The diagram showing wave function and probability distribution for n = 4 and n = 5 states for a particle trapped in a finite square well.

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Chapter 38, Problem 35P

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The diagram showing wave function and probability distribution for n=4 and n=5 states for a particle trapped in a finite square well.

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(A) An electron is confined between two impenetrable walls 0.200 nm apart. Determine the energy levels for the states n = 1, 2, and 3.

(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U ---E- Electron X L L

What would you NOT expect in the limits that the quantum well has infinite width, e.g. quantum confinement goes away? the lowest possible energy will be larger than zero O there will be no wave solution anymore (probability amplitude goes to zero) O energy separation would be zero; and rather than discrete energy levels there will be a continuous energy spectrum O the Correspondence principle will be fulfilled