Quantum mechanics and hydrogen atom
Consider an electron of mass m moves with the velocity v in a hydrogen atom. If an electron is at a distance r from the proton, then the potential energy function of the electron can be written as follows:
Isotopes of Hydrogen Atoms
To understand isotopes, it's easiest to learn the simplest system. Hydrogen, the most basic nucleus, has received a great deal of attention. Several of the results seen in more complex nuclei can be seen in hydrogen isotopes. An isotope is a nucleus of the same atomic number (Z) but a different atomic mass number (A). The number of neutrons present in the nucleus varies with respect to the isotope.
Mass of Hydrogen Atom
Hydrogen is one of the most fundamental elements on Earth which is colorless, odorless, and a flammable chemical substance. The representation of hydrogen in the periodic table is H. It is mostly found as a diatomic molecule as water H2O on earth. It is also known to be the lightest element and takes its place on Earth up to 0.14 %. There are three isotopes of hydrogen- protium, deuterium, and tritium. There is a huge abundance of Hydrogen molecules on the earth's surface. The hydrogen isotope tritium has its half-life equal to 12.32 years, through beta decay. In physics, the study of Hydrogen is fundamental.
![### Problem Statement
A photon having 40 keV scatters from a free electron at rest. What is the maximum energy that the electron can obtain?
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This problem involves the Compton scattering phenomenon, where a photon scatters off a stationary electron. The goal is to determine the maximum energy that the electron can gain as a result of this interaction.
### Explanation
In Compton scattering, the maximum energy transfer to the electron occurs when the photon is scattered backwards, i.e., at an angle of 180 degrees. The energy of the scattered electron can be determined using the Compton formula for the wavelength shift of the photon:
\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]
Where:
- \( \lambda \) is the initial wavelength of the photon.
- \( \lambda' \) is the wavelength of the scattered photon.
- \( h \) is Planck's constant.
- \( m_e \) is the electron's rest mass.
- \( c \) is the speed of light.
- \( \theta \) is the scattering angle (180 degrees for maximum energy transfer).
The energy of the photon before and after scattering can be related to the energy transferred to the electron. Using energy and momentum conservation laws, we derive the maximum kinetic energy \( K_{max} \) of the electron:
\[ K_{max} = E_{\text{photon}} - E'_{\text{photon}} \]
Where \( E_{\text{photon}} \) is the initial energy of the photon and \( E'_{\text{photon}} \) is the energy of the photon after scattering.
For maximum energy transfer:
\[ E'_{\text{photon}} = \frac{E_{\text{photon}}}{1 + \frac{E_{\text{photon}}}{m_e c^2} (1 - \cos \theta)} \]
For \( \theta = 180^\circ \):
\[ E'_{\text{photon}} = \frac{E_{\text{photon}}}{1 + \frac{2E_{\text{photon}}}{m_e c^2}} \]
Given that \( E_{\text{photon}} = 40\ \text{keV} \):
\[ K_{max} = 40\ \text{keV} - \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F093e7116-3e01-49ad-9157-7a22a5caeb16%2F19ab6c8d-9865-4617-8178-6dfbd0de7f75%2F6cbnxp5_processed.png&w=3840&q=75)
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