(a) The one-dimensional time-independent Schrodinger equation is -(h-bar2/2m)(d2?(x)/dx2) + U(x)?(x) = E?(x) Give the meanings of...

(a) The one-dimensional time-independent Schrodinger equation is
-(h-bar²/2m)(d²?(x)/dx²) + U(x)?(x) = E?(x)
Give the meanings of the symbols in this equation.
(b) A particle of mass m is contained in a one-dimensional box of width a. The potential energy U(x) is infinite at the walls of the box (x = 0 and x = a) and zero in between (0 < x < a). Solve the Schrodinger equation for this particle and hence show that the normalized solutions have the form ?_n(x) = (2/a)^1/2 sin (n?x/a) , with energy E_n = h²n²/8ma², where n is an integer (n > 0).
(c) For the case n = 3, find the probability that the particle will be located in the region a/3 < x < 2a/3 .

(d) Sketch the wave-functions and the corresponding probability density distributions for the cases n = 1, 2 and 3.

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genius_generous answered 2 years ago

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