Show that, in ħ=c=1 units, the Compton wavelength of an...

Show that, in $\hbar=c=1$ units, the Compton wavelength of an electron is $m^{-1}$, the Bohr radius of the hydrogen atom is $(\alpha m)^{-1}$, and the velocity of an electron in its lowest Bohr orbit is simply $\alpha ; m$ is the mass of the electron or, more precisely, the reduced mass $m_e m_p /\left(m_e+m_p\right)$.

Key Concepts

Fine Structure Constant

The fine structure constant, denoted by ?, is a dimensionless constant that quantifies the strength of electromagnetic interactions. It plays a key role in determining scales in atomic physics, such as the Bohr radius and the electron velocity in the hydrogen atom, effectively linking quantum mechanical effects with electromagnetic force strength.

Natural Units

Natural units are a system of units in physics where constants such as the reduced Planck constant (?) and the speed of light (c) are set to 1. This greatly simplifies equations by eliminating these constants from the expressions, making relationships between physical quantities clearer and often dimensionless.

Compton Wavelength

The Compton wavelength is a quantum mechanical property defined as the wavelength of a particle associated with its mass. In natural units, the Compton wavelength for a particle of mass m simplifies to m?¹, highlighting the inverse proportionality between mass and wavelength.

Bohr Radius

The Bohr radius represents the characteristic size of the hydrogen atom in the Bohr model. When expressed in natural units, it takes the form (?m)?¹, where ? is the fine structure constant, showing that the radius depends inversely on both the electron mass and the electromagnetic coupling strength.

Reduced Mass

In systems like the hydrogen atom where two bodies interact, the reduced mass is used to simplify the problem into an effective one-body problem. It is defined in terms of the individual masses and plays an essential role in accurately determining the scales of quantum properties, such as the Compton wavelength and Bohr radius, in atomic systems.

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