Model a hydrogen atom as an electron in a cubical box with...

Model a hydrogen atom as an electron in a cubical box with side length $L$. Set the value of $L$ so that the volume of the box equals the volume of a sphere of radius $a$ = 5.29 $\times$ 10$^{-11}$ 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.

Key Concepts

Particle in a Box Model

This concept involves solving the Schrödinger equation for a particle confined within impenetrable boundaries (a potential well). The energy eigenvalues are quantized and depend on the size and shape of the well as well as quantum numbers corresponding to the spatial dimensions. It provides a simple model to understand quantum confinement effects and discretization of energy levels.

Quantum Energy Quantization

Quantum mechanics dictates that particles confined to a finite region can only occupy discrete energy states. For a particle in a three?dimensional box, the energy levels are derived from the boundary conditions and depend inversely on the square of the box dimensions. This quantization is fundamental for comparing computed energy differences in various quantum systems such as the hydrogen atom versus a confined electron.

Bohr Model of the Hydrogen Atom

The Bohr model is an early quantum theory that describes how an electron orbits a nucleus in circular paths with quantized angular momentum. It provides discrete energy levels for the hydrogen atom, with the energy differences determined by the Coulombic binding between the electron and the proton. This model introduces the Bohr radius as a characteristic length scale.

Volume Equivalence Between Geometric Shapes

Setting the volume of the box equal to that of the sphere involves equating the volume of a cube (side length cubed) to the volume of a sphere (calculated as (4/3)?a³). This concept allows one to relate models with different geometry but equivalent confinement scales, thereby bridging the gap between an idealized cubical well and the spherical symmetry of the actual atomic system.

Transcript

00:01 So in this problem, we are trying to model a hydrogen atom as an electron inside a cubicle box of side l.

00:12 And the given condition is the volume of a cube is equal to the volume of a sphere with the radius equal to this number.

00:21 And this number is basically a famous bars radius.

00:24 So in this question, basically they are asking us to calculate the separation, energy separation between ground state and first excited a stage using this model and then compare it with the hydrogen model given by the powers.

00:41 So okay, so we will start working on it by writing the energies ground straight using a cube which is 1 -1 -1 means nx equal to 1 and y equal to 1 and z equal to 1 equal to 3 pi squared h bar square over 2m o square and similarly so basically this is your ground state in the current model and if you want to write the first excited state so first excited state have three states basically with same energy one is this, which is 6x5 square h bar square over 2 ml squared.

01:44 And other two possibilities are 1 2, 1, and 1 ,12, which is first excited states.

02:05 So let's try to calculate the separation.

02:08 To calculate the separation, we can write it as e211, negative e111, which will give us 3 ,5, square, which bar square over 2n over square.

02:31 So now to calculate the value of this term, we need the value of l.

02:37 To calculate the value of l, we are going to use the condition already given in the problem.

02:43 This, the volume of a cube is equal to the volume of a sphere of radius a.

02:48 Now just plug this number in and calculate l...

Please give Ace some feedback