My question is: from the point of view of classical mechanics, when a wave encounters a barrier, it is totally transmitted through the barrier, while in quantum mechanical there is also a part of the wave that is reflected? Or is it the opposite? If I calculate the transmission and reflection coefficients in the classically accessible and inaccessible regions I conclude yes. However, I understood that the tunneling effect is the phenomenon whereby, in quantum mechanics, the wave can pass through the barrier as if there was a Tunnel.
In both quantum and classical physics, a wave incident on a barrier is, in general, partially transmitted and partially reflected.
In classical physics, if you had a particle incident on a barrier with energy larger than the particle's energy, the particle would never penetrate the barrier and get to the other side.
In quantum mechanics, the particle is represented by a wave function and even when the energy of the particle is less than the barrier energy the wave has a non-zero value in the barrier and a non-zero value after the barrier. The square amplitude of that wave gives the probability of finding the particle, so there is a non-zero probability of finding it on the other side of the barrier. In such cases, the square amplitude of the wave drops exponentially with distance into the barrier. That is, if the probability of the particle penetrating through 1 metre of barrier is 0.1, then the probability after 2 metres is 0.1 x 0.1 = 0.01, the probability of penetrating three metres is 0.1 x 0.1 x 0.1 = 0.001 and so on. A wave whose amplitude decreases exponentially with distance is called an evanescent wave.
In classical physics, there are also evanescent waves but they just describe an amount of energy that varies continuously and that amount of energy drops exponentially. In quantum physics, the energy of a finite system can only have a finite number of values. The amplitude of the wave gives the probability and not the energy.
For a detailed account of tunnelling, see
Classically, a particle impinging on a potential is always reflected or transmitted. Which one occurs is determined by whether the particle's kinetic energy is less than or greater than the maximum of the potential barrier. If the particle has enough energy, it passes through the potential; if not, it rebounds.
Quantum mechanically, the situation is more complex. Normally, there is both a transmitted and reflected component to the outgoing wave. There is always some transmission; even if the energy is small, there will be a small amount of transmission. However, sometimes there is no reflection; in cases of "resonant" scattering, the wave is entirely transmitted, with no reflection. This requires the energy to be tuned to one of discrete set of values.
Usually though, there is both some transmission and some reflection. If the energy is large compared to the potential barrier height, it will be mostly transmission. If the the energy is small compared to the barrier, it will be mostly reflection. These mirror was happens classically. When one measures the position of a particle afterward, the transmission and reflection amplitudes squared give the probabilities for an individual particle to make it through the barrier or not.
If you have a particle wave impinging on a finite width potential barrier, you will always have a quantum mechanical reflection and transmission, for a particle energy below or above the barrier height. Classically, there is only a transmission for an energy above the barrier height and only a reflection for an energy below the barrier height.
The quantum mechanical wave transmission coefficient for an energy below the barrier height gives the probability that the particle will be transmitted. This effect is called tunneling. The reflection coefficient gives the probability that it will be reflected. Quantum mechanically, there is also a probability for reflection even when the particle energy is above the poetential energy barrier height.
In quantum mechanics we have the particle-wave duality. So if you are considering a particle confronted by some potential barrier one can convert it to the wave mechanics description, compute the probability for the wave function to penetrate the wall and then go back to the particle point-of-view, with the conclusion that the particle can tunnel through the barrier.
Now what shall we compare this to in the classical case? One can compare this scenario to the classical case of a ball being through at a wall. As we know the ball doesn't just tunnel through the wall.
Alternatively, one can consider a classical wave, such as light in a frustrated internal reflection scenario. Then one finds that the mathematics is precisely the same as in the quantum case. A classical wave does not in genral represent a probability amplitude for particles as in the quantum case, but other than that it gives the same mathematical result. However, in the case of light, one can even convert to the quantum case and interpret it as the probability amplitude to find photons. So the classical case and the quantum case is exactly equivalent (apart from the particle interpretation).