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The wave-particle duality concept resolves the ultimate question about the nature of electrons: are they waves, or are they particles? Remarkably, they are both. This duality is key in comprehending the behavior of electrons under different circumstances.

In some situations, electrons behave like particles, with traits like mass and localized paths. For instance, in the photoelectric effect, electrons act as energy packets that can dislodge themselves from a material when struck by light. Meanwhile, electrons also demonstrate wave properties that allow them to spread through space. This is seen in the double-slit experiment where electrons create an interference pattern, mimicking waves traversing through slits.

Comprehending this dual nature can seem tricky because in our day-to-day life, objects don't show such behavior. However, on the subatomic level, this duality explains functions within atoms, aiding in our broader understanding of the universe.

Quantum mechanics is the branch of physics that describes the peculiar behavior of particles at the nanoscale. It’s the theoretical foundation for understanding phenomena like the dual nature of electrons.

Unlike classical mechanics, quantum mechanics provides a set of principles that takes into account the wave-particle duality of electrons. One significant principle is the uncertainty principle, which suggests that the precise position and momentum of an electron can’t be known simultaneously. This means electrons don’t stick to defined paths like planets around the sun, as traditional physics would suggest.

Instead, quantum mechanics uses mathematical tools, like wave functions, to determine the probability of an electron’s location. This explains why we might observe wave-like patterns even when electrons are treated as particles. Recognizing these principles helps us make predictions and develop technologies that rely on the behavior of tiny particles.

The photoelectric effect is a powerful demonstration of the particle nature of light and electrons. In this phenomenon, light shone on certain materials can cause the ejection of electrons from that material's surface.

Albert Einstein explained this by proposing that light is made of small packets of energy called photons. When photons with sufficient energy collide with an electron, they transfer their energy to the electron, knocking it off the surface. This shows electrons behaving like particles.

The importance of the photoelectric effect goes beyond just understanding electron behavior. It was pivotal in the development of quantum mechanics, fundamentally challenging classical ideas about light and matter. It's the basis of many modern devices, such as solar panels, which rely on this effect to convert light into electrical energy.

The double-slit experiment is one of the most demonstrative experiments showcasing wave-particle duality. When electrons are fired through a pair of slits, they don’t just create two distinct bunches on a screen, which would suggest particle behavior.

Instead, the electrons form an interference pattern, with several light and dark bands, resembling the behavior of waves. This wave-like interference arises even when electrons travel one at a time through the slits, suggesting that each electron interferes with itself.

This experiment reveals the complex wave nature of electrons and is fundamental in supporting quantum mechanics doctrines. It forces us to reconsider our understanding of particles, as they can exist in states that aren't strictly as particles or as waves but as probabilities and potentials, governed by wave functions, until they are observed.