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Write the Hamiltonian for the As atom. Use summation notation and specify limits
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- Instead of x=0 to a, assume that the limits on the 1-D box were x=+(a/2) to (a/2). Derive acceptable wavefunction for this particle-in-a-box. You may have to consult an integral table to determine the normalization constant. What are the quantized energies for the particle?Why does the concept of antisymmetric wavefunctions not need to be considered for the hydrogen atom?For the system in exercise 9.1, determine the Hamiltonian equation of motion.
- Verify that the wavefunctions in equation 10.20 satisfy the three-dimensional Schrdinger equation.What are x,y, and z for 111 of a 3-D particle-in-a-box? The operators for y and z are similar to the operator for x, except that y is substituted for x wherever it appears, and the same for z. What point in the box is described by these average values?a Construct Slater determinant wavefunctions for Be and B. Hint: Although you need only include one p orbital for B, you should recognize that up to six possible determinants can be constructed. b How many different Slater determinants can be constructed for C, assuming that the p electrons spread out among the available p orbitals and have the same spin? How many different Slater determinants are there for F?
- Show that the correct behavior of a wavefunction for He is antisymmetric by exchanging the electrons to show that (1,2)=(2,1).An anharmonic oscillator has the potential function V=12kx2+cx4 where c can be considered a sort of anharmonicity constant. Determine the energy correction to the ground state of the anharmonic oscillator in terms of c, assuming that H is the ideal harmonic oscillator Hamiltonian operator. Use the integral table in Appendix 1 in this book.Set up and evaluate numerically the integral that shows that Y11 and Y11 are orthogonal.
- Show that 2 and 3 for the harmonic oscillator are orthogonal.For an unbound or free particle having mass m in the complete absence of any potential energy that is, V=0, the acceptable one-dimensional wavefunctions are =Aei(2mE)1/2x/h+Bei(2mE)1/2x/h, where A and B are constants and E is the energy of the particle. Is this wavefunction normalizable over the interval x+? Explain the significance of your answer.Assume that for a particle on a ring the operator for the angular momentum, p, is i(/). What is the eigenvalue for momentum for a particle having unnormalized equal to e3i? The integration limits are 0 to 2. What is the average value of the momentum, p for a particle having this wavefunction? How are these results justified?