30.3 Bohr's Theory of the Hydrogen Atom

The planetary model of the atom was used immediately by the great Danish physicist.

He published his theory of the simplest atom, hydrogen, in 1913, based on the planetary model of the atom.
Many questions had been asked about atomic characteristics.
Much is known about atoms, but little is known about the laws of physics.
New and broadly applicable principles in quantum mechanics were established by Bohr's theory.

The atomic spectrum and size of the hydrogen atom were explained by the planetary model of the atom.

His contributions to the development of atomic physics and quantum mechanics, his personal influence on many students and colleagues, and his personal integrity, especially in the face of Nazi oppression, earned him a prominent place in history.
The energies of some small systems are quantized.
The emission and absorption of atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms and atoms There must be a connection between the spectrum of an atom and its structure, something like the resonance frequencies of musical instruments.
Many great minds tried to come up with a theory, but no one came up with one.
After Einstein's proposal of quantized energies directly proportional to their wavelength, it became clear that electrons in atoms can only exist in a single trajectory.

A discharge tube, slit, and grating produce a line spectrum from left to right.

The emission line spectrum for iron is shown in part (b).
The lines imply quantized energy states for the atoms.
The line spectrum for each element is unique, providing a powerful and much used analytical tool, and many line spectrum were well known for many years before they could be explained with physics.
The simplest atom has a relatively simple spectrum.
The hydrogen spectrum had been observed in a number of different places.
The series is named after researchers who studied them in detail.

A series is associated with a constant.

Part of the Balmer series is visible with the rest of the UV.
The rest are all IR.
There is an unlimited number of series, although they are difficult to see as they get deeper into theIR.
The constant is positive, but must be greater than.

That can approach infinite.

The formula in the wavelength equation was just a recipe designed to fit data and was not based on physical principles.
The formula for his series was the first to be devised, and it was later found to describe all the other series by using different values.
The deeper meaning was comprehended by Bohr.
We see the interplay between theory and experiment in physics again.
An equation was found to fit the experimental data, but the theoretical foundation was missing.

The hydrogen spectrum has several series named for those who contributed the most.

The Balmer series is in the visible spectrum, while the Lyman series is in the UV and the Paschen series is in the IR.

We need to identify the physical principles involved in the Integrated Concept problem.

We need to know the wavelength of light and the conditions for an interference maximum for the pattern from a double slit.
Part (a) deals with a topic of the present chapter, while part (b) considers the wave interference material of Wave Optics.

That is required in the Balmer series.

The wavelength equation can be applied to the calculation.

The wavelength is similar to the second line in the Balmer series.

The same simple recipe predicted all of the hydrogen spectrum lines in subsequent experiments.

To get constructive interference for a double slit, the path length difference must be an integral multiple of the wavelength.

In this example, the number is the order of the interference.

This number is similar to the one used in the interference examples of the introduction to quantum physics.

Basic physics, the planetary model of the atom, and some very important new proposals were used to derive the formula for the hydrogen spectrum.

The first proposal was that only certain orbits were allowed.
electrons can move to a higher orbit by absorbing energy and then dropping to a lower one by emitting energy The amount of energy absorbed or emitted can be quantized.
The primary methods of transferring energy into and out of atoms are photon absorption and emission.
When an electron moves from one electron to another, the energy of the photon is equal to the change in the electron's energy.

There is a change in energy between the initial and final orbits.

It's logical that energy is involved in changing orbits.
The space shuttle needs a lot of energy to get to a higher altitude.
It is not expected that atomic orbits should be quantized.
This is not observed for satellites or planets that have a proper energy balance.

The planetary model of the atom has the electrons quantized.

There are only certain orbits that are allowed.
The energy carried away from an atom by a photon is quantized by the dropping of an electron from one atom to another.
This is also true for atomic absorption.

These are the allowed energy levels of the electron.

The energy is plotted vertically with the lowest state at the bottom and excited states above.
It is possible to determine the energy levels of an atom using the lines in an atomic spectrum.
Energy-level diagrams are used for many systems.
The physics of the system must be predicted by a theory of the atom.

An energy-level diagram plots energy vertically and is useful in visualization of the energy states of a system.

The way to calculate the electron orbital energies in hydrogen was found by Bohr.

This was an important first step that has been improved upon, but it is worth repeating because it describes many characteristics of hydrogen.

This value can only be equal to, according to quantization.

At the time, he didn't know why the energy in the hydrogen spectrum should be quantized, but using this assumption, he was able to calculate the energies in the hydrogen spectrum, something no one else had done at the time.

We will derive a number of important properties of the hydrogen atom from the classical physics we have covered in the text.

The centripetal force that causes the electron to follow a circular path is supplied by the Coulomb force.
This analysis is valid for any single-electron atom.
The hydrogen-like ion is similar to hydrogen but has a higher energy due to the attraction of the electron and nucleus.

The assumption is that the nucleus is larger than the stationary electron.

The planetary model of the atom is consistent with this.

In an earlier equation, the quantization is stated.

We solve the equation for and substitute it for the one in the above.

The formula that gives the correct size of hydrogen is very impressive.

The radii are shown for the allowed electron orbits in hydrogen.

The equation gave these radii after they were first calculated.
The diameter of a hydrogen atom is verified by the lowest orbit.

If the electron is not moving at fast speeds, the Kinetic energy is familiar.

The potential energy for the electron can be found in the nucleus, which looks like a point charge.
The nucleus has a positive charge, recalling an earlier equation for the potential due to a point charge.
The electron's charge is negative.

The above expression is for energy.

The diagram in Figure 30.20 shows an energy-level diagram for hydrogen and shows how the various hydrogen spectrums are related to transitions between energy levels.

The diagram shows the Lyman, Balmer, and Paschen series of transitions.

The above equation is used to calculate the orbital energies.

The electron is bound to the nucleus so it's negative, like being in a hole without enough energy to escape.

The electric potential energy becomes zero since the free electron gets very large.
To ionize hydrogen, 13.6 eV is needed.
The electron becomes unbound with some energy.
Giving an electron 15 eV in the ground state of hydrogen takes it out of the atom and leaves it with a small amount of energy.

The formula first proposed by Balmer years earlier was used to derive the formula we use today.