Question

An electron is confined to a one-dimensional region in which its ground-state (n = 1) energy is 2.00 eV. (a) What is the length of the region? (b) How much energy is required to promote the electron to its first excited state?

Answer 1

Answer on Question #67636, Physics / Atomic and Nuclear Physics

An electron is confined to a one-dimensional region in which its ground-state $(n = 1)$ energy is 2.00 eV. (a) What is the length of the region? (b) How much energy is required to promote the electron to its first excited state?

Find: $a - ?\Delta E - ?$

Given:

$n_1 = 1$

$n_2 = 2$

$E_{1} = 2.00\mathrm{eV} = 2.00\times 1.6\times 10^{-19}\mathrm{J}$

$\mathrm{m = 9.10\times 10^{-31}kg}$

$h = 6.63 \times 10^{-34} \, \text{J} \times s$

Solution:

Energy of electron:

$\mathrm{E_n = \frac{h^2}{8a^2m}n^2}$ (1)

Of (1) $\Rightarrow$ a = $\sqrt{\frac{h^2n_1^2}{8Em}}$ (2)

Of (2) $\Rightarrow$ a=0.4×10-9m

Of (1) $\Rightarrow$ $\Delta E = \frac{h^2}{8a^2m} (n_2^2 - n_1^2)$ (3)

Of (3) $\Rightarrow$ $\Delta E = 11.32 \times 10^{-19} \, \text{J}$

1 eV - 1.6×10-19 J

$\Delta E - 11.32 \times 10^{-19} \, \text{J}$

$\Delta E = 7 \, \text{eV}$

Answer:

(a) $0.4 \times 10^{-9} \, \text{m}$

(b) $7 \mathrm{eV}$