Consider a state of a 2-electron diatomic molecule AB described by the electronic normalized wave function (1,2) = 4(1,2) [a(1) B(2) – a(2) B(1)] where p(1, 2) is the spatial part of the electronic wave function. (a) What must be the value of the integral (p(1, 2)|4(1,2)) so that the complete (spatial and spin) function v(1, 2) is normalized?. (b) Is the spatial function p(1, 2) symmetric or antisymmetric with respect to the exchange of the space coordinates of electron 1 and 2?
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- A one-dimensional square well of infinite depth and 1 Å width contains 3 electrons. The potential well is described by V = 0 for 0 1 Å. For a temperature of T = 0 K, the average energy of the 3 electrons is E = 12.4 cV in the approxination that one neglects the Coulomb interaction between clectrons. In the same approximation and for T = 0 K, what is the average cuergy for 4 electrons in this potential well?Consider a two-dimensional (2D) lattice having N atoms with mass m. Assume that each atom interacts with only nearest neighbors with force constant k. Thus, take the dispersion relation as 4k qa sin 2 where a is the lattice constant. m a) In the long-wavelength limit, i.e., as q → 0, obtain the density of modes D(w) = dN/dw, that is the number of vibration modes per frequency interval dw. You should work in the Debye model. b) Calculate the total (internal) energy U of the lattice at high temperatures (kgT > hw). c) At high temperature limit, the average potential energy is equal to the average kinetic energy, and thus half the total energy. Find the mean square displacement V(r²) of an atom from its equilibrium position. Comment on the stability of 2D crystals.The normalized 2 p eigen functions of hydrogen atom are 1 1 1 1 r √√ (2a0) ³/2 √√(2a)³/2 2a0 1 1 sin e¯iº, for m = +1, 0, −1 respectively. √ (20) ³/2 -e-r/2a0 e-¹/2a0 r 2a0 r · 2a⁰ sin ei -e-r/2a0 cos , Apply the raising operator L+ = Lx+ iLy and lowering operator to show that the states with m = +2 do not exist.
- Suppose that the ground vibrational state of a molecule is modelled by using the particle-in-a-box wavefunction ψ0 = (2/L)1/2 sin(πx/L) for 0 ≤ x ≤ L and 0 elsewhere. Calculate the Franck–Condon factor for a transition to a vibrational state described by the wavefunction ψ′= (2/L)1/2sin{π(x −L/4)/L} for L/4 ≤ x ≤ 5L/4 and 0 elsewhere.An arbitrary quantm mechanical system is initially in the ground state |0). At t = 0, a perturbation of the form H' (t) = Hoc:/T is applied. Show that at large times the probability that. tlhe system is in state |1) is given by |(0}Ho|1}|2 (A) A + (Ac)? where As is the difference in cnergy of states |0) and |1). Be specific about what assumption, if any, were inade arriving at your conclusion.A diatomic molecule is modeled as a Morse oscillator, and one finds that its energy level differences decrease from E2 - E₁ = 1374.2 cm-¹ to E7 - E6 = 1139.6 cm ¹. (a) Use this information and the quantum energy levels for the Morse oscillator to find the harmonic angular frequency, w, in cm-¹. (b) What is the dissociation energy, D (in kJ/mole) for this Morse oscillator? (Note that the energy units, 11.9627 J/mole = 1 cm-¹.)
- can you help for this questionA deuterium molecule (D2₂) at 30°K is known to be in the state, 1 /26 12/₂) = = |3|1, 1) + 4 |7, 3) + |7, 1) where , m) are eigenstates of the angular momentum operator. (a) If one were to measure L₂, what posible values one would get and what would be their associated probabilities? (b) Repeat (a) but for L². (c) What is the expectation value of the energy (E) of the molecule in this state, assuming purely rotational states. Take c= 30.4 cm-¹, where I=moment of inertia of D₂ and c=speed of light. Express your answer in eV. -pls answer d and e
- A quantum system described by a Hamiltonian Ĥ is in the state | 1/2(191) - 1/21/2) + √/15 Φι √2 14/) N = (1 +21) 19:3) + √ēlga)]. where [on) are the eigenstates of energy such that Ĥ|ón) = nEo|ón), Eo has units of energy, and NER. (a) Find a suitable scalar N such that |) is normalized. (b) Let the energy of y) be measured. Give all possible measurement results and their corresponding probabilities. Assume that the measurement is ideal, i.e., no measurement errors occur.Bhwa),` where a is (a) Show that the partition function of a photon gas is Z = Πα for the vibration modes of the electromagnetic field. (b) Show that the free energy of an electromagnetic radiation is F(V,T) = = -aVT4, 1-e-Bħwa, where a is a constant. The density of states of an electromagnetic radiation is given by V g(w) = 723 w².Consider a three-dimensional infinite-well modeled as a cube of dimensions L x L x L. The length L is such that the ground state energy of one electron confined to this box is 0.50eV. (a) Write down the four lowest energy states and evaluate their corresponding degeneracy. (b) If 15 (total) electrons are placed in the box, find the Fermi energy of the system. (c) What is the total energy of the 15-electron system? (d) How much energy would be required to lift an electron from Fermi energy of part (b) to the first excited state? Need full detailed answers and explanations to understand the concept.