Consider a box with width L centered at x = 0, so that it extends from x = -L/2

Question

Consider a box with width L centered at x = 0, so that it extends from x = -L/2 to x = +L/2. Note that this box is symmetrical about x = 0. Consider possible wave functions of the form psi (x) = A sin kx. Apply the boundary conditions at the wall to obtain the allowed energy levels. Another set of possible wave functions are functions of the form psi (x) = A cos kx. Apply the boundary conditions at the wall to obtain the allowed energy levels. Propose a combination of all the energies in parts (a) and (b). Find the width L of a one-dimensional box that would correspond to the absolute value of the ground state of a hydrogen atom. What is your observation?

Explanation / Answer

a)

(x) = A sinkx

(L/2) = 0 >>>> A sink(L/2) = 0 >>>> kL/2 = n >>>> kn = 2n/L

n = (h/2)(k2/(2m)) = (h/2)((2n/L)2/(2m)) = hn2/mL2

En = (h/2) n = h2n2/2mL2

b)

(x) = A coskx

(L/2) = 0 >>>> A cosk(L/2) = 0 >>>> kL/2 = (2n'-1)(/2) >>>> km = (2n'-1)(/L)

n' = (h/2)(k2/(2m)) = (h/2)(((2n'-1)(/L))2/(2m)) = h(2n'-1)2/(4mL2)

En' = (h/2) n = h2(2n'-1)2/(8mL2)

c)

E = En + En' = h2n2/2mL2 + h2(2n'-1)2/(8mL2)

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