(10) A non-relativisitic particle of mass m moves in a one-dimensional potential well with infinitely hard walls at x = −L and x = +L (in other words, at the walls the potential energy U becomes infinite). In between these walls the potential takes some undetermined form U(x). The total energy of the particle is zero (E = 0). Its wave function is:
x4
ψ(x)=K 1− L for−L<x<+L,
where ψ(x) = 0 elsewhere and K is a constant.
(i) Use the Schr ̈odinger equation to determine the potential energy U as a function of x between
the walls, at which x = ±L.
(ii) Determine the value of the constant K. A useful integral is: xndx = 1 xn+1.
n+1
(iii) At what value of x is the probability density the greatest for finding this particle? The first
and second derivative tests can be useful, or you could make a graph.
10 A Non Relativisitic Particle Of Mass M Moves In A One Dimensional Potential Well With Infinitely Hard Walls At X 1 (26.93 KiB) Viewed 35 times
(10) A non-relativisitic particle of mass m moves in a one-dimensional potential well with infinitely hard walls at r=-L and x = +L (in other words, at the walls the potential energy U becomes infinite). In between these walls the potential takes some undetermined form U(r). The total energy of the particle is zero (E = 0). Its wave function is: (2) = K[1- ()*] for - L<x<+L, where (r) = 0 elsewhere and K is a constant. (i) Use the Schrödinger equation to determine the potential energy U as a function of ar between the walls, at which z=L. (ii) Determine the value of the constant K. A useful integral is: fx"dr = +1 (iii) At what value of r is the probability density the greatest for finding this particle? The first and second derivative tests can be useful, or you could make a graph.
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