1. Consider the two-particle Hamiltonian given by...Show that the
exchange operator ℰ12 commutes with H as long as the two-particle
potential has the property that V(r1, r2) = V(r2, r1). Get solution
2. Two identical spin-1/2 particles with mass m are in a one-dimensional infinite square-well potential with width a, so V(x) = 0 for 0 ≤ x ≤ a, and there are infinite potential barriers at x = 0 and x = a. The particles do not interact with each other; they see only the infinite square-well potential.(a) Calculate the energies of the three lowest-energy singlet states.(b) Calculate the energies of the three lowest-energy triplet states.(c) Suppose that the particles are in a state with wave function...where x1 is the position of particle 1 and x2 is the position of particle 2. Are the particles in a triplet spin state or a singlet spin state? Explain. Get solution
3. Two identical spin-0 particles with mass m are confined inside a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c, where a ≤ b ≤ c, and the potential barriers at the walls of the box are infinitely high. The particles do not interact with each other; they see only the potential of the box. Write down the normalized wave function ψ(x1, y1, z1, x2, y2, z2) for the ground state, and indicate the energy of the ground state. Get solution
4. (a) Two identical spin-1/2 particles are confined inside of the rectangular box from Exercise 13.3. The particles do not interact with each other; they see only the potential of the box. Write down the normalized spatial part of the lowest-energy singlet wave function. What is the energy of this state?(b) Write down the normalized spatial part of the lowest-energy triplet wave function. What is the energy of this state?(c) What is the total spin s of the ground-state wave function for the system? Get solution
5. Two identical spin-1/2 particles are confined to an infinite one-dimensional square well of width a with infinite potential barriers at x = 0 and x = a. The potential is V(x) = 0 for 0 ≤ x ≤ a. Suppose that the particles interact weakly by the potential V1(x) = Kδ(x1 − x2), where x1 and x2 are the positions of the two particles, K is a constant, and δ is the Dirac delta function. This represents a very short-range weak force between the two particles.(a) Using first-order perturbation theory, find the perturbation to the energy of the lowest-energy singlet state.(b) Show that the first-order perturbation to the energy of the lowest-energy triplet state is zero.(c) What is the physical reason for the answer in part (b)? Get solution
6. Two identical spin-1/2 particles are in the one-dimensional simple harmonic oscillator potential V(x) = (1/2) Kx2. The particles do not interact with each other; they see only the harmonic oscillator potential. The particles are in the lowest-energy triplet state (s = 1).(a) Write down the normalized spatial part of the wave function.(b) Calculate the energy of this state.(c) If the positions of both particles are measured, what is the probability that both particles will be located on the right-hand side of the minimum in the potential (i.e., the probability that both particles have x > 0)? Get solution
7. Show that the n shell in an atom can hold 2n2 electrons. Get solution
8. Sodium has Z = 11. Determine the ground-state electron configuration. Get solution
9. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Determine the ground-state L, S, and J values for(a) calcium which has the electron configuration...(b) yttrium which has the electron configuration... Get solution
10. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Consider the excited state of beryllium with the electron configuration...Determine all possible L, S, and J values. Note that the Pauli exclusion principle for the two n = 3 electrons can be ignored; why? Get solution
11. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Zirconium (Z = 40) consists of closed subshells plus 2 electrons in an unfilled d subshell. Derive the set of allowed L, S, and J values, and determine which state has the lowest energy. Get solution
12. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.The electron configuration for nitrogen is...Calculate L, S, and J for the ground state. Get solution
2. Two identical spin-1/2 particles with mass m are in a one-dimensional infinite square-well potential with width a, so V(x) = 0 for 0 ≤ x ≤ a, and there are infinite potential barriers at x = 0 and x = a. The particles do not interact with each other; they see only the infinite square-well potential.(a) Calculate the energies of the three lowest-energy singlet states.(b) Calculate the energies of the three lowest-energy triplet states.(c) Suppose that the particles are in a state with wave function...where x1 is the position of particle 1 and x2 is the position of particle 2. Are the particles in a triplet spin state or a singlet spin state? Explain. Get solution
3. Two identical spin-0 particles with mass m are confined inside a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c, where a ≤ b ≤ c, and the potential barriers at the walls of the box are infinitely high. The particles do not interact with each other; they see only the potential of the box. Write down the normalized wave function ψ(x1, y1, z1, x2, y2, z2) for the ground state, and indicate the energy of the ground state. Get solution
4. (a) Two identical spin-1/2 particles are confined inside of the rectangular box from Exercise 13.3. The particles do not interact with each other; they see only the potential of the box. Write down the normalized spatial part of the lowest-energy singlet wave function. What is the energy of this state?(b) Write down the normalized spatial part of the lowest-energy triplet wave function. What is the energy of this state?(c) What is the total spin s of the ground-state wave function for the system? Get solution
5. Two identical spin-1/2 particles are confined to an infinite one-dimensional square well of width a with infinite potential barriers at x = 0 and x = a. The potential is V(x) = 0 for 0 ≤ x ≤ a. Suppose that the particles interact weakly by the potential V1(x) = Kδ(x1 − x2), where x1 and x2 are the positions of the two particles, K is a constant, and δ is the Dirac delta function. This represents a very short-range weak force between the two particles.(a) Using first-order perturbation theory, find the perturbation to the energy of the lowest-energy singlet state.(b) Show that the first-order perturbation to the energy of the lowest-energy triplet state is zero.(c) What is the physical reason for the answer in part (b)? Get solution
6. Two identical spin-1/2 particles are in the one-dimensional simple harmonic oscillator potential V(x) = (1/2) Kx2. The particles do not interact with each other; they see only the harmonic oscillator potential. The particles are in the lowest-energy triplet state (s = 1).(a) Write down the normalized spatial part of the wave function.(b) Calculate the energy of this state.(c) If the positions of both particles are measured, what is the probability that both particles will be located on the right-hand side of the minimum in the potential (i.e., the probability that both particles have x > 0)? Get solution
7. Show that the n shell in an atom can hold 2n2 electrons. Get solution
8. Sodium has Z = 11. Determine the ground-state electron configuration. Get solution
9. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Determine the ground-state L, S, and J values for(a) calcium which has the electron configuration...(b) yttrium which has the electron configuration... Get solution
10. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Consider the excited state of beryllium with the electron configuration...Determine all possible L, S, and J values. Note that the Pauli exclusion principle for the two n = 3 electrons can be ignored; why? Get solution
11. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Zirconium (Z = 40) consists of closed subshells plus 2 electrons in an unfilled d subshell. Derive the set of allowed L, S, and J values, and determine which state has the lowest energy. Get solution
12. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.The electron configuration for nitrogen is...Calculate L, S, and J for the ground state. Get solution
