Gallium arsenide is piezoelectric, so it is possible to generate coupled mechanical and electrical surface acoustic waves (SAWs) by applying a high-frequency voltage to a transducer on the surface of GaAs. By combining SAWs with existing low-dimensional nanostructures one can create a series of dynamic quantum dots corresponding to the minima of the travelling electric wave, and each dot carries a single electron at the SAW velocity (∼ 2800 m/s). These devices may be of use in developing future quantum information processors, and also offer an ideal environment for probing the quantum mechanical behaviour of single electrons. This thesis describes a numerical and theoretical study of the dynamics of an electron in a range of geometries. The numerical techniques for solving the time-dependent Schr ̈dinger equation with an arbitrary time-dependent potential will be described in Chapter 2, and then applied in Chapter 3 to calculate the transmission of an electron through an Aharonov-Bohm (AB) ring. It will be seen that an important property of the techniques used in this thesis is that they can be easily adapted to study realistic geometries, and we will see features in the AB oscillations which do not arise in simplified analytic descriptions. In Chapter 4, we will then study a device consisting of two parallel SAW channels separated by a controllable tunnelling barrier. We will use numerical simulations to investigate the effect of electric and magnetic fields upon the electron dynamics, and develop an analytic model to explain the simulation results. From the model, it will be apparent that it is possible to use this device to rotate the state of the electron to an arbitrary superposition of the first two eigenstates. We then introduce coherent and squeezed states in Chapter 5, which are ex- cited states of the quantum harmonic oscillator. Coherent and squeezed electronic states may be of use in quantum information processing, and could also arise due to unwanted perturbations in a SAW device. We will discuss how these states can be controllably generated in a SAW device, and also discuss how they could then be detected. In Chapter 6 we describe how to use the motion of a SAW to create a rapidly- changing potential in the frame of the electron, leading to a nonadiabatic excita- tion. The nonadiabatically-excited state oscillates from side to side within a 1D channel on a few-picosecond timescale, and this motion can be probed by placing a tunnelling barrier at one side of the channel. Numerical simulations will be performed to show how this motion can be controlled, and the simulation results will be seen to be in good agreement with recent experimental work performed by colleagues. Finally, we will show that this device can be used to measure the initial state of an electron which is an arbitrary superposition of the first two eigenstates.