1 Answer all questions in this section.

  Value of this section: 60 marks

1. Calculate the probability that for a harmonic oscillator in its ground state. Give your answer to four significant digits. [5 marks]

   Maple hints: The number is Pi in Maple and is sqrt(u). Maple will need to know that b is a positive parameter.

2. Explain briefly why there is no such thing as a molecule. [5 marks]
3. Suppose that you apply a pulse to a set of protons in a Fourier transform NMR spectrometer and then immediately start acquiring an FID. What happens? Explain briefly. [5 marks]
4. In the self-consistent field (SCF) method, the electronic Hamiltonian is broken up into a set of one-electron Hamiltonians. Write down the one-electron SCF Hamiltonian for an electron in a hydrogen molecule. [5 marks]
5. The Hamiltonian is normally just the sum of the kinetic and potential energies. However, for systems in which the potential energy depends on the velocity, the Hamiltonian must be constructed slightly differently. For example, the classical Hamiltonian for a charged particle in a magnetic field is

   

   where is a vector which depends on the configuration (shape, intensity) of the magnetic field and q is the charge of the particle. Write down the quantum mechanical Hamiltonian operator corresponding to this classical Hamiltonian. Explain briefly why it would not be possible to write this operator in the usual form

   

   [5 marks]

6. Calculate the expectation value of for a hydrogen atom in its ground state. [10 marks]

   Maple hints: The number is Pi in Maple, is sqrt(u), and the exponential function is exp(u). Maple will need to know that is a positive parameter.

7. Calculate the delocalization energy of 2-vinyl-1,3-butadiene:

   

   Only the carbon skeleton is shown. Report your answer to four significant figures in dimensionless form. The orbital energy of ethene is -1. [10 marks]

   Maple hints: Type with(linalg): to load the linear algebra package. To create a square matrix, type name := matrix(d,d); where d is the dimension of the matrix. To place elements in your matrix, use entermatrix(). To find eigenvalues of a matrix, use evalf(Eigenvals(name));. Note that you don't need the second argument of Eigenvals() if you don't want the eigenvectors.  

8. Suppose that a single particle in a one-dimensional universe is subjected to the potential energy

   

   Find the best possible wavefunction of the form

   

   After completing the calculation, normalize your wavefunction. [15 marks]

   Maple hints: Maple will need to know that a is positive. The exponential function is exp(u) in Maple. Maple may give more than one solution but close inspection should reveal that only one is physically reasonable.