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Illustration Wave function in the classically forbidden / allowed region

Wave function in the forbidden region
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Example of a wave function that penetrates into a classically forbidden region (red). The wave function oscillates in the classically allowed region (blue) between \(x_1\) and \(x_2\). In classically forbidden region the wave function runs towards positive or negative infinity. For certain total energies \(W\) of the particle, the wave function decreases exponentially. This property of the wave function enables the quantum tunneling.

In the classically allowed region between \(x_1\) and \(x_2\), the signs of the wave function \(\mathit{\Psi}(x)\) and the curvature \(\frac{\partial^2 \mathit{\Psi}(x) }{\partial x^2}\) (second spatial derivative of the wave function) are opposite at the location \(x\). This behavior leads to the oscillation of the wave function in the classically allowed region. In the classically forbidden region, the wave function and its curvature always have the same sign (both positive or both negative at the location \(x\)).