Im going to take as a example a quantum well (i.e 2DEG). Literature says that motion of electrons is quantised in the vertical direction for a 2DEG, therefore motion in the vertical direction is forbidden. The way I understand this is , if motion in the vertical direction is quantised, it means the particle can take certain (discrete) energy values in that direction, meaning its k-vector in that direction can take certain values therefore it can move in that direction. I know I am wrong in my reasoning, I just do not know where? Some light please :)
As long as your particle stays within the quantum well your reasoning is not wrong. The point is that the very definition of a quantum well is that the discrete energy states are confined in this narrow region, i.e. even though you can see the particle with the finite k-vector as 'moving' within the quantum well, its wave function is still confined to the quantum well since outside of it there are NO available states, i.e. the particle will scatter back from the wall of the quantum well.
If you look at an eigenstate of a one-dimensional well of finite depth (corresponding to the perpendicular direction of your 2DEG) you will find that it goes as a cosine when the particle is inside the well, and decays exponentially when it is outside. The latter means that it cannot get outside the well (it has ``complex momentum'' or evanescent waves, but that precisely means it is only rarely found there, and does not stay there permanently). On the other hand, within the well, you will find both momentum going in one direction and in the other, so the particle is not moving in one direction more than in the other. But it is, in a sense, bouncing around within the well.
To all this, one should always add the standard caveat: QM is not really meant to be visualised. But what I have said above is not, I believe, all that wrong
I think you are looking at a 2D electron gas case. There is no third physical direction here, which is supposedly a vertical one. For convenience of plotting wave functions a third direction may be used which corresponds to amplitude of wave function. Alternatively if you plot eigenvalues of available quantum states (within the well) then a vertical axis may be used to plot energy of that state. As an example zero point energy of a harmonic oscillator is 1/2 hbar w, next available energy is 3/2 hbar w and so on.
Wikipedia shows a plot which tries to put a lot of information at one place in a picture of a 1D well. Here vertical axis corresponds to energy eigenvalues. Shapes of wave functions are also displayed but for the displayed shapes of wave functions, vertical direction is used to plot probability amplitude.
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