefunction of an electron in a stationary state is defined as *(x) = √√ sin (""). L where n is the principal quantum number and L is the width of the potential. The width of the potential is L = 1 x 10-¹0 m. (a) Explain the meaning of the term normalised wavefunction and why normalisation is important. (b) Use the wavefunction defined above with n = 2 to determine the probability that an electron in the first excited state will be found in the range between x = 0 and x = 1 × 10-¹¹ m. Use an appropriate trigonometric identity to simplify your calculation. (c) Use the time-independent Schrödinger Equation and the wavefunction defined above to find the energies of the first two stationary states. You may assume that the electron is trapped in a potential defined as V(x) = 0 for 0≤x≤L ∞ for elsewhere.Explain each stepA hypothetical molecule oscillates with a natural frequency of 1.4 × 1013 Hz. Part (a) What is the energy difference, in electron volts, between adjacent harmonic oscillator states of the hypothetical molecule? Part (b) What is the quantum number of the state of the hypothetical molecule that has an energy of 0.75 eV? Round your answer to the nearest integer.