We have measured the low temperature current-voltage characteristics (
I(V)) of GaAs/AlAs resonant tunnelling diodes with sub-micron lateral
dimensions. Additional peaks in I(V) are observed due to resonant tunn
elling via one-dimensional quantum wire states. In the presence of a m
agnetic field oriented perpendicular to the current and parallel to th
e wire the peaks show a complex splitting evolving into a regular seri
es at high field with up to 20 resonances. For the smallest device we
are able to deduce the probability density of the lowest three bound s
tates from the magnetic field dependence of the current and show that
the confining potential is close to parabolic. For a magnetic field wh
ich is perpendicular to both the current and the wire a much weaker de
pendence on magnetic field is observed confirming the one-dimensional
nature of our device. Finally, in the presence of a field oriented par
allel to the current a continuous transition from electrostatic (at lo
w field) to magnetic confinement (at high field) is observed.