Dual Nature



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                         Wave function is a mathematical tool to describe a physical system, in Quantum Physics. Each particle (sub atomic) is represented by a wave function ψ (position, time) in the manner that the probability of finding of the particle at that position at the same time. The values of wave function are probability amplitudes (complex number), the squares of absolute values of which, give the probability distribution that the system will be in any of the possible states.

The wave function used in Schrodinger’s equation which predicts the future behavior of the dynamic system.

Schrödinger’s equation can be written as:

The above equation can be interpreted as that the left-hand side, the rate of change of slope, is the curvature – so the curvature of the function is proportional to [V(x) - E]ψ(x). This means that if E > V(x), for ψ(x) positive ψ(x) is curving negatively, for ψ(x) negative ψ(x) is curving positively. In both cases ψ(x), is always curving towards the x-axis, so, for E > V(x), ψ(x) has a kind of stability: its curvature is always bringing it back towards the axis, and so generating oscillations.

If a plane wave coming in from the left encounters a step at the origin of height V0 > E, the incoming energy, there will be total reflection, but with an exponentially decaying wave penetrating some distance into the step. This, by the way, is a general wave phenomenon, not confined to quantum mechanics. If a light wave traveling through a piece of glass is totally internally at the surface, there will be an exponentially decaying electromagnetic field in the air outside the surface. If another piece of glass with a parallel (flat) surface is brought close, some light will “tunnel through” the air gap into the second piece of glass. We are considering here the quantum analogue of this classical behavior.

Let then we replace the step with a barrier,

V = 0 for x < 0, call this region I

V = V0 for 0 < x < L, this is region II

V = 0 for L < x, region III.

In this situation, the wave function will still decay exponentially into the barrier (assuming the barrier is thick compared to the exponential decay length), but on reaching the far end at x = L, a plane wave solution is again allowed, so there is a nonzero probability of finding the particle beyond the barrier, moving with its original speed. This phenomenon is called tunneling, since in the classical (particle) picture the particle doesn’t have enough energy to get over the top of the barrier.