Question

Model a hydrogen atom as an electron in a cubical box with side length LL. Set the value of LL so that the volume of the box equals the volume of a sphere of radius a=5.29×10−11a=5.29\$$times10^{-11} m$$, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

Accepted Answer

Step 1: Start by equating the volume of the cubical box to the volume of the sphere. The volume of a sphere is given by $$ V_{sphere} = \frac{4}{3} \pi a^3 $$, where $$ a  is $$the Bohr radius. The volume of the cubical box is $$ V_{cube} = L^3 . $$Set $$ L^3 = \frac{4}{3} \pi a^3 $$ and solve for $$ L $$: $$ L = \left( \frac{4}{3} \pi a^3 ight)^{1/3} . $$Step 2: Write the energy levels for a particle in a cubical box. The energy levels are given by $$ E_{n_x, n_y, n_z} = \frac{h^2}{8mL^2} (n_x^2 + n_y^2 + n_z^2) $$, where $$ h  is $$Planck's constant, $$ m  is $$the mass of the electron, $$ L  is $$the side length of the box, and $$ n_x, n_y, n_z $$ are quantum numbers. Step 3: Calculate the ground state energy. For the ground state, $$ n_x = n_y = n_z = 1 . $$Substitute these values into the energy formula: $$ E_{1,1,1} = \frac{h^2}{8mL^2} (1^2 + 1^2 + 1^2) = \frac{3h^2}{8mL^2} . $$Step 4: Calculate the first excited state energy. For the first excited state, one of the quantum numbers increases to 2 while the others remain 1. For example, $$ n_x = 2, n_y = 1, n_z = 1 . $$Substitute these values into the energy formula: $$ E_{2,1,1} = \frac{h^2}{8mL^2} (2^2 + 1^2 + 1^2) = \frac{6h^2}{8mL^2} . $$Step 5: Find the energy separation between the ground and first excited states. Subtract the ground state energy from the first excited state energy: $$ \Delta E = E_{2,1,1} - E_{1,1,1} = \frac{6h^2}{8mL^2} - \frac{3h^2}{8mL^2} = \frac{3h^2}{8mL^2} . $$Finally, compare this result to the energy separation calculated using the Bohr model, which is $$ \Delta E_{Bohr} = 13.6 	ext{ eV} 	imes (1 - \frac{1}{4}) = 10.2 	ext{ eV} .$$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Quantum Mechanics and Particle in a Box

Bohr Model of the Hydrogen Atom

Volume and Energy Relationships