4-74. In the ground state of the hydrogen atom, according to Bohr's theory, what are (a) the quantum number, (b) the orbit radius, (c) the angular momentum, (d) the linear momentum, (e)
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- 2. The 4p radial wavefunction for the H-atom is (1/32/15) × R(20 - 10p + p²) x Z3/2 x ep/2 Using this information determine the radius of the nodes in units of Bohr radius for the 4p orbitals.In the Bohr model of the atom, we assume that nanoscale particles behave according to classical physics (This assumption is not entirely justified, but does work surprisingly well). In this model, an electron (mass m = 9.11 x 10-³1 kg) orbits a nucleus at a distance that depends on the principal quantum number of the electron (1s orbital, 2s orbital, etc) and the composition of the nucleus. If the electron orbits at a distance of 9.90 x 10-¹1 m due to a Coulomb force of 2.35 x 10-8 N, how many revolutions per second does the electron make? 2.569In accordance with the Bohr's model, find the quantum number that characterises the earth's revolution around the sun in an orbit of radius 1.5 x 10" m with orbital speed 3 x 104 m/s. (Mass of earth = 6.0 x 1024 kg.)
- Why is the analysis of the helium atom much more complex than that of the hydrogen atom, either in a Bohr type of model or using the Schrödinger equation?3) Determine the energy of a photon emitted when an electron relaxes from the excited state ν=4 to ground state ν=1 of a harmonic oscillator if the force constant κ is 285 N m–1.For a hydrogen atom, calculate the energy of an electron with principal quantum number n = 3. (your answer should be in units of 10^-19 J)
- 4. (a) Calculate the de Broglie wavelength of An electron travelling at 4 x 106 m/s and (i) (ii) A car of mass 1.1 × 106 g travelling at 15 m/s. (b) Comment on the significance of the relative magnitude of your answers in part (a). Note: the diameter of an atom is on the order of 10-10 m.1. (a) Why are crystals used in the measurement of matter waves? [1] (b) Which physical phenomena do matter waves exhibit when scattered off of crystals? [1] 2. For de Broglie waves (of free particles), show that: (a) the angular frequency and the wave number are related by [3] ω = ~k 2 2m ; (1) (b) vph is equal to halve the speed of the particle; [3] (c) and vg is equal to the speed of the particle. [2] 3. The wavefunction of a certain quantum system is given by Ψ(x, t) = Ae− mω 2~ x 2−iEt/~ , E = 1 2 ~ω, (2) where n = 1, 2, 3, · · · , ω is a constant and −∞ < x < ∞. (a) Show that [3] A = mω π~ 1/4 (3) (b) Compute ∆x and ∆p. [Recall: ∆q = p hq 2i − hqi 2 ] [14] 1(b) A photon is emitted by a doubly ionised lithium atom (Li²+) when an electron makes a transition to the ground state. The wavelength of the photon is measured to be 10.83 nanometres. Determine the principal quantum number and the energy of the initial state The atomic number of lithium is Z = 3.