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Quantum tunneling is a fundamentally important concept within quantum mechanics, describing an unusual phenomenon where particles, such as electrons, are able to pass through a barrier that, according to classical physics, they should not be able to. This baffling behavior arises from the wave-like nature of particles postulated by quantum mechanics.

Unlike a ball that hits a wall and bounces back in the everyday macroscopic world, an electron has a nonzero probability of 'tunneling' through a potential barrier, even if its energy is less than the height of the barrier. This probability is quantified by the transmission coefficient, which reflects the likelihood of the particle being on the other side of the barrier. The surprising aspect of tunneling is that it enables particles to access regions of space that would be forbidden in classical scenarios, which is pivotal in numerous quantum processes, including those in nuclear fusion and semiconductor physics.

In quantum mechanics, a potential barrier is essentially a region where a particle experiences a force opposing its motion. This force is due to a potential energy that is higher than the particle's kinetic energy. As with the exercise regarding the electron encountering a barrier, the potential energy of the barrier, denoted as U₀, represents a hurdle that classically the particle wouldn't have the energy to overcome.

A potential barrier can be visualized like a hill of a certain height and width. Classically, a ball rolling towards it would need enough kinetic energy to climb over. In contrast, quantum particles have a 'smearing' of their position, described by their wavefunction, which allows for the probability of them being detected beyond the barrier, reflecting the enigmatic tunneling effect.

Energy levels are a quintessential aspect of quantum mechanics, providing a discrete spectrum of energy states that quantum systems, such as atoms and molecules, can occupy. What separates quantum mechanics from classical physics is that these energy levels are quantized; particles can only exist in specific energy states and no in-between values are allowed.

For instance, an electron bound to an atom can only reside in certain orbitals, each corresponding to a distinct energy level. Transitions between these levels occur via the absorption or emission of photons, with energies equivalent to the difference in energy levels. This discrete nature of energy levels is imperative for understanding the electronic structures of atoms, the behavior of materials, and the functioning of lasers, LEDs, and other quantum devices.

Planck's constant (\( h \) ) is a fundamental constant that plays a pivotal role in quantum mechanics. It relates the energy of a photon to its frequency through the equation E = hf, where E is energy, h is Planck's constant, and f is the frequency. Named after Max Planck, who discovered it in 1900, this constant signifies the size of the quantum 'jumps' in energy that electrons make within an atom.

With a value of approximately 6.62607015 × 10^-34 m² kg / s, Planck's constant also sets the scale for the Heisenberg uncertainty principle, which limits the precision with which pairs of physical properties like position and momentum can be known simultaneously. It is deeply woven into the fabric of quantum theory and has practical applications such as determining the energy of an electron transitioning between energy levels (as in the exercise), which helps us understand and predict the behavior of quantum systems.