Relativistic effects in the hydrogen atom are quite small and can
generally be neglected. Relativistic effects are only important in
heavier elements in the periodic table, for which the electrons are
confined to smaller atomic orbitals (and hence have higher
"velocities" -- though it when discussing bound electrons in an atom,
it really isn't correct to think about the electron having a classical
velocity). Even for atoms in which relativistic effects are
important, the apparent increase in mass of the atomic electrons is
very small. The Dirac formulation of quantum mechanics, which is a
fully relativistic theory, takes these effects into account.
One can make a crude assessment of whether relativistic effects are
important in the hydrogen atom this by considering the bound electron
as a classical point particle and calculating it's velocity from the
known binding energy of the electron. If the velocity is small
relative to the speed of light, then relativistic effects are not
going to be important.
According to classical electrodynamics, an electron with charge -e and
mass m moving with speed v in a circular orbit of radius r around a
stationary proton with charge +e experiences a centripital force equal
to the electrostatic attractive (Coulomb) force:
m*v^2/r = e^2/(4*pi*Q*r^2)
where Q is the permittivity of the vacuum.
Multiplying both sides by r/2 yields:
m*v^2/2 = e^2/(8*pi*Q*r).
Inspection of this equation shows that the left hand side is simply
the classical expression for the kinetic energy (K) of the electron
and the right hand side is simply the classical electrostatic
potential energy (U) divided by minus two, so the equation could also
be written as:
K = -U/2
The total energy of the electron is the sum of the kinetic and
potential energies:
E_tot = K + U
Substituting the relationship between K and U to eliminate U yields:
E_tot = K - 2*K = -K
The binding energy of an electron in an atom is defined as minus the
total energy, and corresponds to the amount of energy required to move
the electron to an infinite distance from the atom at which position
it is at rest. The binding energy (B) of an electron in the ground
state of the hydrogen atom is measured to be B= 13.6 electron volts =
2.18 * 10^-18 Joules = -E_tot
Substituting this in to the above equation yields:
-2.18 * 10^-18 J = -K = -m*v^2/2
solving for the velocity:
v = sqrt(2 * 2.18*10^-18 J /m)
the rest mass of the electron is 9.11*10^-31 kg, plugging this in for
m and solving for v gives:
v = 2.19 * 10^6 meters/sec
The speed of light (c) is 2.998 * 10^8 meters/sec, so the classical
velocity of the electron in the ground state in a hydrogen atom would
only be 2.19*10^6/2.998*10^8 = 0.0073 = 0.73% of the speed of light.
The "relativistic mass" (m_r) is related to the rest mass (m) of a
particle by:
m_r = m * sqrt(1/(1-(v/c)^2))
Plugging in the value of v/c from above we get:
m_r/m = sqrt(1/(1-0.0073^2)) = 1.000027
The relativistic mass of the hydrogen electron is only 1.000027 times
larger than the mass of an electron at rest. For most purposes, this
difference is negligable. |
