[8A.2] Describe the features of the solution of the particle in a one-dimensional box that appear in the solutions of the particle in two- and three-dimensional boxes. What concept applies to the latter but not to a one-dimensional box.
![card image](../uploads/514/67/6f845e79_182c571a84b__8000_00001756.png)
[8A.1(b)] Determine the linear momentum and kinetic energy of a free proton described by the wavefunction e^−ikx with k=5nm−1.
![card image](../uploads/515/68/6f845e79_182c571a84b__8000_00001758.png)
[8A.3(b)] Calculate the energy separations in joules, kilojoules per mole, electronvolts, and reciprocal centimetres between the levels (i) n = 3 and n = 1, (ii) n = 7 and n = 6 of an electron in a box of length 1.50 nm.
![card image](../uploads/516/69/6f845e79_182c571a84b__8000_00001760.png)
[8.4(b)] Calculate the probability that a particle will be found between 0.65L and 0.67L in a box of length L when it has (i) n = 1, (ii) n = 2. Take the wavefunction to be a constant in this range.
![card image](../uploads/517/70/6f845e79_182c571a84b__8000_00001762.png)
![card image](../uploads/518/71/6f845e79_182c571a84b__8000_00001764.png)
[8A.7(b)]
![card image](../uploads/519/72/6f845e79_182c571a84b__8000_00001766.png)
![card image](../uploads/520/73/6f845e79_182c571a84b__8000_00001768.png)
[8A.11(b)]
![card image](../uploads/521/74/6f845e79_182c571a84b__8000_00001770.png)
surprised to find degeneracy in a box with unequal lengths. Symmetry
is a matter of degree.
This box is less symmetric than a square
box, but it is more symmetric than boxes whose
sides have a
non-integer or irrational ratio. Every state of a square box except
those with
n1=n2 is degenerate (with the
state that has n1 and n2 reversed). Only a few
states in this box
are degenerate. In this system, a state
(n1, n2) is degenerate with a state
(n2/2, 2n1) as long as
the latter state (a) exists
(i.e., n2/2 must be an integer) and (b) is distinct from
(n1, n2). A box
with incommensurable sides,
say, L and 21/2L, would have no degenerate levels.
![card image](../uploads/522/75/6f845e79_182c571a84b__8000_00001772.png)
[8A.12(b)]
![card image](../uploads/523/76/6f845e79_182c571a84b__8000_00001774.png)