Module Descriptors

Applied Mathematics

MP356: Quantum Mechanics I (5 ECTS)

(This course will be run every other year.) This is an introductory course to 1- dimensional quantum mechanics. The course covers topics such as the Schrodinger equation and wave functions, infinite and finite square well potentials, the harmonic oscillator, wave packets, vector spaces and the uncertainty principle.

Taught in Semester(s) I. Examined in Semester(s) I.

Workload: 36 hours (24 Lecture hours, 12 Tutorial hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. Calculate position and momentum expectation values for simple 1-d wave functions.
  2. Solve for energy eigenvalues and eigenfunctions of the Schrodinger Equation for the 1-d infinite potential well.
  3. Solve for energy eigenvalues and eigenfunctions of the Schrodinger Equation for the 1-d Harmonic oscillator
  4. Solve for bound and free energy eigenvalues and eigenfunctions of the Schrodinger Equation for a finite potential well and finite potential barrier.
  5. Calculate 1-d free wave packet solutions using the Fourier transform.
  6. Calculate inner products, norms, unitary base changes for a vector space.
  7. Derive the Cauchy-Schwarz inequality and uncertainty principle for 1-d quantum systems.
  8. Apply the uncertainty principle in selected cases such as showing that the minimal uncertainty wave function is a Gaussian wave packet.

Indicative Content

This is an introductory course in quantum mechanics in 1 spatial dimension. Topics covered include:

  1. Introduction to the 1-d Schrodinger equation and wave functions. The probabilistic interpretation of the wave function and expectation values for observables.
  2. The infinite square well potential. Finding the energy eigenvalues and eigenfunctions. Computing some expectations and showing orthogonality of wave functions.
  3. The Harmonic Oscillator. Finding the energy eigenvalues and eigenfunctions by solving differential equations. Solving by an alternative algebraic method.
  4. Solve the Schrodinger equation for a finite square potential well and barrier illustrating scattering and transmission.
  5. Wave packets solution for the free Schrodinger equation using the Fourier transform.
  6. Introduction to vector spaces and Hilbert spaces including inner products, norms, orthonormal bases and the Gram–Schmidt process, linear transformations, matrices and vectors.
  7. The Heisenberg uncertainty principle and examples.

Module Resources

Introduction to Quantum Mechanics, David J. Griffiths, Pearson Education

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