Question

An electron is trapped in a one-dimensional infinite potential well. For what of the following is the corresponding energy difference equal to the energy difference?E65 between the levels n = 6 and n = 5?

(a) the higher quantum number


(b) the lower quantum number


(c) Show that no pair of adjacent levels has an energy difference equal to 2?E65. (Do this by giving the value that the level is equal to.)

Answer 1

Answer:

a.

The higher quantum number is n+1 = 12 +1 =13

b.

The higher quantum number is n = 12

c.

$E_{n+1} -E_{n}=E_{6}=6^{2}E_{1}=(2n+1)E_{1} \\n=17.5$

this is not integer so impossible to find pair of adjacent level

Explanation:

Let the quantum number pair will be n and n+1

equation of energy of particle in one dimensional box is = $\frac{n^{2}h^{2}}{8mL^{2}}$

$E_{n+1} -E_{n}=\frac{(n+1)^{2}h^{2}}{8mL^{2}}-\frac{n^{2}h^{2}}{8mL^{2}}$

$E_{n+1} -E_{n}=\frac{(2n+1)^{}h^{2}}{8mL^{2} }$

$E_{n+1} -E_{n}=(2n+1)E_{1}$

Which leads to

2n+ 1 =25

n= 12

a.

The higher quantum number is n+1 = 12 +1 =13

b.

The higher quantum number is n = 12

c.

$E_{n+1} -E_{n}=E_{6}=6^{2}E_{1}=(2n+1)E_{1} \\n=17.5$

this is not integer so impossible to find pair of adjacent level