Calculate the probability of finding an electron in the lowest bound state inside the 4 nm well, using equation (4.21). This requires the ratio of coefficients DIC, which can be calculated from equation (4.7) or (4.8) after (and hence k and K ) has been found. Explain qualitatively how this fraction depends on the width of the well. (equation refer to the book Physics of Low dimensional)

1H2) = DpoCZ} Jr" urn); with *2\": . f (4.6) It must be possible to normalize the wave functions so the positive exponential In equation (4.5) holds for z 0. As we know that the wave functions are either even or odd we can concentrate on 2 :> 0 with the negative exponential and use symmetry to nd iii?) for z 4; g ' The wave functions (4.3) and (4.6) must now be matched at 2:312. Conttnmty of 1M2) requires eta/2): c[::|("3'i)= Dexp(;xa). (4.7) Similarly, matching the derivatives gives dtlr sin ka 1 _ :5. \"Mel le-rwl) dz This must be modied ifthe effective masses in the two materials are different. and will be described in Section 4.9. an no - _ - I f lvlzllzdz /f lll'lzzdz. .1421} a/2 -oo Making the well deeper causes 5, B, and K all to rise. and the state becomes better bound. (v) The character of the wave functions changes as the state becomes better bound. When the state has Only just become bound in the well and 9 = 21(21 1).rr. the derivative is almost zero at the edge of the well. matching to a slowly decaying wave in the barriers. increasing V0 increases its kinetic energy to give 9 Rt Elna. at which point the amplitude of the wave function is almost zero at the bonndan'es, and the exponential tail has only a small amplitude. Most of these results apply to any potential well. Ca mScanner