We report the observation of quantum size efFects in the thermal conductance and Peltier coefficient of a quantum point contact. Our experimental method involves a novel usage of quantum point contacts as local electron gas thermometers.PACS numbers: 73.50.Lw, 72.20.Pa, 73.40.Kp An electrical current I through a conductor is accompanied by a thermal current Q. These currents are driven by the differences in voltage (6V) and temperature (hT) across the conductor. In the regime of linear response, this is conventionally expressed as [1] (1)Here R is the electrical resistance of the conductor, 8 the thermopower, II the Peltier coefficient, and rc the thermal conductance. The cross coefficients S and II are related by the Kelvin-Onsager relation II = ST.(3)In recent years, epitaxial growth and nanofabrication techniques have made it possible to study the regime of quantum ballistic transport [2]. Ideally, in this regime, transport is governed by collisions of charge carriers with the boundaries of the conductor, rather than with impurities or defects. In addition, the lateral size of the conductor can be made comparable to the Fermi wavelength, so that it acts as an electron waveguide.An experimental realization of such an electron waveguide is the quantum point contact (QPC), an electrostatically defined narrow constriction in the twodimensional electron gas (2DEG) of an (Al, Ga)As heterostructure. Electron transport through the point contact is ballistic since the mean free path is much larger than the size of the constriction. In the wider regions on either side of the QPC the transport is diffusive. These regions have a very small resistance (compared to that of the QPC) and may be regarded as electron gas reservoirs [2]. The most striking manifestation of quantum ballistic transport is the quantization in units of 2e /It of the electrical conductance of a QPC [3,4]. This quantization is observed as a series of plateaus in a plot of the conductance versus the voltage on the gate electrodes defining the constriction; the conductance quantum 2e2/It corresponds to a perfectly transmitted one-dimensional (1D) subband or transverse mode in the QPC. Theoretically, it has been argued that quantum size effects in a QPC due to the 1D subband structure should occur as well for the thermal conductance, the thermopower, and the Peltier coefficient [1, 5 -8]. Indeed, quantum oscillations in the thermopower, which line up with the steps between the plateaus in the conductance, have recently been observed experimentally [9]. The conductance and thermopower of a QPC can both be measured, on applying the appropriate current or temperature difference, simply with a voltmeter. A measurement of the Peltier coefficient and the thermal con ductance requires, in addition, a means of detecting the heat flow Q through the constriction. In this Letter, we present a method by which this is possible. We infer Q from the resulting change in the local temperature bT of the electron gas on one or both sides of the QPC.These temperature chan...