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

CP 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, the box, the equals the volume of a sphere of radius $a=5.29 \times 10^{-11} \mathrm{m},$ 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

The particle in a box (or infinite potential well) model is a basic quantum mechanical system that describes a particle confined to a finite region in space with impenetrable boundaries. In this model, the energy levels become quantized due to the boundary conditions that the wavefunction must vanish at the edges of the box, leading to discrete energy states determined by the dimensions of the box and the quantum numbers associated with each state.

Volume Equivalence and Geometric Considerations

Volume equivalence is a method used to relate different geometries by equating their volumes. In this context, it involves setting the volume of a cubical box equal to the volume of a sphere to establish an effective size of the confining region. This calculation is important in scaling the dimensions of one model (the cubical box) to align with a physical length scale (such as the Bohr radius) that is inherently spherical in nature.

Bohr Model of the Hydrogen Atom

The Bohr model is an early quantum theory that describes the hydrogen atom by assuming electrons move in circular orbits with quantized angular momentum. It provides discrete energy levels for the electron that depend on the principal quantum number, offering a simple explanation for the spectral lines of hydrogen. Although it lacks the complete rigor of modern quantum mechanics, the Bohr model yields energy separations that are useful for comparison with other models.

Energy Level Separation

Energy level separation refers to the difference in energy between consecutive quantum states in a system. For both the particle in a box and the hydrogen atom in the Bohr model, calculating the energy difference between the ground state and the first excited state helps in understanding the scale of quantum transitions. Comparing these separations across different models reveals how assumptions about confinement and potential energy affect the quantized energy spectra.

Transcript

00:01 We're asked to calculate the separation between the ground state and the first excited state, and then we're going to compare that to the separation in the energy using the bors model.

00:09 So the energy expression for a particle in a one -dimensional box having length l is e sub -n, where n is the energy level, is equal to n squared times planks constant squared divided by 8 times m, where m is the mass of the particle, divided by l squared, where l is length of the box.

00:27 So it is given that the volume of a cubical box, since it's a cube, the length are all the same, l cube is equal to four thirds pi a cubed if the volume is equal to that of a sphere.

00:38 So here, a is the boars radius that was given in the value was given in the question, 5 .29 times 10 to the minus 11 meters.

00:46 So if we substitute that in, we can get a value for l, since l is going to be equal to four thirds times pi a cubed all raised to the one third.

01:21 So there's our expression for l, and if we plug those values in, we get a value for the length of the box of 8 .53 times 10 to the minus 11 meters, or no, just 10 to the minus 11 meters for the length.

01:50 Okay, so now we can use the formula for the energy of the different excited states to get the change in energy between the same.

01:58 Second in the first excited state.

02:03 So delta e is going to be equal to e sub 2 minus e sub 1.

02:12 So you simply replace n with 2 and with 1 in these two expressions.

02:22 So when n is equal to 2 and above expression we have 2 squared which is 4 divided by 8.

02:31 So that's 1 half...

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