Calculate Quantization Of k For Allowed Energies Of Antisymmetric Waves

Question

Consider a particle with energy E bound to a finite square wall of height V and width 2L situated on –L ≤ x ≤ +L. Because the potential energy is symmetric about the midpoint of the well, the stationary state waves will be either symmetric or anti-symmetric about this point.

(a) Show that for E < V, the condition for smooth joining of the interior and exterior waves lead to the following equation for the allowed energies of the antisymmetric waves: tan kL = kδ where k = √ 2mE/ℏ² and δ is the penetration length. This equation cannot be solved analytically, but show graphically that k is indeed quantized.

c) Find a similar relationship for the symmetric case. Check that in the limit where the penetration length is 0 (i.e. V goes to + ∞), we recover quantization conditions for k corresponding to the particle in a box as derived in class.

d) Sketch the wave function ψ(x) and the probability density | ψ (x)²| for the n = 3 state of a particle in a finite potential.

Summary

The question belongs to Physics and it is discusses about calculation of quantization of k for allowed energies of antisymmetric waves.

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