One important quantum mechanics experiment, which using quantum effects explain the dependency of specific heat capacity on temperature, an effect which is not present in the Dulong-Petit law.
This is the solid-state analogue to the black-body radiation problem. It is also therefore a quantum mechanics-specific phenomenon.
It can be seen as the limit case of an Einstein solid at high temperatures. At lower temperatures, the heat capacity depends on temperature.
Wikipedia mentions that it is completely analogous to Planck's law.
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Specific heat capacity, often simply referred to as specific heat, is a physical property of a substance that measures the amount of heat energy required to raise the temperature of a unit mass of that substance by one degree Celsius (or one Kelvin). The specific heat capacity is typically denoted by the symbol \( c \) and is expressed in units such as joules per kilogram per degree Celsius (J/kg·°C) or joules per kilogram per Kelvin (J/kg·K).