Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 8.4, Problem 8.12P
To determine
Show that the antisymmetric state can be expressed in terms of molecular orbital’s.
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Suppose that the out-of-plane distortion of an AB3 planar molecule is described by a potential energy
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If the force constant for a 1-dimensional harmonic oscillator decreases (and all other parameters and properties remain constant), how does the energy gap between n = 3 and n = 4 states change?
(A) The energy gap does not change.(B) The energy gap increases.(C) The energy gap decreases.(D) The energy gap change cannot be determined.
Enumerate all states of the hydrogen atom corresponding to the principal quantum
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Chapter 8 Solutions
Introduction To Quantum Mechanics
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- Consider an object containing 6 one-dimensional oscillators (this object could represent a model of 2 atoms in an Einstein solid). There are 4 quanta of vibrational energy in the object. (a) How many microstates are there, all with the same energy? (b) If you examined a collection of 38000 objects of this kind, each containing 4 quanta of energy, about how many of these objects would you expect to find in the microstate 000004?arrow_forward(a) What is the magnitude of the orbital angular momentum in a state with L=3 ? (b) What is the magnitude of its largest projection on an imposed z axis?arrow_forwardWhether the energy gap for a system decreases, is constant, or increases as the initial quantum number for the transition increases can be determined by calculating the second derivative of the energy level expression with respect to the quantum number. Calculate the second derivative with respect to the quantum number of the energy level expression for a linear rotor, EJ = hBJ(J+1) and also for the harmonic oscillator, Ev = hυ\upsilonυ(v+1/2).arrow_forward
- (a) “An electron is caught by the interatomic void site of a Cu crystal and exhibits 0.3 and 0.7 probabilities at a time in the middle point of the trap” Discuss the validity of this observation and explain why.(b) “The state function of a hydrogen atom is broken at a space point on its way towards it mean position when the atom is oscillating” Justify the possibility of this.arrow_forwardEstimate kBT at room temperature, and convert this energy into electronvolts (eV). Using this result, answer the following: (a) Would you expect hydrogen atoms to be ionized at room temperature? (The binding energy of an electron in a hydrogen atom is 13.6 eV.) (b) Would you expect the rotational energy levels of diatomic molecules to be excited at room temperature? (It costs about 10−4 eV to promote such a system to an excited rotational energy level.)arrow_forwardWhat do we get if we multiply the angular momentum raising and lowering operators together?arrow_forward
- Find expectation value of position and for an electron in the ground state of hydrogen across the radial wave function. Express your answers in terms of the Bohr radius a.arrow_forwardConsider a microscopic spring-mass system whose spring stiffness is 50 N/m and the mass is 4 × 10¯24 kg. What is the smallest amount of vibrational energy that can be added to this system? In a collection of these microscopic oscillators, the temperature is high enough that the ground state and the first three excited states are occupied. What are the possible energies of photons emitted by these oscillators? Sketch an energy level diagram.arrow_forwardShow the solution that proves that Prandtl number is dimensionlessarrow_forward
- Estimate kBT at room temperature, and convert this energy into electronvolts (eV). Using this result, answer the following: (a) Would you expect hydrogen atoms to be ionized at room temperature? (The binding energy of an electron in a hydrogen atom is 13.6 eV.) (b) Would you expect the rotational energy levels of diatomic molecules to be excited at room temperature? (It costs about 10-4 eV to promote such a system to an excited rotational energy level.)arrow_forwardSolve the 3-dimensional harmonic oscillator for which V(r) = 1/2 mω2(x2 + y2 + z2), by the separation of variables in Cartesian coordinates. Assume that the 1-D oscillator has eigenfunctions ψn(x) that have corresponding energy eigenvalues En = (n+1/2)ħω. What is the degeneracy of the 1st excited state of the oscillator?arrow_forwardTwo identical particles of spin ½ are moving under the influence of a onedimensional harmonic oscillator potential. Assuming that the two-particle system is in a triplet spin state, find the energy levels, the wave functions, and the degeneracies corresponding to the three lowest states.arrow_forward
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