My question: why are options A and B correct? An electron is moving in the poten

Question

My question: why are options A and B correct?


An electron is moving in the potential U(x) shown to the right. This potential is an even function of x. The energies, E1 = -1.5 eV and E2 = +0.5 eV, of the two lowest energy states are indicated by the dashed lines. Five x values are labeled A-F in the figure. 1(x) is the wave function of the ground state, and 2(x) is the wave function of the first excited state.

Which one of the following statements is not true?

(a)   1(A) = 1(F)
(b)   2(A) = -2(F)
(c)   Both 1 and 2 are proportional to sin(kx) in the regions A < x < B and D < x < F (though 'k' might be different for 1 and 2)
(d)   d1/dx = 0 at x = C
(e)   d|2|2/dx > 0 for x < A

Explanation / Answer

At the lower energy state the two symmetric potential wells behave as if they are separate entities since the energy required by a particle to tunnel from one well to another is very high and thus the probability of tunnelling is low.

Thus the wavefuctions of the lower energy state , i.e wavefuction at A= wavefuction at f (considering them to behave as two separate potential wells)

Whereas at the higher energy state tunnelling is possible and also the probability of tunnelling is high so the two potential wells do not behave as separate entities but as a single double well problem and since it is a symmetric potential the signs of the wavefuction at A and F are opposite.

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